geometry problems from final round of All-Siberian Open School Olympiad with aops links in the names
2007 - 2021
finals
2007 All-Siberian 11.3
The circle inscribed in the triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Let $M$ and $N$ be the midpoints of the sides $AB$ and $BC$, respectively. It turned out that $MP = NQ$. Does it follow that the triangle $ABC$ is isosceles?
2008 All-Siberian 10.3
b) Find the locus of the midpoints of segments $A'B'$.
2008 All-Siberian 11.3
We will call two unequal triangles alike , if $ABC$ and $A'B'C$ can be designated in such a way that the equalities $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ are fulfilled. Are there three pairwise alike triangles?
2009 All-Siberian 9.2
In an isosceles triangle $ABC$ with a base $AC$, the length of the midline parallel to the side $AB$ is equal to the length of the height drawn from the vertex $C$. Find the angles of the triangle .
2009 All-Siberian 9.5
On the sides of the $ABC$ triangle, three similar triangles $ABQ, BCR$ and $ACP$ are constructed (if similar, their vertices are in the order in which they are written), as shown in the figure. Prove that the quadrilateral $PQBR$ is a parallelogram.
2009 All-Siberian 10.2
On the side $AC$ of the acute-angled triangle $ABC$, an arbitrary internal point $P$ is taken. Points $S$ and $T$ are the bases of perpendiculars dropped from point $P$ to $AB$ and $BC$, respectively. Prove that the perpendicular bisector of segment $ST$ bisects the segment $BP$.
2009 All-Siberian 10.5
Let line $\ell$ be the bisector of the angle $BOC$ between the diagonals $AC$ and $BD$ of the trapezoid $ABCD$ with bases $AD$ and $BC$. We denote by $B_1$ and $C_1$, points that are symmetric to the vertices $B$ and $C$ with respect to $\ell$. Prove that $\angle BDB_1= \angle CAC_1$.
2009 All-Siberian 11.3
In the parallelogram $ABCD$, the lengths of the sides $AB ,BC$ are respectively $10 , 15$ cm respectively and $\cos DAB=3/5$ Inside $ABCD$, a point $M$ is chosen such that $MC = 3\sqrt{10}$, and the distance of $M$ from line $AD$ is $5$ cm. Find the length of A$M$.
2009 All-Siberian 11.6
Two different tangents are drawn to a circle at points $A$ and $B$, the distance from the point M of the circle to these tangents is equal to $p$ and $q$, respectively. Find the distance from point $M$ to line $AB$.
2010 All-Siberian 9.4
Let $ABCD$ be a parallelogram, the circle inscribed in the triangle $ABD$ is tangent to the sides $AB$ and $AD$ respectively, at points $M$ and $N$, the circle inscribed in the triangle $ACD$ touches the sides $AD$ and $DC$ respectively at points $P$ and $Q$. Prove that the lines $MN$ and $PQ$ are perpendicular.
2010 All-Siberian 10.5
Chords $AB$ and $AC$ are drawn in a circle, the bisector of the angle $BAC$ intersects the circle at point $D$, point $E$ is the base of the perpendicular from $D$ on line $AB$. Prove that the length $AE$ is equal to the half-sum of the lengths $AB$ and $AC$
2010 All-Siberian 11.4
Four different straight lines $I, m, n$ and $p$ pass through a certain point on the plane, they are indicated clockwise. It is known that the angle between $I$ and $m$ is equal to the angle between $n$ and $p$. From the arbitrary point $A$ of the plane that does not belong to these lines, the perpendiculars $AL,AM, AN$, and $AP$ were lowered onto the lines $I, m, n$, and $p$, respectively. Prove that lines $LP$ and $MN$ are parallel
2011 All-Siberian 8.3
Let the angle between the extensions of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ be $90$ degrees, and the length of the segment $PQ$ connecting the midpoints of the sides $AD$ and $BC$, equal to half the difference of these sides. Prove that $ABCD$ is a trapezoid.
2011 All-Siberian 9.3
Let the lengths of the bases of the right trapezoid $ABCD$ be $6$ cm and $3$ cm, a circle having as diameter the - perpendicular to the bases - side $CD$, touches the side $AB$ at point $P$, and the diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OP$.
2011 All-Siberian 10.4
On the sides of $AB, BC$ and $AC$ of an acute-angled triangle $ABC$ marked points $P,Q$ and $R$, respectively, so that $\angle APR = \angle BPQ$, $\angle BQP = \angle CQR$, $\angle CRQ= \angle ARP$. Prove that $P,Q$ and $R$ are the bases of the heights of the triangle $ABC$.
2011 All-Siberian 11.2
A trapezoid $ABCD$ with bases $AD$ and $BC$ is given, the lengths of the sides of $AB, BC, CD$ and $DA$ of which are $3$ cm, $7$ cm, $5$ cm and $13$ cm, respectively. We denote by $P$ the intersection point of the bisectors of the angles of the $BAD$ and $ABC$, and by $Q$ the intersection of the bisectors of the angles of the $ADC$ and $BCD$. Find the length of the segment $PQ$.
2012 All-Siberian 8.4
Find the angles of the triangle $ABC$, in which $AB = BC$, and the height of the $BH$ is half to the bisector of $AK$.
2012 All-Siberian 9.3
The line tangent to the circumcircle of the triangle $ABC$ at point $A$, intersects the line $BC$ at point $K$. On the line $BC$ from point $K$ to the points $B$ and $C$, a segment $KM$ is laid out, the length of which is equal to the length $AK$. Prove that $AM$ is the bisector of the angle of $BAC$.
2012 All-Siberian 10.4
A circle is inscribed in the angle with vertex $O$, touching its sides at points $A$ and $B$, respectively. A straight line is drawn from point $A$ parallel to $OB$ for the second time intersecting the circle at point $C$, and the segment $OC$ again intersects the circle at point $E$. Let line A$E$ intersect the segment $OB$ at point $K$. Prove that $K$ is the midpoint of the $OB$.
2012 All-Siberian 11.4
Two intersecting circles of radii $\sqrt2$ cm and $\sqrt{17}$ cm are given, the distance between the centers of which is $5$ cm.The straight line intersects these circles at points $A, B, C$ and $D$ as shown in the figure, and the lengths of segments $AB, BC$ and $CD$ are equal. Find the length of these segments.
2013 All-Siberian 7.4
The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at point $O$. It is known that the perimeter of triangle $ABC$ is equal to the perimeter of triangle $ABD$. Besides, the perimeter of $ACD$ is equal to the perimeter of the triangle $BCD$. Prove that $AO = OB$.
2013 All-Siberian 8.4
A triangle $ABC$ and points $D$ and $E$, outside the triangle, are given such that the angles $ADB$ and $CEB$ are right. Prove that the length of the segment $DE$ is not greater than the half-perimeter of the triangle $ABC$.
2013 All-Siberian 9.3
In the acute-angled triangle $ABC$, a point $H$ is chosen such that the radii of the circumscribed circles of the triangles $AHB, BHC$, and $CHA$ are equal. Prove that $H$ is the intersection point of the heights of the triangle $ABC$.
2013 All-Siberian 10.3
The circles with the centers $O_1$ and $O_2$ intersect at two points $A$ and $B$. Let $P$ and $Q$ be the intersection points of the circle circumscribed around the triangle of $O_1AO_2$ with the first and second circles, respectively. Prove that segments $O_1Q$ and $O_2P$ intersect at point $B$.
2013 All-Siberian 10.5
In an acute-angled triangle $ABC$, points $A_1,B_1,C_1$ are the bases of heights dropped from the vertices $A, B, C$, respectively, and $H$ is the point of intersection of heights. Let point $M$ be the midpoint of the $AH, Q$ be the intersection point of the segments $BH$ and $A_1C_1$, and $P$ be the intersection point of the straight line $B_1M$ and side $AB$. Prove that the line $PQ$ is perpendicular to the side $BC$.
2013 All-Siberian 11.3
The perimeter of the triangle $ABC$ is $24$ cm, and the segment connecting the point of intersection of its medians with the point of intersection of its bisectors is parallel to the side $AC$. Find the length of segment $AC$.
2013 All-Siberian 11.4
Point $M$ is marked on the sphere. Consider all triples of points $A, B, C$ on a sphere other than $M$, such that the segments $MA, MB, MC$ are pairwise perpendicular, and for each such triple we consider a plane passing through $A, B, C$. Prove that all such planes pass through some common point.
2014 All-Siberian 8.4
Two equal segments $AB$ and $CD$ are perpendicular, and the point $C$ lies inside the segment $AB$. Point $X$ is such that the triangles $XAD$ and $XBC$ are isosceles with a vertex in $X$. Prove that these triangles are right-angled.
2014 All-Siberian 9.2
Two lines passing through two different vertices of a triangle break it into three triangles and a quadrangle. Can the areas of all triangles coincide?
2014 All-Siberian 9.5
The lengths of all sides of the not necessarily convex pentagon $ABCDE$ are equal to $a$, the angle between a pair of diagonals with a common vertex is $30^o$. Prove that the length of some diagonal of the pentagon is also equal to $a$.
2014 All-Siberian 10.2
Is it possible to find such convex and non-convex quadrangles in the plane, the lengths of the sides of which in some order and the lengths of the diagonals of which in some order coincide?
2014 All-Siberian 10.3
In the inscribed quadrilateral ABCD the lengths $AB$ and $AC$ are equal, as well as the lengths $BC$ and $CD$. Let the point $P$ be the midpoint of the arc $CD$ that does not contain $A$, and let the $Q$ be the intersection point of $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.
2014 All-Siberian 11.1
The lengths of the sides of the inscribed quadrangle in the clockwise order are $6,3, 5,4$, respectively. Find the angle between sides of length $6$ and $3$.
2015 All-Siberian 7.4
The following figure is given (see figure, all angles are right). Using a ruler without divisions, divide it into two polygons of equal area.
2015 All-Siberian 8.4
On the sides of the triangle $AB, BC, AC$ of the triangle $ABC$ in an arbitrary way points $C_1, A_1, B_1$ are selected respectively. Let $K_1, K_2, K_3$ be the midpoints of $AA_1, BB_1, CC_1$. Prove that these points cannot lie on one line
2015 All-Siberian.9.2
In the parallelogram $ABCD$ on the side $AD$, an arbitrary point $M$ is taken and through $M$ drawn lines parallel to the diagonals intersecting the sides $AB$ and $CD$ at the points $P$ and $Q$, respectively. Prove that the areas of the triangles $MPB$ and $MQC$ are equal.
2015 All-Siberian 10.3
In an equilateral triangle $ABC$ on the sides $AB$ and $AC$ taken points $P$ and $Q$ respectively, such that $AP: PB = CQ: QA = 2$. Let $O$ be the intersection point of the segments $CP$ and $BQ$, prove that the angle $AOC$ is right ..
2015 All-Siberian 11.2
On sides $AB$ and $AC$ of an equilateral triangle $ABC$ with side $10$, points $P$ and $Q$ are taken, respectively, such that the segment $PQ$ touches the circle inscribed in the triangle and its length is $4$. Find the area of the triangle $APQ$.
2016 All-Siberian 7.3
The $ABC$ triangle is given, the $AB$ side is divided into $4$ equal segments $AB_1 = B_1B_2 = B_2B_3 = B_3B$, and the $AC$ side is divided into $5$ equal segments $AC_1 = C_1C_2 = C_2C_3 = C_3C_4 = C_4C$. How many times the area of triangle $ABC$ is greater than the sum of the areas of triangles $C_1B_1C_2, C_2B_2C_3, C_3B_3C_4, C_4BC$ ?
2016 All-Siberian 8.2
A convex quadrangle $ABCD$ with $AD=3$ is given. The diagonals $AC$ and $BD$ intersect at point $E$, and it is known that the areas of the triangles $ABE$ and $DCE$ are $1$. Find the side $BC$ if it is known that the area $ABCD$ does not exceed $4$.
2016 All-Siberian 9.4
In a right triangle $ABC$, let point $K$ be the midpoint of the hypotenuse $AB$ and lie point $M$ lie on the side $BC$ such that $BM: MC = 2$. Let segments $AM$ and $CK$ intersect at point $P$. Prove that the line $KM$ touches the circumscribed circle of the triangle $AKP$.
2016 All-Siberian 10.3
Two circles intersect at points $P$ and $M$. On the first circle, an arbitrary point $A$ is selected, different from $P$ and $M$ and lying inside the second circle, the rays $PA$ and $MA$ re-intersect the second circle at points $B$ and $C$, respectively. Prove that the line passing through $A$ and the center of the first circle is perpendicular to the $BC$.
2016 All-Siberian 11.3
In the triangle $ABC$, the segments $AK, BL, CM$ are the heights, $H$ is their intersection point, $S$ is the intersection point of $MK$ and $BL, P$ is the midpoint of the segment $AH, T$ is the intersection point of the line $LP$ and the side $AB$. Prove that the line $ST$ is perpendicular to the side $BC$.
2017 All-Siberian 7.4
An angle of $120$ degrees lies in a triangle with sides of length $a, b$ and opposite side $c$. Prove that a triangle can be made up of segments of length $a, c$, and $a + b$.
2017 All-Siberian 8.4
In the triangle $ABC $, bisector $BE$ was drawn. It turned out that $BC + CE = AB$. Prove that there are two angles in the triangle $ABC$, one of which is two times larger than the other.
2017 All-Siberian 9.1
On the sides $AB$ and $AD$ of the square $ABCD$, inside it are drawn the equilateral triangles $ABK$ and $ADM$, respectively. Prove that the triangle $CKM$ is also equilateral.
2017 All-Siberian 10.3
In the quadrangle $ABCD$, the equal diagonals $AC$ and $BD$ intersect at point $O$, and points $P$ and $Q$ are the midpoints of the sides $AB$ and $CD$, respectively. Prove that the angle bisector of $AOD$ is perpendicular to the segment $PQ$.
2017 All-Siberian 11.3
Inside the acute-angled triangle $ABC$, a point $P$ was chosen that is different from $O$ - the center of the circumscribed circle of triangle $ABC$, such that $\angle PAC=\angle PBA$ and $\angle PAB = \angle PCA$. Prove that the angle of the $APO$ is right .
2017 All-Siberian 11.4
Prove that the edges of an arbitrary tetrahedron (triangular pyramid) can be broken in some way into three pairs so that there exists a triangle whose side lengths equal to the sum of the lengths of the edges of the tetrahedron in these pairs.
2018 All-Siberian 7.4
On the plane through the same distance are $2018$ parallel lines. Each line has one point. Points $B_1$ and $B_2$ are taken arbitrarily on the first two lines. Then point $B_3$ is taken so that $B_1B_2 = B_2B_3, B_4$ so that $B_1B_3 = B_3B_4$, ..., $B_I$ so that $B_1B_{I-1} = B_{I-1}B_I$ ,..., $B_{2018}$ so that $B_1B_{2017} = B_{2017}B_{2018}$. Moreover, if the next point can be selected in two ways, then for the odd number the right point is selected, for the even - the more left point (see figure). Prove that the location of point $B_{2018}$ depends only on the location of point $B_1$.
2018 All-Siberian 8.3
Find the angle $DAC$, if it is known that $AB = BC$ and $AC = CD$, and the lines on which lie points $A, B, C, D$ are parallel, and the distances between adjacent lines are equal. Point $A$ to the left of $B, C$ to the left of $B, D$ to the right of $C$ (see figure).
2018 All-Siberian 9.2
Exist intersection points in some acute-angled triangle $ABC$, bisector of angle $A$, altitude drawn from vertex $B$ and median drawn from $C$, that are the vertices of the vertices of a non-degenerate equilateral triangle?
2018 All-Siberian 10.3
Different lines $a$ and $b$ intersect at point $O$. Consider all possible segments $AB$ of length $l$, the ends of $A$ and $B$ of which lie on $a$ and $b$, respectively, and denote by $P$ the point the intersection of the perpendiculars on the straight lines $a$ and $b$, drawn from $A$ and $B$, respectively. Find the locus of points $P$.
2018 All-Siberian 11.5
Let $A$ and $B$ be two different fixed points of a circle, $C$ an arbitrary a point on this circle other than $A$ and $B$, and $MP$ is a perpendicular dropped from the midpoint $M$ of chord $BC$ on the chord $AC$. Prove that the lines $PM$ pass through some common point $T$ with any choice of $C$.
2019 All-Siberian 7.2
In triangle $ABC$, $\angle A=60^o$. Points $M, N$, and $K$ lie on sides $BC, AC$, and $AB$ respectively, with $BK = KM = MN = NC$. It turned out that $AN = 2AK$. Find angles $B$ and $C$.
2019 All-Siberian 8.4
In the convex quadrilateral $ABCD$, it is known that $AD = BC$ and
$\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$.
Prove that from segments $DB, CA$ and $DC$ you can make a right triangle. 2019 All-Siberian 9.4
On the extension of the median $AM$ of an isosceles triangle $ABC$ with base $AC$,
take point $P$ such that $ \angle CBP=\angle BAP$. Find the angle $ACP$. 2018-19 All-Siberian 10.5
The quadrilateral $ABCD$ is inscribed in a circle with $BC=DC$ and $AB =AC$.
Let the point $P$ be the midpoint of the arc $CD$ not containing point $A$, and $Q$ is the
intersection point of the diagonals $AC$ and $BD$. Prove that lines $PQ$ and $AB$ are perpendicular. 2019 All-Siberian 11.4
In a right triangle $ABC$, point $M$ is the midpoint of the hypotenuse $BC$,
and the points $P$ and $T$ divide the legs AB and AC in ratios $AP: PB = AT: TC = 1: 2$.
We denote by $K$ the intersection point of the segments $BT$ and $PM$, and by $E$ the
source: sesc.nsu.ru
The circle inscribed in the triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Let $M$ and $N$ be the midpoints of the sides $AB$ and $BC$, respectively. It turned out that $MP = NQ$. Does it follow that the triangle $ABC$ is isosceles?
2008 All-Siberian 10.3
Let $H$ be the orthocenter of triangle $ABC$. On the circumscribed circle of the triangle $AHC$, a point $P$ is arbitrarily chosen. Let $A'$ be the point of intersection of lines of $AP$ and $BC$, $C'$ be the intersection point of the lines $CP$ and $AB$.
a) Prove that the ratio of the lengths of the segments $AA'/ BB'$ does not depend on the choice of point $P$.b) Find the locus of the midpoints of segments $A'B'$.
2008 All-Siberian 11.3
2009 All-Siberian 9.2
In an isosceles triangle $ABC$ with a base $AC$, the length of the midline parallel to the side $AB$ is equal to the length of the height drawn from the vertex $C$. Find the angles of the triangle .
2009 All-Siberian 9.5
On the sides of the $ABC$ triangle, three similar triangles $ABQ, BCR$ and $ACP$ are constructed (if similar, their vertices are in the order in which they are written), as shown in the figure. Prove that the quadrilateral $PQBR$ is a parallelogram.
On the side $AC$ of the acute-angled triangle $ABC$, an arbitrary internal point $P$ is taken. Points $S$ and $T$ are the bases of perpendiculars dropped from point $P$ to $AB$ and $BC$, respectively. Prove that the perpendicular bisector of segment $ST$ bisects the segment $BP$.
Let line $\ell$ be the bisector of the angle $BOC$ between the diagonals $AC$ and $BD$ of the trapezoid $ABCD$ with bases $AD$ and $BC$. We denote by $B_1$ and $C_1$, points that are symmetric to the vertices $B$ and $C$ with respect to $\ell$. Prove that $\angle BDB_1= \angle CAC_1$.
In the parallelogram $ABCD$, the lengths of the sides $AB ,BC$ are respectively $10 , 15$ cm respectively and $\cos DAB=3/5$ Inside $ABCD$, a point $M$ is chosen such that $MC = 3\sqrt{10}$, and the distance of $M$ from line $AD$ is $5$ cm. Find the length of A$M$.
2009 All-Siberian 11.6
Two different tangents are drawn to a circle at points $A$ and $B$, the distance from the point M of the circle to these tangents is equal to $p$ and $q$, respectively. Find the distance from point $M$ to line $AB$.
2010 All-Siberian 9.4
Let $ABCD$ be a parallelogram, the circle inscribed in the triangle $ABD$ is tangent to the sides $AB$ and $AD$ respectively, at points $M$ and $N$, the circle inscribed in the triangle $ACD$ touches the sides $AD$ and $DC$ respectively at points $P$ and $Q$. Prove that the lines $MN$ and $PQ$ are perpendicular.
2010 All-Siberian 10.5
Chords $AB$ and $AC$ are drawn in a circle, the bisector of the angle $BAC$ intersects the circle at point $D$, point $E$ is the base of the perpendicular from $D$ on line $AB$. Prove that the length $AE$ is equal to the half-sum of the lengths $AB$ and $AC$
2010 All-Siberian 11.4
Four different straight lines $I, m, n$ and $p$ pass through a certain point on the plane, they are indicated clockwise. It is known that the angle between $I$ and $m$ is equal to the angle between $n$ and $p$. From the arbitrary point $A$ of the plane that does not belong to these lines, the perpendiculars $AL,AM, AN$, and $AP$ were lowered onto the lines $I, m, n$, and $p$, respectively. Prove that lines $LP$ and $MN$ are parallel
2011 All-Siberian 8.3
Let the angle between the extensions of the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ be $90$ degrees, and the length of the segment $PQ$ connecting the midpoints of the sides $AD$ and $BC$, equal to half the difference of these sides. Prove that $ABCD$ is a trapezoid.
2011 All-Siberian 9.3
Let the lengths of the bases of the right trapezoid $ABCD$ be $6$ cm and $3$ cm, a circle having as diameter the - perpendicular to the bases - side $CD$, touches the side $AB$ at point $P$, and the diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OP$.
2011 All-Siberian 10.4
On the sides of $AB, BC$ and $AC$ of an acute-angled triangle $ABC$ marked points $P,Q$ and $R$, respectively, so that $\angle APR = \angle BPQ$, $\angle BQP = \angle CQR$, $\angle CRQ= \angle ARP$. Prove that $P,Q$ and $R$ are the bases of the heights of the triangle $ABC$.
A trapezoid $ABCD$ with bases $AD$ and $BC$ is given, the lengths of the sides of $AB, BC, CD$ and $DA$ of which are $3$ cm, $7$ cm, $5$ cm and $13$ cm, respectively. We denote by $P$ the intersection point of the bisectors of the angles of the $BAD$ and $ABC$, and by $Q$ the intersection of the bisectors of the angles of the $ADC$ and $BCD$. Find the length of the segment $PQ$.
2012 All-Siberian 8.4
Find the angles of the triangle $ABC$, in which $AB = BC$, and the height of the $BH$ is half to the bisector of $AK$.
2012 All-Siberian 9.3
The line tangent to the circumcircle of the triangle $ABC$ at point $A$, intersects the line $BC$ at point $K$. On the line $BC$ from point $K$ to the points $B$ and $C$, a segment $KM$ is laid out, the length of which is equal to the length $AK$. Prove that $AM$ is the bisector of the angle of $BAC$.
2012 All-Siberian 10.4
A circle is inscribed in the angle with vertex $O$, touching its sides at points $A$ and $B$, respectively. A straight line is drawn from point $A$ parallel to $OB$ for the second time intersecting the circle at point $C$, and the segment $OC$ again intersects the circle at point $E$. Let line A$E$ intersect the segment $OB$ at point $K$. Prove that $K$ is the midpoint of the $OB$.
2012 All-Siberian 11.4
Two intersecting circles of radii $\sqrt2$ cm and $\sqrt{17}$ cm are given, the distance between the centers of which is $5$ cm.The straight line intersects these circles at points $A, B, C$ and $D$ as shown in the figure, and the lengths of segments $AB, BC$ and $CD$ are equal. Find the length of these segments.
The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at point $O$. It is known that the perimeter of triangle $ABC$ is equal to the perimeter of triangle $ABD$. Besides, the perimeter of $ACD$ is equal to the perimeter of the triangle $BCD$. Prove that $AO = OB$.
2013 All-Siberian 8.4
A triangle $ABC$ and points $D$ and $E$, outside the triangle, are given such that the angles $ADB$ and $CEB$ are right. Prove that the length of the segment $DE$ is not greater than the half-perimeter of the triangle $ABC$.
2013 All-Siberian 9.3
In the acute-angled triangle $ABC$, a point $H$ is chosen such that the radii of the circumscribed circles of the triangles $AHB, BHC$, and $CHA$ are equal. Prove that $H$ is the intersection point of the heights of the triangle $ABC$.
2013 All-Siberian 10.3
The circles with the centers $O_1$ and $O_2$ intersect at two points $A$ and $B$. Let $P$ and $Q$ be the intersection points of the circle circumscribed around the triangle of $O_1AO_2$ with the first and second circles, respectively. Prove that segments $O_1Q$ and $O_2P$ intersect at point $B$.
2013 All-Siberian 10.5
In an acute-angled triangle $ABC$, points $A_1,B_1,C_1$ are the bases of heights dropped from the vertices $A, B, C$, respectively, and $H$ is the point of intersection of heights. Let point $M$ be the midpoint of the $AH, Q$ be the intersection point of the segments $BH$ and $A_1C_1$, and $P$ be the intersection point of the straight line $B_1M$ and side $AB$. Prove that the line $PQ$ is perpendicular to the side $BC$.
2013 All-Siberian 11.3
The perimeter of the triangle $ABC$ is $24$ cm, and the segment connecting the point of intersection of its medians with the point of intersection of its bisectors is parallel to the side $AC$. Find the length of segment $AC$.
2013 All-Siberian 11.4
Point $M$ is marked on the sphere. Consider all triples of points $A, B, C$ on a sphere other than $M$, such that the segments $MA, MB, MC$ are pairwise perpendicular, and for each such triple we consider a plane passing through $A, B, C$. Prove that all such planes pass through some common point.
2014 All-Siberian 8.4
Two equal segments $AB$ and $CD$ are perpendicular, and the point $C$ lies inside the segment $AB$. Point $X$ is such that the triangles $XAD$ and $XBC$ are isosceles with a vertex in $X$. Prove that these triangles are right-angled.
2014 All-Siberian 9.2
Two lines passing through two different vertices of a triangle break it into three triangles and a quadrangle. Can the areas of all triangles coincide?
2014 All-Siberian 9.5
The lengths of all sides of the not necessarily convex pentagon $ABCDE$ are equal to $a$, the angle between a pair of diagonals with a common vertex is $30^o$. Prove that the length of some diagonal of the pentagon is also equal to $a$.
2014 All-Siberian 10.2
Is it possible to find such convex and non-convex quadrangles in the plane, the lengths of the sides of which in some order and the lengths of the diagonals of which in some order coincide?
2014 All-Siberian 10.3
In the inscribed quadrilateral ABCD the lengths $AB$ and $AC$ are equal, as well as the lengths $BC$ and $CD$. Let the point $P$ be the midpoint of the arc $CD$ that does not contain $A$, and let the $Q$ be the intersection point of $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.
2014 All-Siberian 11.1
The lengths of the sides of the inscribed quadrangle in the clockwise order are $6,3, 5,4$, respectively. Find the angle between sides of length $6$ and $3$.
2015 All-Siberian 7.4
The following figure is given (see figure, all angles are right). Using a ruler without divisions, divide it into two polygons of equal area.
On the sides of the triangle $AB, BC, AC$ of the triangle $ABC$ in an arbitrary way points $C_1, A_1, B_1$ are selected respectively. Let $K_1, K_2, K_3$ be the midpoints of $AA_1, BB_1, CC_1$. Prove that these points cannot lie on one line
2015 All-Siberian.9.2
In the parallelogram $ABCD$ on the side $AD$, an arbitrary point $M$ is taken and through $M$ drawn lines parallel to the diagonals intersecting the sides $AB$ and $CD$ at the points $P$ and $Q$, respectively. Prove that the areas of the triangles $MPB$ and $MQC$ are equal.
2015 All-Siberian 10.3
In an equilateral triangle $ABC$ on the sides $AB$ and $AC$ taken points $P$ and $Q$ respectively, such that $AP: PB = CQ: QA = 2$. Let $O$ be the intersection point of the segments $CP$ and $BQ$, prove that the angle $AOC$ is right ..
2015 All-Siberian 11.2
On sides $AB$ and $AC$ of an equilateral triangle $ABC$ with side $10$, points $P$ and $Q$ are taken, respectively, such that the segment $PQ$ touches the circle inscribed in the triangle and its length is $4$. Find the area of the triangle $APQ$.
2016 All-Siberian 7.3
The $ABC$ triangle is given, the $AB$ side is divided into $4$ equal segments $AB_1 = B_1B_2 = B_2B_3 = B_3B$, and the $AC$ side is divided into $5$ equal segments $AC_1 = C_1C_2 = C_2C_3 = C_3C_4 = C_4C$. How many times the area of triangle $ABC$ is greater than the sum of the areas of triangles $C_1B_1C_2, C_2B_2C_3, C_3B_3C_4, C_4BC$ ?
2016 All-Siberian 8.2
A convex quadrangle $ABCD$ with $AD=3$ is given. The diagonals $AC$ and $BD$ intersect at point $E$, and it is known that the areas of the triangles $ABE$ and $DCE$ are $1$. Find the side $BC$ if it is known that the area $ABCD$ does not exceed $4$.
2016 All-Siberian 9.4
In a right triangle $ABC$, let point $K$ be the midpoint of the hypotenuse $AB$ and lie point $M$ lie on the side $BC$ such that $BM: MC = 2$. Let segments $AM$ and $CK$ intersect at point $P$. Prove that the line $KM$ touches the circumscribed circle of the triangle $AKP$.
2016 All-Siberian 10.3
Two circles intersect at points $P$ and $M$. On the first circle, an arbitrary point $A$ is selected, different from $P$ and $M$ and lying inside the second circle, the rays $PA$ and $MA$ re-intersect the second circle at points $B$ and $C$, respectively. Prove that the line passing through $A$ and the center of the first circle is perpendicular to the $BC$.
2016 All-Siberian 11.3
In the triangle $ABC$, the segments $AK, BL, CM$ are the heights, $H$ is their intersection point, $S$ is the intersection point of $MK$ and $BL, P$ is the midpoint of the segment $AH, T$ is the intersection point of the line $LP$ and the side $AB$. Prove that the line $ST$ is perpendicular to the side $BC$.
2017 All-Siberian 7.4
An angle of $120$ degrees lies in a triangle with sides of length $a, b$ and opposite side $c$. Prove that a triangle can be made up of segments of length $a, c$, and $a + b$.
2017 All-Siberian 8.4
In the triangle $ABC $, bisector $BE$ was drawn. It turned out that $BC + CE = AB$. Prove that there are two angles in the triangle $ABC$, one of which is two times larger than the other.
2017 All-Siberian 9.1
On the sides $AB$ and $AD$ of the square $ABCD$, inside it are drawn the equilateral triangles $ABK$ and $ADM$, respectively. Prove that the triangle $CKM$ is also equilateral.
2017 All-Siberian 10.3
In the quadrangle $ABCD$, the equal diagonals $AC$ and $BD$ intersect at point $O$, and points $P$ and $Q$ are the midpoints of the sides $AB$ and $CD$, respectively. Prove that the angle bisector of $AOD$ is perpendicular to the segment $PQ$.
2017 All-Siberian 11.3
Inside the acute-angled triangle $ABC$, a point $P$ was chosen that is different from $O$ - the center of the circumscribed circle of triangle $ABC$, such that $\angle PAC=\angle PBA$ and $\angle PAB = \angle PCA$. Prove that the angle of the $APO$ is right .
2017 All-Siberian 11.4
Prove that the edges of an arbitrary tetrahedron (triangular pyramid) can be broken in some way into three pairs so that there exists a triangle whose side lengths equal to the sum of the lengths of the edges of the tetrahedron in these pairs.
2018 All-Siberian 7.4
On the plane through the same distance are $2018$ parallel lines. Each line has one point. Points $B_1$ and $B_2$ are taken arbitrarily on the first two lines. Then point $B_3$ is taken so that $B_1B_2 = B_2B_3, B_4$ so that $B_1B_3 = B_3B_4$, ..., $B_I$ so that $B_1B_{I-1} = B_{I-1}B_I$ ,..., $B_{2018}$ so that $B_1B_{2017} = B_{2017}B_{2018}$. Moreover, if the next point can be selected in two ways, then for the odd number the right point is selected, for the even - the more left point (see figure). Prove that the location of point $B_{2018}$ depends only on the location of point $B_1$.
Find the angle $DAC$, if it is known that $AB = BC$ and $AC = CD$, and the lines on which lie points $A, B, C, D$ are parallel, and the distances between adjacent lines are equal. Point $A$ to the left of $B, C$ to the left of $B, D$ to the right of $C$ (see figure).
Exist intersection points in some acute-angled triangle $ABC$, bisector of angle $A$, altitude drawn from vertex $B$ and median drawn from $C$, that are the vertices of the vertices of a non-degenerate equilateral triangle?
2018 All-Siberian 10.3
Different lines $a$ and $b$ intersect at point $O$. Consider all possible segments $AB$ of length $l$, the ends of $A$ and $B$ of which lie on $a$ and $b$, respectively, and denote by $P$ the point the intersection of the perpendiculars on the straight lines $a$ and $b$, drawn from $A$ and $B$, respectively. Find the locus of points $P$.
2018 All-Siberian 11.5
Let $A$ and $B$ be two different fixed points of a circle, $C$ an arbitrary a point on this circle other than $A$ and $B$, and $MP$ is a perpendicular dropped from the midpoint $M$ of chord $BC$ on the chord $AC$. Prove that the lines $PM$ pass through some common point $T$ with any choice of $C$.
2019 All-Siberian 7.2
In triangle $ABC$, $\angle A=60^o$. Points $M, N$, and $K$ lie on sides $BC, AC$, and $AB$ respectively, with $BK = KM = MN = NC$. It turned out that $AN = 2AK$. Find angles $B$ and $C$.
2019 All-Siberian 8.4
In the convex quadrilateral $ABCD$, it is known that $AD = BC$ and
$\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$.
Prove that from segments $DB, CA$ and $DC$ you can make a right triangle. 2019 All-Siberian 9.4
On the extension of the median $AM$ of an isosceles triangle $ABC$ with base $AC$,
take point $P$ such that $ \angle CBP=\angle BAP$. Find the angle $ACP$. 2018-19 All-Siberian 10.5
The quadrilateral $ABCD$ is inscribed in a circle with $BC=DC$ and $AB =AC$.
Let the point $P$ be the midpoint of the arc $CD$ not containing point $A$, and $Q$ is the
intersection point of the diagonals $AC$ and $BD$. Prove that lines $PQ$ and $AB$ are perpendicular. 2019 All-Siberian 11.4
In a right triangle $ABC$, point $M$ is the midpoint of the hypotenuse $BC$,
and the points $P$ and $T$ divide the legs AB and AC in ratios $AP: PB = AT: TC = 1: 2$.
We denote by $K$ the intersection point of the segments $BT$ and $PM$, and by $E$ the
intersection point of the segments of $CP$ and $MT$, and by $O$ the intersection point of the
segments of $CP$ and $BT$. Prove that the quadrilateral $OKME$ is cyclic.
2020 All-Siberian 7.3
The points $A, B, C, D, X $ are located on the plane. Some lengths of segments are known:
$AC = 2$, $AX = 5$, $AD = 11$, $CD = 9$, $CB = 10$, $DB = 1$, $XB = 7$.
Find the length of the segment $CX$.
2020 All-Siberian 8.4
Of the same isosceles triangles, which the angle opposite the base is $45^o$, and the lateral side is $1$, folded the shape as shown. Find the distance between points $A$ and $B$
2020 All-Siberian 9.4
On the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$ are marked points $P$ and $Q$, respectively, such that the segments $BQ$ and $DP$ divide the area of the quadrangle in half. Prove that the segment $PQ$ passes through the midpoint of the diagonal $AC$
2020 All-Siberian 10.2
On the sides of $AB, BC, AC$ equilateral triangle $ABC$ mark points $P$ and $Q, R, S$, respectively, such that $AP = CS$, $BQ = CR$. Prove that the angle between the segments $PR$ and $QS$ is $60^o$ .
2020 All-Siberian 11.4
Let points $O$ and $I$ be the center of the circumscribed and inscribed circles triangle $ABC$, respectively. It is known that the angle $AIO$ is right, and $\angle CIO =45^o$. Find the aspect ratio $AB: BC: CA$.
2020 All-Siberian 7.3
The points $A, B, C, D, X $ are located on the plane. Some lengths of segments are known:
$AC = 2$, $AX = 5$, $AD = 11$, $CD = 9$, $CB = 10$, $DB = 1$, $XB = 7$.
Find the length of the segment $CX$.
2020 All-Siberian 8.4
Of the same isosceles triangles, which the angle opposite the base is $45^o$, and the lateral side is $1$, folded the shape as shown. Find the distance between points $A$ and $B$
2020 All-Siberian 9.4
On the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$ are marked points $P$ and $Q$, respectively, such that the segments $BQ$ and $DP$ divide the area of the quadrangle in half. Prove that the segment $PQ$ passes through the midpoint of the diagonal $AC$
2020 All-Siberian 10.2
On the sides of $AB, BC, AC$ equilateral triangle $ABC$ mark points $P$ and $Q, R, S$, respectively, such that $AP = CS$, $BQ = CR$. Prove that the angle between the segments $PR$ and $QS$ is $60^o$ .
2020 All-Siberian 11.4
Let points $O$ and $I$ be the center of the circumscribed and inscribed circles triangle $ABC$, respectively. It is known that the angle $AIO$ is right, and $\angle CIO =45^o$. Find the aspect ratio $AB: BC: CA$.
What is the largest number of non-intersecting diagonals that can be drawn in a convex $n$ -gon (diagonals with a common vertex are allowed)?
In triangle $ABC$, point $M$ is the midpoint of the side $BC$, $H$ is the base of the altitude drawn from the vertex $B$. It is known that the angle $MCA$ is twice the angle $MAC$, and the length of the $BC$ is $10$ cm. Find the length of the segment$ AH$.
Point $M$ is the midpoint of the side $AB$ of the triangle $ABC$. On the segment $CM$, points $P$ and $Q$ are selected so that $P$ is closer to $M, Q$ is closer to $C$ and $CQ = 2PM$. It turned out that $BQ = AC$. Find the value of the angle $APM$.
Let $Q$ and $P$ be the bases of the perpendiculars drawn from the vertex $B$ of the triangle $ABC$ to the bisectors of its angles $A$ and $C$, respectively. Prove that line $PQ$ is parallel to the side $AC$ .
qualifying full time round
Is there a right-angled triangle whose sidelengths are prime numbers?
Let $O$ be the center of the circumscribed circle of triangle $ABC$. The circle circumscribed around the triangle $AOB$ intersects the line $AC$ for the second time at point $A'$. The circle circumscribed around the triangle $COB$ intersects the line $AC$ at point $C$ for the second time.
a) Prove that if the angle $\angle ABC$ is obtuse, then $A'C <AC / 2$.
b) Prove that $A'C <AC / 2$ if and only if $\tan (\angle ABC) = \tan (\angle CAB) + \tan (\angle ACB)$.
In a triangle $ABC$, the lengths of the sides are such that $AB <BC <AC$. Which the angle of the triangle closest to the point of intersection of its angle bisectors?
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $AB$ . Prove that $AD + BD = DC$.
Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular dropped from point $F$ to $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$.
KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse
Inside a right-angled triangle with $3$ cm and $4$ cm legs there are two equal circles so that the first touches the hypotenuse and small leg, the second touches the hypotenuse, the greater leg and the first circle. Find the radius of the circles.u
Let $ABC$ be a triangle and $P$ be any point on $(ABC)$. Let $X,Y,Z$ be the feet of the perpendiculars from $P$ onto lines $BC,CA,$ and $AB$. Prove that points $X,Y,Z$ are collinear.
Through an arbitrary point $M$ of the diagonal $AC$ of the parallelogram $ABCD$ parallel to its sides straight lines are drawn, intersecting sides $AB, BC, CD$ and $AD$ at points $P, R, S$ and $Q$, respectively. Prove that the areas of parallelograms $PBRM$ and $MSDQ$ are equal.
Let point $M$ be the midpoint of side $BC$ of triangle $ABC$. It is known that the measures of the angles $ACB$ and $AMB$ are equal to $30$ and $45$ degrees, respectively. Find the measure of angle $ABC$.
Inside the square $ABCD$ find the locus of points $M$ such that the sum of the distances from $M$ to vertices $A$ and $C$ is equal to the sum of the distances from $M$ to vertices $B$ and $D$.
Regular pentagon and regular $20$-gon are inscribed in the same circle. Which is greater, the sum of the squares of the lengths of all sides of the pentagon, or the sum of the squares of the lengths of all sides of the $20$-gon?
Prove that in an arbitrary triangular pyramid there is a vertex, such that of the three edges containing it, you can make a triangle.
In the quadrilateral $ABCD$, the angles at the vertices $B$ and $D$ are right, sides $AB$ and $BC$ are equal, and the length of the perpendicular drawn from vertex $B$ on side $AD$ is $1$ cm. Find the area of a quadrilateral $ABCD$.
In rectangle $ABCD$, point $E$ is the foot of the perpendicular from vertex $B$ on the diagonal $AC$, points $P$ and $Q$ are the midpoints of the segments $AE$ and $CD$, respectively. Prove that $BPQ$ is a right angle.
In triangle $ABC$, the angles $B$ and $C$ are $40$ degrees, and the bisector of angle $B$ intersects side $AC$ at point $D$. Prove that $BC = BD + AD$.
The rectangle is split by two vertical and two horizontal lines by $9$ rectangles, some perimeters of which are shown in the figure. Find the perimeter of the top left rectangle.
The perpendicular bisector of the side $BC$ of triangle $ABC$ intersects the side $AB$ at the point $D$, and the extension of the side $AC$ beyond point $A$, at the point $E$. Prove that $AD<AE$.
In a trapezoid $ABCD$ with bases $AD$ and $BC$, angle $A$ is right , $E$ is the intersection point of the diagonals, and the point $F$ is the projection of $E$ on the side $AB$. Prove that the angles $DFE$ and $CFE$ are equal.
Segments $AM$ and $BH$ are the median and the altitude, respectively, of the triangle $ABC$. It is known that $AH=1$ and $2\angle MAC=\angle MCA$. Find the length of side $BC$.
Points $D$ and $E$, respectively, are marked on the sides $AB$ and $BC$ of triangle $ABC$, such that $\angle ACB =2 \angle BED$. Prove that $AC + EC> AD$.
Two different points $A$ and $B$ are marked on the line $\ell$. Consider all possible pairs of circles touching each other and the line $\ell$ at points $A$ and $B$ respectively. For each pair, let $M$ be the midpoint of the segment tangent to these circles, not lying on $\ell$. Find the locus of points $M$.
Inside a right-angled triangle with sides of $3, 4$ and $5$ cm, there are two circles, the ratio of the radii of which is $9$ to $4$. The circles touch each other externally, both touch the hypotenuse, one touches also one leg, the other touches also the other. Find the radii of the circles.
The quadrilateral is $ABCD$ such that $\angle BCD=\angle ABC= 120^o$ and $BC+CD=AD$. Prove that $AB=CD$ .
In an isosceles triangle $ABC$ with the base of the $AC$, the length of the angle bisector $AK$ is twice more than the length of the altitude $AH$. Find the angles of this triangle.
The diameter is drawn through the midpointof the chord $AB$ of some circle. Denote $C$ a point of its intersection with the circle, and $E$ the point of intersection of its extension with the tangent to circle drawn at point $A$. In this case, the midpoint of $AB$ lies between $C$ and the center of the circle . Prove that $AC$ is the bisector of the angle $BAE$.
In trapezium $ABCD$, the length of the lateral side $AB$ is $5$ cm, the bisector of angle $A$ intersects the lateral side $CD$ in its midpoint $P$, the length of the segment $AP$ is $4$ cm. Find the length of the segment $BP$ .
Two equal intersecting circles with centers $A$ and $B$ are located so that the center of each lies outside the other. Let's denote their points of intersection as $P$ and $Q$, $L$ the second point of the intersection of the ray $AP$ with the second circle , and $M$ the intersection point of the ray $AB$ with the second circle , such that $B$ lies between $A$ and $M$. Prove that the angle $LBM$ is three times the angle $LAM$.
On the side $AB$ of the square $ABCD$, point $P$ is chosen so that $AP: PB = 1: 2$. Through $P$ and the center of the square passes a straight line $m$. Prove that for any point $K$ located inside the square on the straight line $m$, the distances from $K$ to the sides $AB, AD, BC$ and $CD$, taken in the specified order form an arithmetic progression.
Arrange four players on the football field so that the pairwise distances between them were equal to $1, 2, 3, 4, 5$ and $6$ meters.
In a convex quadrilateral $ABCD$, the angle $CBD$ is equal to the angle $CAB$, and the angle $ACD$ is equal to the angle $BDA$. Prove that then the angle $ABC$ is equal to the angle $ADC$
Inside a semicircle of radius $12$, there is a circle of radius $6$, and a small semicircle, touching each other in pairs, like shown in the figure. Find the radius of the small semicircle.
Given a triangle $ABC$. Points $D$ and $E$ are taken on sides $AB$ and $BC$ , respectively, so that the angle $ACB$ is twice the angle $BED$ . Prove that $AC + EC> AD$.
The angle bisector splits a triangle into two triangles with equal perimeters. Prove that the original triangle is isosceles.
Let $M$ and $N$ be the points of tangency of the incircle of triangle $ABC$ with sides $AB$ and $AC$, and $P$ is the point of intersection of the straight line $MN$ with the bisector of angle $B$. Prove that the angle $BPC$ is right.
In a semicircle with a radius of $18$ cm, a semicircle of radius $9$ cm is built on one of the halves of the diameter, and a circle is inscribed, touching the larger semicircle from the inside, the smaller semicircle from the outside and the second half of the diameter. Find the radius of this circle.
The rectangle is cut into several rectangles, the perimeter of each of which is an integer number of meters divisible by $4$. Is it true that the perimeter of the original rectangle an integer number divisible by $4$ ?
Find the perimeter of a parallelogram if the bisector of one of its angles divides a side of the parallelogram into segments $7$ and $14$.
Through the points of tangency of the inscribed circle with the sides of the triangle, we draw straight lines, respectively parallel to the bisectors of opposite angles. Prove that these lines intersect at one point.
Median $AM$ of the triangle $ABC$ divides the segment $PR$, parallel to the side $AC$ with ends on the sides $AB$ and $BC$, in lengths of $5$ cm and $3$ cm, counting from the side $AB$. What is the length of side $AC$?
Two circles intersect at points $A$ and $B$, and the center $O$ of the first of them lies on the second. On the second circle, a certain point $C$ is selected, the segment $CO$ intersects the first circle at point $P$. Prove that $P$ is the center of the incircle of the triangle $ABC$.
Danil drew several straight lines through one point. From all the angles formed he considered only the angles with integer degrees. Danil claims that among them angles with odd measures there are exactly $15$ more than even ones. Could this be true?
In triangle $ABC$, points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively. Segments AM and $BN$ meet at point $O$. Prove that the sum of the angles $AMB$ and $ANB$ is greater than the angle $AOB$.
On the plane, a segment $AB$ of length $1$ is given and an arbitrary point $M$ is given on it. Using segments $AM$ and $MB$ as sides, are drawn the squares $AMCD$ and $MBEF$, lying on one side of $AB$. Let $P$ and $Q$ be the intersection points of the diagonals of these squares, respectively. Find the locus of the midpoints of the segments $PQ$, while the point $M$ moves along the entire segment $AB$.
In a right trapezoid $ABCD$, the sum of the lengths of the bases $AD$ and $BC$ is equal to its height $AB$. In what ratio does the bisector of the angle $ABC$ divide the lateral side $CD$?
A segment $AB$ is given on the plane and an arbitrary point $M$ is given on it. Using segments $AM$ and $MB$ as sides, are drawn the squares $AMCD$ and $MBEF$, lying on the same side of $AB$. Let $N$ be the point of intersection of lines $AF$ and $BC$. Prove that for any position of the point $M$ on segment $AB$, each straight line $MN$ passes through some point $S$, common for all such lines.
Given a triangle $ABC$ with point $D$ on side $BC$ and point $H$ on side $AC$. $DK$ is the angle bisector of the triangle $BDA$. It turned out that the angles $CHD$ and $HDK$ are right. Find $HC$ if $AC = 2$.
Point $M$ is the midpoint of the hypotenuse $BC$ of the right-angled triangle $ABC$, and point $P$ divides the leg $AC$ in ratio $AP: PC = 1: 2$. Prove that the angles $PBC$ and $AMP$ are equal.
A circle is inscribed in the square $ABCD$, touching its sides $AB, BC, CD, DA$ at points $P, Q, R ,S$, respectively. Points $M$ and $N$ are taken on the segments $AP$ and $AS$ such that the segment $MN$ touches the incircle. Prove that the line segments $MC$ and $NR$ are parallel.
Can the bisectors of two adjacent external angles of a triangle (adjacent to some side of it) intersect on its circumcircle?
Vasya drew a hexagon, and then selected two of its vertices and drew a straight line through them. This straight line cut a heptagon from the hexagon. How could this be?
The altiude BD was drawn in right-angled triangle $ABC$ with right angle $B$ and angle $A$ equal to $30$. Then, in triangle $BDC$, the median $DE$ was drawn, and in triangle $DEC$, the angle bisector $EF$ was drawn. Find the ratio $FC:AC$.
On the side $AC$ of triangle $ABC$, point $P$ is chosen such that $PC = 2AP$. Point $O$ is the center of the inscribed circle of the triangle $PBC$, $E$ is the touchpoint of this circle with the straight line $PB$. It turned out that $PB=BC$. Prove that line $AE$ is parallel to line $PO$.
Prove that the difference in the lengths of the diagonal $A_1A_4$ and side $A_1A_2$ of a regular decagon $A_1A_2...,A_{10}$ is equal to the radius of its circumcircle.
On the sides $AB$ and $AC$ of the triangle $ABC$, points $M$ and $P$ are selected, respectively, such that the segment $PM$ is parallel to the side $BC$ . The perpendicular from the point $M$ on the straight line $AB$ is drawn , and perpendicular from the point $P$ on the $AC$ is drawn, their intersection point is denoted by $T$. Prove that points $A, T$ and $O$ , the center of the circumscribed circle of triangle $ABC$ , lie on one straight line.
The leaf was folded, as in the picture, and flattened. When the leaf was unfolded, then there were four fold lines, which divided the leaf into $4$ angles . Two adjacent angles turned out to be equal to $57$ and $83$ degrees. What are the angles $\alpha$ and $\beta$ indicated on the figure?
Equilateral triangles are arranged as shown in figure. Prove that line $BK$ is parallel to line $AC$.
On the side $AC$ of an equilateral triangle ABC as on the diameter in outside, a semicircle is constructed, divided by points $P$ and $Q$ into three equal arcs. Prove that the intersection points $M$ and $N$ of the side $AC$ with the segments $BP$ and $BQ$ respectively, the $AC$ is divided into three equal segments.
In the triangle $ABC$, the segments $AM$ and $CP$ are the bisectors of the angles $A$ and $C$ respectively, with $AP+CM=AC$. Find the angle $B$.
On the legs $CA, CB$ and hypotenuse $AB$ of the right-angled triangle $ABC$ outside equilateral triangles $ACM$, $BCH$ and $ABP$, respectively, are constructed. Prove that the lengths of the segments $CP$ and $MH$ are equal.
What is the largest number of disjoint diagonals that can be drawn in convex $n$-gon (diagonals with a common vertex are allowed) ?
In triangle $ABC$, point $M$ is the midpoint of the side $BC$, $H$ is the foot of the altitude , drawn from the vertex $B$. It is known that the angle of the $MCA$ is twice the angle of the $MAC$, and the length of the $BC$ is equal to $10$ cm. Find the length of the segment $AH$.
Point $M$ is the midpoint of side $AB$ of triangle $ABC$. On $CM$ lie points $P$ and $Q$ so that $P$ is closer to $M$, $Q$ is closer to $C$, and $CQ = 2PM$. It turned out that $BQ = AC$. Find the measuere of the angle $APM$.
Let $Q$ and $P$ be the feet of the perpendiculars drawn from the vertex $B$ of the triangle $ABC$ into the bisectors of its angles $A$ and $C$, respectively. Prove that the line $PQ$ parallel to the side $AC$.
Given a triangle $ABC$ with an angle $BAC$ of $30$. In this triangle let $BD$ be the median. The angle $BDC$ turned out to be $45^o$. Find the angle $ABC$.
In a convex quadrilateral $ABCD$, the radii of the circles inscribed in the triangles $ABC, BCD, CDA$ and $DAB$ are equal. Prove that the diagonals $AC$ and $BD$ of this quadrilaterals are equal.
Let $AH,BP$ and $CT$ be the altitudes, and $M$ be the midpoint of the side $BC$ in an acute-angled triangle $ABC$. Line $PM$ intersects the extension of side $AB$ beyond the vertex $B$ at point $Y$, and line $TH$ intersects the extension of the side $AC$ beyond the vertex $C$ at the point $X$. Prove that the lines $BC$ and $XY$ are parallel.
$9$ vertices of a regular $20$-gon are colored red. Prove that there are always three red vertices forming an isosceles triangle.
Two circles intersect at points $A$ and $B$. Through point $A$, we draw a tangent to the first circle intersecting the second at point $C$. Through point $B$, we draw a tangent to the second circle intersecting the first at point $D$. Find the angle between lines $AD$ and $BC$.
correspondence round
The inscribed circle of triangle $ABC$ touches the sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Let $M$ and $N$ be midpoints of sides $AB$ and $BC$, respectively. It turned out that $MP=NQ$. Does it follow that triangle $ABC$ is isosceles?
In the larger semicircle there is a smaller one, the diameter of which is lies on the diameter of the larger one, as shown in the figure. Chord of length $AB$, parallel to the diameters of the semicircles, and tangent to the smaller semicircle, equal to $4$ cm. Find the area of part of the larger semicircle, lying outside the smaller one.
On each side of a regular convex hexagon inside it construct an equilateral triangle. A circle is circumscribed around each such triangle. Prove that these six circles together cover the entire hexagon.
A semicircle of radius $2$ is inscribed with a circle $\omega$ of radius $1$, touching a semicircle in the midpoint $O$ of its diameter $AB$. Find the radius of the circle tangent to $\omega$ and the semicircle as shown in the image.
On different sides of the angle with the vertex $O$, points $A$ and $B$ are taken such that the sum of the lengths $OA$ and $OB$ is constant. Prove that the circumcircles of all possible triangles $AOB$ pass, bedises point $O$, through one more common point.
Three identical balls of radius $1$ are located on the plane, each of which touches the other two and the plane. Find the radius of the fourth ball touching each of these three and the plane.
In an acute-angled triangle $ABC$, the altitudes $BD$ and $AE$ meet at point $P$. Prove that $AB^2 = AP\cdot AE + BP \cdot BD$
Inside a triangle with side lengths $a, b$, and $c$ are drawn three line segments parallel to the sides so that all sides of the hexagon marked in the figure with bold lines, are equal. Find the length of the sides of this hexagon.
A circle is inscribed in an arbitrary triangle. Three tangents parallel to the sides of the triangle, cutting off from the large three small triangles. Prove that the sum of the radii of the circles inscribed in small triangles is equal to the radius of the circle inscribed in original triangle.
A tangential polygon is intersected by a straight line that divides it into two parts having equal area and equal perimeter. Prove that this line goes through the center of the inscribed circle of this polygon.
On the sides $AB, BC, CD$ and $DA$ of the parallelogram $ABCD$, the points $P, Q, R$ and $S$ such that the segment $PR$ is parallel to the side $AD$, the segment $QS$ is parallel to the side $AB$, and their intersection point $M$ lies on the diagonal $AC$. Prove that areas of parallelograms $PBQM$ and $SMRD$ are equal.
In the trapezoid, one of the lateral sides is twice as large as the other, and the sum of the angles at the greater base is $120$ degrees. Find these angles.
Midpoints of consecutive cides of an arbitrary convex quadrilateral $ABCD$ form a convex quadrilateral $PQRS$. Find the the ratio of area $ABCD$ and $PQRS$.
In the quadrilateral $ABCD$, the length of the side $AB$ is $12$ cm, the sine of the angle $BAC$ is $0.33$, the sine of the angle $ADB$ is $0.44$. The angles $BAD$ and $BCD$ have sum $180$ degrees. Find the length of the side $BC$ .
A point $M$ is taken inside an arbitrary rectangle $ABCD$. Prove that that among the segments $AM, BM, CM$, and $DM$ there are three, independent of the choice of $M$, of which you can make a triangle.
In a convex quadrilateral $ABCD$, the lengths of the sides $AB$ and $BC$ are equal to $1$ cm, and the measures of angles $ABC$ and $ADC$ are equal to $102$ and $129$ degrees, respectively. Find the length of the diagonal $BD$.
Three arbitrary different points $A, B$ and $C$ are marked on the circle. Using a compass and a ruler, construct the fourth point $D$ on the circle so that it, together with $A, B$ and $C$ are the vertices of the tangential quadrilateral.
On the segment $AB$, we arbitrarily choose a point $M$ and on one side of $AB$, construct squares $AMCD$ and $BMFE$. The circumscribed circles of these squares intersect in points $M$ and $N$.
a) Prove that lines $FA$ and $BC$ meet at point $N$.
b) Prove that all segments $MN$ pass through one point, independent of the choice of $M$.
$P$ and $Q$ are the midpoints of the bases$ AD$ and $BC$ of trapezoid $ABCD$, respectively. It turned out that $AB = BC$, and the point $P$ lies on the bisector of angle $B$. Prove that $BD = 2PQ$.
In triangle $ABC$, the angle $A$ is $30$ degrees, and the length of the median drawn from vertex $B$ is equal to the length of the altitude drawn from vertex $C$. Find the angles $B$ and $C$.
In an isosceles triangle $ABC$ with a base $AC$, the angle between the angle bisector and the altitude drawn from $A$ is $60$ degrees. Find the angles of the triangle $ABC$.
The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.
Is it possible from $12$ squares and $19$ equilateral triangles with unit sides to create a convex polygon? All squares and all triangles must be used.
You are given an isosceles triangle $ABC$ with base $AC$. From the midpoint $D$ of the base , the perpendicular $DH$ is drawn on the side $BC$. The segments $AH$ and $BD$ intersect at point $E$. Which of segments $BH$ and $BE$ is longer?
In an isosceles triangle $ABC$ with a base $AC$, the angles at the base are equal to $40^o$, and the segment $AD$ is the bisector of the angle $BAC$. Prove that $AC = AD + BD$.
In the quadrilateral $ABCD$, the lengths of the sides $AB$ and $BC$ are equal to $1$ cm, and the angles $ABC$ and $ADC$ are $106^o$and $127^o$, respectively. Find the length of the diagonal $BD$.
In triangle $ABC$, point $K$ is the middle of the side $BC$, and point $L$ is the middpoint of the median $AK$. It is known that the center of the circumcircle of triangle $KCL$ lies on side $AC$ and the circle intersects this side at point $M$ such that $AC: AM = 3: 1$. Find the aspect ratio $AB: BC: AC$
A non isosceles triangle is divided into two parts by some straight line. Prove that these parts cannot be congruent.
Anya drew a square $ABCD$. She then built an equilateral triangle $ABM$ so that the vertex $M$ lies inside the square. The diagonal $AC$ meets the triangle at the point $K$. Prove that $CK = CM$.
Find in an arbitrary triangle $ABC$ a point $M$ such that if we construct circles using segments $MA, MB$ and $MC$, as diameters, then the lengths of their pairwise common chords will be equal.
In a convex quadrilateral, the points $P, Q, R, S$ are the midpoints of the sides $AB,BC,CD,DA$, respectively, and $K, L, M, N$ are the intersection points of the segments $AQ$ and $DP, AQ$ and $BR,CS$ and $BR, CS$ and $DP$ respectively. Prove that the area of the quadrilateral $KLMN$ is the sum of the areas of triangles $AKP$, $BLQ$, $CMR$ and $DNS$.
Area of a quadrilateral formed by the midpoints of the bases and diagonals trapezium, four times less than the area of the trapezoid itself. Find the ratio of the lengths of the bases of the trapezoid.
Prove that in an arbitrary acute-angled triangle $ABC$ there exists a point $M$ such that the angles $MAB, MBC$ and $MCA$ are equal.
Consider all graphs of quadratic functions of the form $y = x^2 + px + q$, intersecting coordinate axes at three different points. Prove that all circles circumscribed around triangles with vertices at these points pass through one common point.
The side lengths of the pentagon $ABCDE$ are equal to $1$. Let the points $P, Q, R, S$ be the midpoints of the sides $AB, BC, CD, DE$, respectively, and the points $K$ and $L$ are the midpoints segments $PR$ and $QS$, respectively. Find the length of the line segment $KL$.
Inside triangle $ABC$, point $O$ is chosen in such a way that the angles $OAC$ and $OCA$ are equal. In addition, straight lines $AO$ are drawn through point $O$ until they intersect with $BC$ at point $L$ and $CO$ until the intersection with $AB$ at point $K$. It turned out that $AK = CL$. Is it nesessary that $AB = BC$?
Diagonals were drawn in the parallelogram, and then the bisectors of all the angles formed by them until they intersect with the sides of the parallelogram. These the points are named $A, B, C$, and $D$, respectively. Prove that $ABCD$ is a rhombus.
In a trapezoid, one side is twice as large as the other, and the sum of the angles at the larger base is $120$ degrees. Find the angles of the trapezoid.
Two circles externally touch each other at point $P$. A line touches the first of them at point $A$ and intersects the second at points $B$ and $C$ ($B$ between $A$ and $C$). Prove that AP is the bisector of the angle adjacent to the angle $BPC$.
An arbitrary point $M$ is taken inside the circle, different from the center of the circle. For each chord of the circle passing through $M$ and different from the diameter, denote by $C$ the point of intersection of the tangents to the circle, drawn through the ends of this chord. Prove that the locus of points $C$ is a straight line.
A square with a side of $100$ cm was drawn on the board.Alexey crossed it with two straight lines parallel to one pair of sides of the square. After that Danil crossed the square two straight lines parallel to another pair of sides of the square. As a result, the square broke into $9$ rectangles, and it turned out that the lengths of the sides of the central section are $40$ cm and $60$ cm. Find the sum of the areas of the rectangles at corners.
In triangle $ABC$, angle $A$ is half the angle $C$, and point $D$ on side $AC$ is the foot of the altitude from $B$. Prove that the difference of the line segments into which $D$ divides $AC$, equals to one of the sides of triangle $ABC$.
In the quadrilateral $ABCD$, the points $P, Q, R, S$ are the midpoints of the sides $AB, BC, CD, DA$ respectively, and $T$ is the intersection point of the segments $PR$ and $QS$. Prove that the sum of the areas of the quadrangles $APTS$ and $CRTQ$ is equal to half the area of the quadrilateral $ABCD$.
A quadrilateral $ABCD$ is inscribed in a circle. Prove that the intersection points og medians of triangles $ABC$, $BCD$, $CDA$ and $DAB$ lie on the same circle.
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, and $P, Q, R,S$ the points of intersection of medians of triangles $AOB$, $BOC$, $COD$ and $DOA$, respectively. To find the ratio of area of quadrangles $PQRS$ and $ABCD$.
The point inside the convex pentagon was connected to its vertices, as a result of which the pentagon was divided into five congruent non-isosceles triangles. Prove that these triangles are right-angled.
Find the angle $ACB$ shown in the picture if you know that all triangles drawn with dotted lines are equilateral.
The inscribed circle of triangle $ABC$ touches its sides $AB, BC$ and $CA$ at points $P, K$ and $M$, respectively, and points $T$ and $X$ are the midpoints of the segments $MP$ and $MK$. Prove that quadrilateral $ATXC$ is cyclic .
On the extension of the diameter $AB$ of the semicircle beyond point $B$, an arbitrary point $C$ is taken, through which a tangent to this semicircle is drawn, tangent to it at point $E$. Let the bisector of the angle $BCE$ intersect the chords $AE$ and $BE$ of the semicircle at points $K$ and $M$, respectively. Prove that triangle $KEM$ is isosceles.
In triangle $ABC$, point $P$ is taken such that the sum of the angles $PBA$ and $PCA$ is equal to the sum of the angles $PBC$ and $PCB$. Prove that the distance from vertex $A$ to point $P$ is not less than the distance from $A$ to point $I$, the center of the circle inscribed in $ABC$, and if these distances are equal, then $P$ coincides with $I$.
In an equilateral triangle $ABC$, through a random point inside it, draw three straight lines: parallel to $AB$ until intersection with $BC$ and $CA$, parallel to $BC$ before crossing with $AB$ and $CA$, parallel to $CA$ until intersection with $BC$ and AB. Prove that the sum of three of the obtained segments is equal to the double of the side of the triangle $ABC$.
Inside an isosceles triangle $ABC$ with equal sides $AB = BC$ and an angle of $80$ degrees at apex $B$, point $M$ is taken such that the angle $MAC$ is $10$ degrees and the $MCA$ is $30$ degrees. Find the angle $AMB$.
Outside the parallelogram $ABCD$, a point $M$ is taken such that the angle $MAB$ is equal to the angle $MCB$ and both triangles $MAC$ and $MCB$ are located outside parallelogram $ABCD$. Prove that the angle $AMB$ is equal to the angle $DMB$.
An arbitrary point $M$ lies on the base $AC$ of the isosceles triangle $ABC$, and through it are drawn straight lines parallel to the lateral sides of the triangle intersecting sides $AB$ and $BC$ at points $P$ and $T$ respectively. Prove that a point $E$ symmetric to $M$ with respect to line $PT$, lies on the circumscribed circle of triangle $ABC$.
Draw $6$ rays from one point so that exactly $4$ acute corners are formed. The angles are considered not only between adjacent rays, but formed by any two rays.
Two pillars are at a distance of $4$ meters. The large pillar is $4$ meters high. A wire is stretched between the tops of these pillars. After a strong wind small the pillar fell towards the larger one. What happened to the wire: it broke, sagged, stayed taut? Explain the answer.
In a triangle, the length of the altitude dropped to side $a$ is equal to $h_a$, and respectively to side $b$ is $h_b$. Prove that if $a> b$ then $a + h_a> b + h_b$.
Let $O$ be the center of the circumscribed circle of an acute-angled triangle $ABC$, and points $A_1, B_1, C_1$ obtained by reflection of $O$ wrt to the sides of $BC , AC$ and $AB$ respectively. Prove that straight lines $AA_1, BB_1, CC_1$ intersect in one point.
Let $E$ be the point of intersection of the diagonals of the convex quadrilateral $ABCD$, and $A_1, B_1, C_1, D_1$ and $A_2, B_2, C_2, D_2$ are the points of intersection of medians and altitudes in triangles $AEB, BEC, CED$ and $DEA$, respectively. Prove that quadrilaterals $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ are similar.
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