geometry problems from Olympiad of Youth Mathematical School of St. Petersburg State University (Russia) with aops links in the names
collected inside aops: here
2005 - 2021
(only the problems found get posted)
Ordinary Type Problems
correspodence round
In a convex pentagon, the $k$ diagonals have a length less than $1$ cm, and the remaining $5-k$ of the diagonals - are more than $2$ cm long. What can $k$ be equal to? (Find all possible meanings of $k$ and prove that there are no others).
One night, three flying saucers landed on the field, red, green and blue. Each of them densely covered a
certain triangular area with a powder of its color. The next morning, surprise villagers found that the
intersection of red and green has the shape of a triangle, the intersection of red and blue is quadrangular,
and the intersection of green and blue is pentagonal. Can the intersection of all three sections be
hexagonal?
Kostya drew a triangle and accurately measured the lengths of three sides and three medians in it. He got six different numbers. He told these numbers to Sasha, without specifying which of them are the lengths of the sides, and which are the lengths of the medians. Prove that Sasha can determine at least one of the numbers which is the length of the side.
How to cut the kite of the figure in $6$ identical triangles? Remember to prove that the triangles are indeed equal.
In a right-angled triangle $ABC$, angle $B$ is right and angle $A$ is $30^o$. Let $BH$ be the altitude and $AM$ the median. Point $N$ lies on the segment $AC$, where $AN = BM$, and point $L$ is symmetric to $H$ wrt line $AB$. Prove that triangle $LMN$ is equilateral.
Four points are marked inside the square and nine segments are drawn (see figure). The lengths of all line segments are $1$. Line segments drawn from points to the sides of the square, perpendicular to them. Find the side length of the square.
In an acute-angled triangle $ABC$, the points $B_1$ and $C_1$ are the midpoints of the sides $AC$ and $AB$, respectively, and $B_2$ and $C_2$ are the feet of the altitudes drawn on these sides. It is known that the angle between straight lines $B_1C_2$ and $B_2C_1$ is $60$ degrees greater than angle $A$. What is it equal to?
In convex quadrilateral $ABCD$, angle $A$ is $40^o$, angle $D$ is $45^o$, and the bisector of angle $B$ divides $AD$ in half. Prove that $AB> BC$.
$ABCDE$ and $BEFGH$ are regular pentagons on a plane.
a) Find the value of the angle $CFD$.
b) Prove that eight more points can be added to the eight points mentioned in the condition so that the resulting $16$ points serve as the vertices of six regular pentagons..
In triangle $ABC$, angle $A$ is $50^o$, $BH$ is altitude, point $M$ lies on $BC$ such that $BM = BH$. The perpendicular bisector of the segment $MC$ meets $AC$ at the point $K$. It turns out that $AC = 2 HK$. Find the angles of triangle $ABC$.
In triangle $ABC$, angle $B$ is right. $M$ is the midpoint of the side $BC$ and a point $K$ was found on the hypotenuse such that $AB = AK$ and $\angle BKM = 45^o$. In addition, on the sides $AB$ and $AC$ there were points $N$ and $L$, respectively, such that $BC = CL$ and $\angle BLN = 45^o$. In what ratio does point $N$ divide side $AB$?
final round
Is it possible to mark points $A, B, C, D, E$ on a straight line in such a way that $AB = 14$, $BC = 16$, $CD = 12$, $DE = 8$, $AE = 16 cm$?
There are three identical tiles in the shape of a right-angled triangle with angles of $30^o$ and $60^o$. Vasya made one big triangle out of all of them. Petya claims that he can make a different triangle from the same tiles (also all three). Is Petya right?
In a convex hexagon $ABCDEF$ exist a point $M$ such that $ABCM$ and $DEFM$ are parallelograms . Prove that exists a point $N$ such that $BCDN$ and $EFAN$ are also parallelograms.
In $ \vartriangle ABC$, it is known that $\angle A = 20^o$, $\angle B = 50^o$, $CH$ altitude of $ \vartriangle ABC$, $HE$ angle bisector $\vartriangle AHC$. Find $\angle BEH$.
In triangle $ABC$, it is known that $\angle C = 3\angle A$, and $AB = 2 BC$. Prove that $\angle B = 60^o$.
Inside (or on the border) of a square with side $2$, you took $5$ points. What is the largest number of triangles with vertires at these points may have an area greater than $1$?
In the triangle $ABC$, $AB = BC$. On rays $CA, AB$ and $BC$, points $D, E$ and $F$ are marked, respectively, so that $AD = AC$, $BE = BA$, $CF = CB$. Find $\angle ADB + \angle BEC + \angle CFA$.
In pentagon $ABCDE$, the angle $A$ is $80^o$, $\angle B = 140^o$, $\angle C = 90^o$, $\angle D = 130^o$. There is a point $X$ on the $AE$ side such that $BA + AX = XE + ED = 1$. Calculate the length of the line segment $CX$.
$ABCDEFGHIJK$ and $GIMNOPQRSTU$ are two regular $11$-gons (the vertex order is counterclockwise). Prove that $AP = BQ = IC$.
The perimeters of triangles $ABC$ and $DEF$ are $239$ and $533$, respectively. Can triangles $ABD, BCE, CAF$ be equilateral?
All angles of an convex pentagon with equal sides are different. Prove that the largest and the smallest of them are adjacent.
Point $M$ is the midpoint of the side $AC$ of the triangle $ABC$. Point $N$ lies on the segment $AM$ such that the angle $MBN$ is equal to the angle $CBM$ . On the extension of the segment $BN$ beyond the point $N$, point $K$ is selected such that the angle $BMK$ is right. Prove that $BC = AK + BK$.
$I$ is the center of the inscribed circle $\omega$ of the triangle $ABC$. The circumscribed circle of the triangle $AIC$ intersects $\omega$ at points $P$ and $Q$. Prove that if $PQ$ and $AC$ are parallel, then triangle $ABC$ is isosceles.
A line intersects the sides $AB$ and $BC$ of the square $ABCD$ at the points $X$ and $Y$, and the extensions of the sides $AD$ and $CD$ at the points $Z$ and $T$. Prove that the triangles $CXZ$ and $AYT$ have the same area.
An isosceles triangle $ABC$ with base $AC$ is given. The angle $BAC$ is $37$ degrees. $X$ is the point of intersection of the altitude from the vertice $A$ with a straight line passing through $B$ parallel to the base, $M$ is a point on the line $AC$ such that $BM = MX$. Find the measure of the angle $MXB$.
On the base $AE$ of trapezoid $ABCE$, point $D$ is selected such that $S_{ABCD} = S_{CDE}$. It is known that $ABCD$ is a parallelogram, and its diagonals meet at point $O$. On the segment $DE$, point $T$ is selected. Prove that if $OT \parallel BE$, then $OD \parallel CT$.
In a right-angled triangle $ABC$, point $M$ is the midpoint of the hypotenuse $AB$. On the ray $CM$ mark the points $X$ and $Y$. It is known that lines $XA$ and $YA$ form the same angles with line $AC$. Prove that lines $XB$ and $YB$ form the same angles with line $BC$.
On the diagonal $BD$ of the isosceles trapezoid $ABCD$, there is a point $E$ such that the base $BC$ and the segment $CE$ are the legs of a right-angled isosceles triangle. Prove that lines $AE$ and $CD$ are perpendicular.
Point $D$ is marked on the lateral side $AB$ of an isosceles triangle $ABC$, point $E$ is marked on the lateral side $AC$, and point $F$ is marked on the extension of the base $BC$ beyond point $B$, such that $CD = DF$. Point $P$ is selected on line $DE$, and point $Q$ is selected on segment $BD$ so that $PF\parallel AC$ and $PQ\parallel CD$. Prove that $DE = QF$.
In a triangle $ABC$, the angle $A$ is right. Let $BM$ be the median of the triangle, $D$ the midpoint of $BM$. It turned out that $\angle ABD = \angle ACD$. And what are these angles equal to?
Through the center $O$ of a circle circumscribed about a regular pentagon $ABCDE$, are drawn straight lines parallel to sides $AB$ and $BC$. Their intersection points with $BC$ and $CD$ are denoted by $F$ and $G$ respectively. What is the ratio of the areas of the OFCG to the pentagon $ABCDE$?
Points $X$ and $Y$ are marked in triangle $ABC$, so that the rays $AX, CY$ intersect on the extension of segments $AX$ and $CY$ and are perpendicular to straight lines $BY$ and $BX$, respectively. Sum of distances from $X$ and $Y$ to line $AC$ is less than $BH$. Prove that $AX + CY <AC$
Research Type Problems
Correspodence Round
The axis of symmetry of a set on a plane is a straight line such that for any point from this set, a point symmetric to it with respect to this line also lies in this set.
1. The angle between two straight lines is eight degrees. Is there a polygon in which each of these lines is an axis of symmetry?
2. $AD$ is a non-convex quadrilateral. It is known that each of the triangles $ABC, ABD, ACD, BCD$ has an axis of symmetry. Prove that one of them is the axis of symmetry of the whole quadrilateral.
3. Is the statement of the previous item true if $ABCD$ is a convex quadrilateral?
4. The set of points on the plane has exactly $100$ axes of symmetry. What is the smallest number of points this set can consist of?
A trapezoid is a (flat) quadrilateral in which two sides are parallel and the other two are not (ie, a parallelogram is not a trapezoid).
1. Is there a $6$-hedron in which all faces are trapeziums?
2. Is there a polyhedron $ABCDEFGH$, whose faces $ABFE, BCGF, CDHG$ and $DAEH$ are isosceles trapezoids, and $AE = BF = CG \ne DH$?
3. The $6$-hedron $ABCDEFGH$ has all faces, trapeziums. Is it true that among them there are two parallel ones?
4. $6$-hedron $ABCDEFGH$ has all faces , isosceles trapezoids. Is it true that it is inscribed?
1. Is it possible that the distances from one point inside the $\vartriangle ABC$ to the straight lines $AB, BC$ and $AC$, respectively, are $7, 5$ and $9$ cm, and the distances from another point inside the same triangle to the same straight lines are $8, 9$ and $11$ cm, respectively?
2. Draw three straight lines and mark three points on the drawing, so that the distances from the first point to the straight lines are $0, 0$ and $12$ cm, from the second point - $0, 15$ and $24$ cm, respectively, from the third point - $20, 0$ and $24$ cm respectively. Try to describe the construction of the drawing using a ruler with divisions, a compass and a protractor.
3. Several straight lines are drawn on the plane and one point is marked. Prove that there is a point on the plane that is at a greater distance to each of the drawn lines than the marked point.
4. Three straight lines are drawn on the plane. It is known that for any numbers $a, b$ and $c$, if on the plane there is a point with distances to these straight lines: $a$ - to the first, $b$ - to the second, and $c$ - to the third, then there is also a point, the distance from which to these straight lines: $b$ - to the first, $c$ - to the second, and a - to the third. Pick the shape formed by these lines. Give all (up to similarity and congruence) options and justify why there are no others.
Consider a convex quadrilateral $ABCD$ and a point $E$ on the side $BC$ such that the areas satisfy the condition: $S_{AED} = S_{ABE} + S_{ECD}$.
1. Give an example of a quadrilateral for which there is more than one such point $E$.
2. It is known that the $\angle B$ and $\angle C$ are right, and the corners $\angle A$ and $\angle D$ are not right. Prove that $BE = EC$.
3. It is known that $E$ lies on the bisector of the angles $\angle A$ and $\angle D$. Prove that $AD = AB + CD$.
4. It is known that $BE = EC$, $O$ is the point of intersection of the diagonals. Prove $S_{AOD} = S_{BOC}$.
1. Given a triangle. Prove that you can construct a triangle whose side lengths are the sines of the angles of the given triangle.
2. Prove that among the pairwise products of the tangents of the angles of an acute-angled triangle, one number is at least $3$, the other is at least $2$, and the remaining number is at least $1$.
3. Given an acute-angled triangle, the cosine of the smaller angle is not more than $2/3$. Prove that there is a triangle whose side lengths are equal to the cotangents of the angles of this triangle.
4. It is known that there is a triangle, the lengths of the sides of which are equal to the tangents of the angles of this triangle. In addition, there is a triangle whose side lengths are equal to the cotangents of the angles of this triangle. Prove that all angles of this triangle lie in the interval $(\pi/4, \pi/2)) $
For each three vertices of the $n$-gon (where $n> 4$), the center of the circle passing through them is marked.
1. Any center coincides with one of two given points, $A$ and $B$. Prove that the original $n$-gon is cyclic.
2. It is known that any center coincides with one of six given points. Prove that the original $n$-gon is cyclic.
3. For each natural $n$, give an example of an $n$-gon with exactly $\frac{n^2-3n+4}{2}$ different centers.
4. Is there a natural $n$, such that some $n$-gon has more than one but fewer than $\frac{n^2-3n+4}{2}$ different centers?
Consider four points on the plane: $A, B, C$ and $D$, and the following three points: $P$ , the point of intersection of the lines $AB$ and $CD$, $Q$ , the point of intersection of the lines $AD$ and $BC$, and $R$, the point of intersection of the lines $AC$ and $BD$. Let's call the resulting points accompanying to points $A, B, C$ and $D$, and the quadrilateral $APQR$ also accompanying to $ABCD$.
1. Let the points $A, B, Q$ and $R$ be given on the plane, and none of these four points lie on one straight line. Prove that the remaining points $C, D$ and $P$ can be uniquely reconstructed.
2. Prove that for any $APQR$ on the plane there is at least one nonconvex quadrilateral $ABCD$ for which $APQR$ is accompanying .
3. Let $APQR$ be given on the plane, and $RM$ be is its median. Find on this median infinitely many such positions of point $A$ that you can build a convex quadrilateral $ABCD$, for which $APQR$ will be accompanying and infinitely many positions of the vertex $A$, for which you can construct a non-convex accompanying quadrilateral.
4. Let the position of points $P, Q, R$ and point $A$ inside $APQR$ be given on the plane. Prove that there is at most one quadrilateral $ABCD$ for which $APQR$ is accompanying .
On the third day, Chuk and Gek studied geometry and after lessons started cutting pieces into pieces. They cut the triangle into $9$ triangles as follows (see fig): three straight lines intersecting at one point and passing so that any side is intersected by exactly two of them, and three more segments, connecting the points of intersection of lines with the sides of the original triangle.
1. First, they took an isosceles right-angled triangle as their starting point. What is the largest number of equilateral triangles (among $9$ small ones) they can get?
2. Then the guys took an equilateral original triangle. Can all $9$ triangles be right?
3. The original triangle is equilateral, but of the three lines intersecting inside it, no two are perpendicular. What is the largest number of right-angled triangles can it work now?
A cardboard square was laid on a sheet of paper. The bully pierced the square with a needle (by attaching it to the paper). After that, he turned the square around the needle, drawing on paper the path of each of the vertices of the square, and then threw out the square itself.
1. How many circles can be drawn?
2. Is it true that it is always possible to reconstruct from a picture which two of the four circles correspond to the vertices of the square located diagonally?
3. One of the four circles was erased. What is the maximum number of options for drawing the fourth circle so that the resulting circles can turn out by rotating the vertices of the square?
Final Round
Let's call the circle passing through the midpoints of the sides of the triangle the median circle of this triangle.
1. Prove that the median circle of a triangle touches the circumscribed circle if and only if the triangle is right-angled.
2. Two circles are given with the ratio of radii $1: 2$, and one is strictly inside the other. Prove that for any point on the larger circle there is exactly one triangle, in which this point will be one of the vertices: the larger circle is circumscribed, and the smaller one is the median.
3. Given two intersecting circles with the ratio of the radii $1: 2$. Find the locus of the points which are one of the vertices of a triangle for which the larger circle is the circumcircle, and the smaller one is the median.
4. The radius of the circumscribed circle about a triangle is equal to $1$, the distance between the centers of the circumscribed and the middle one is $\frac{\sqrt3+1}{2}$, one of the angles is $150^o$. Find the area of that triangle.
Recall that the similarity of a figure with a coefficient $k> 0$ is a transformation such that any two points $X$ and $Y$ of the figure are associated with points $X'$ and $Y'$ such that $X'Y'= k \cdot XY$. A figure $\Phi '$ is called similar to a figure $\Phi$ with coefficient $k$ if there is a similarity with coefficient $k$ that transforms $\Phi$ into $\Phi '$.
1. The right-angled triangle $A'B'C$ is inscribed in a similar triangle $ABC$, the names of the vertices are corresponding, while the vertex $A'$ lies on $BC, B'$ on $AC$, and $C'$ on $AB$. Find all possible values for the coefficient of similarity.
2. Triangle $A'B'C'$ is inscribed in triangle $ABC$, similar to it with coefficient $k <1$, the names of the vertices correspond responsible. In this case, vertex $A'$ lies on $BC, B'$ on $AC, C'$ on $AB$, $\angle BC'A'= \theta$, $\angle CAB = \alpha$. Prove that $k\cos \frac{\alpha-\theta}{2}=\frac12$.
3. The heights of the tetrahedron intersect at one point, and the tetrahedron with vertices at the bases of the heights is similar to the original one. Prove that the original tetrahedron is regular.
4. The tetrahedron with vertices at the bases of heights dropped from the vertices of the original tetrahedron turned out to be correspondingly similar to the initial one. Prove that the square of the similarity coefficient is $1 -\sin^2\alpha \sin^2 \beta$, where $\alpha$ is the dihedral angle at some edge, and $\beta$ is the angle between this edge and the opposite one, as between crossing lines.
There is a square with a side length of $1$ km. Two runners run along its diagonals at a speed of $10$ km / h (each on its own diagonal, when it reaches the end of the diagonal, turns around and runs in the opposite direction at the same speed).
1. Prove that at some point the distance between the runners will be at least $\sqrt{2}/2$ km.
2. Prove that someday the distance between the runners will be no more than $1/2$ km.
3. Find how long the minimum of these distances can be.
4. Prove that if at some moment the distance was less than $a$ m, then at some other moment it will be more $(1000 - a)$ m.
Points $A$ and $B$ on the plane are connected by a broken line so that the following conditions are met:
(I) moving from $A$ to $B$ along a given broken line, we keep moving away from $A$,
(II) moving from $A$ to $B$ along this broken line, we are constantly approaching $B$.
In addition, the length of the segment $AB$ is $1$ cm.
1. Prove that any segment of such a broken line, except the initial and final, lies between the bases of the perpendiculars dropped from points $A$ and $B$ to the line containing this segment.
2. Let $A$ and $B$ be connected by a two-link broken line so that conditions (I) and (II) are satisfied. Find the maximum possible length of such a broken line.
3. Let $A$ and $B$ be connected by a three-link broken line so that conditions (I) and (II) are satisfied. Find the maximum possible length of such a broken line.
4. Is it true that polygonal lines satisfying conditions (I) and (II) (and connecting these points $A$ and $B$) can have arbitrarily large length?
1. You are given a circle of radius $1$ with center $O$ and diameter $AB$. Find the largest possible area of a triangle $APB$ if the length of the segment $OP$ is $5$.
2. The diagonal of the quadrilateral passes through the center of its inscribed circle with a radius of $1$. Find the smallest possible perimeter of this quadrilateral.
3. What can be the smallest perimeter of a triangle, the radius of the inscribed circle is $1$, and the length of at least one of the sides is equal to $\ell$?
4. The radius of the inscribed circle of a quadrilateral is $1$, and the length of one of its diagonals is $\ell > 3$. Find the smallest possible value of the perimeter of this quadrilateral.
1. There are $4$ circles $S_1, S_2, S_3, S_4$, which are externally touch each other consecutively in a cycle, i.e., pairs of touching- circles: $S_1$ and $S_2, S_2$ and $S_3, S_3$ and $S_4, S_4$ and $S_1$. Prove that if the centers of the circles form a convex quadrilateral, then it is tangential.
2. Lines $\ell_1$ and $\ell_2$ touch four circles described in the previous paragraph. Line $\ell_1$ separates circles $S_1$ and $S_2$ from $S_3$ and $S_4$, i.e. circles $S_1$ and $S_2$ are on the same side of the straight line $\ell_1$, and $S_3$ and $S_4$ - on the other. Similarly, the line $\ell_2$ separates $S_2$ and $S_3$ from $S_1$ and $S_4$. Prove that the quadrilateral formed by the centers of the circles is a rhombus.
3. There are $8$ spheres that touch the "dice": $S_1, S_2, S_3$ and $S_4$ (consecutively in a cycle) and $S_5, S_6, S_7$ and $S_8$ (also consecutively in a cycle), as well as $S_1$ and $S_5, S_2$ and $S_6, S_3$ and $S_7, S_4$ and $S_8$. There are also three planes, each touching all eight spheres, where the first separates the spheres $S_1 - S_4$ from the spheres $S_5 - S_8$, the second - spheres $S_1, S_2, S_5$ and $S_6$ from $S_3, S_4, S_7$ and $S_8$, and the third - $S_1, S_4, S_5$ and $S_8$ from $S_2, S_3, S_6$ and $S_7$. Prove that the centers of the spheres $S_1, S_2, S_3$ and $S_4$ lie in the same plane.
4. The centers of the eight spheres described in the previous paragraph are the vertices of a hexagon with quadrangular faces. Prove that if this hexagon is inscribed, then it is a cube.
1. Three points move in a straight line at constant speeds so that the sum of the squares of all three pairwise distances between them remains constant. Prove that the speeds of all three points are equal.
2. Three points move in a circle with constant velocity so that the sum of the squares of all three pairwise distances between remains constant (in this problem, the distance between two points on the circle is the smaller of the lengths of the two arcs between them). Prove that all three points are moving at the same speed.
3. Three points move in a circle with constant velocity so that the sum of the squares of all three distances between them is remains constant (in this and the next problem, the distance is is the length of the line segment connecting these points). Moreover, it is known that that not all points move at the same speed. Prove that there are two points that always remain diametrically opposite to each other.
4.Four points move in a circle with constant speeds so that the sum of the squares of the distances between them remains is constant. Prove that among the speeds of these four points no more than two different.
Grade 9: 1,3,4, Grades 10-11: 2,3,4
1. Point $B$ is selected on segment $AC$, then on segments $AB$ and $BC$ are constructed equilateral triangles $ABK$ and $BCL$, lying in one half-plane relative to the straight line $AC$. Points $P, Q,R$ divide the segments $AB, BC, KL$ in a ratio of $1: 2$, respectively. Prove that points $P, Q, R$ form an equilateral triangle.
2. On the sides $AB$ and $BC$ of triangle $ABC$, isosceles triangles $ABK$ and $BCL$ with the same angles at top. Points $P, Q, R$ divide segments $AB, BC, KL$, respectively in the same ratio $k$. Prove that triangle $PQR$ is similar to triangle $ABK$.
3. Two similar triangles $A_1B_1C_1$ and $A_2B_2C_2$ are given, and their orientation coincides (the order of traversing the corresponding the vertices are the same). Points $A_3, B_3, C_3$ are the midpoints of segments $A_1A_2$, $B_1B_2$, $C_1C_2$ respectively. Prove that triangle $A_3, B_3, C_3$ is similar to the first two.
4. Let (in the notation of the previous section) the lines containing the altitudes $A_1H_1$ and $A_2H_2$ of the first two triangles intersect at the point $H$, and the points $H_1$ and $H_2$ lie, respectively on the segments $A_1H$ and $A_2H$. It turned out that these triangles are visible from points $H$ at the same angles. Prove that $A_3H \perp B_3C_3$.
Let $XY$ and $XZ$ be chords of circles $U_1$ and $U_2$, and line $XY$ touches $U_2$, line $XZ$ touches $U_1$, and $T$ is the second pointintersection of these circles.
1. Prove that if $XY = XZ$ then $TY = TZ$.
2. Prove that point $X$ is equidistant from lines $TY$ and $TZ$.
3. Let points $I$ and $J$ be the centers of the incircles $\vartriangle XTY$ and $\vartriangle XTZ$, respectively. Prove that triangles $TIJ$ and $XYT$ are similar (for some vertex order).
4. Let point $O$ be the center of the circumscribed circle of the triangle $\vartriangle XIJ$. Prove that $IOJT$ is cyclic.
5. Let $W$ be the intersection point of the segments $XT$ and $IJ$. Prove those that if the sides $\vartriangle XTY$ are rational, then the length of the segment $WX$ is so is rational.
Grades 9-10: 1-3,5 Grade 11: 2-5
Three circles were drawn on the plane, any two of which intersect at two points. Then all the intersection points were marked, and the circles themselves were erased.
1. Let $6$ points be marked. Can they be the vertices of a regular hexagon?
2. Let $4$ points be marked, and they lie at the vertices of a rhombus with angles $60^o$ and $120^o$. The radius of the smaller circle is $1$. What are the radii of the other two?
3. Let $6$ points be marked. Can different sets of circles correspond to them?
4. Suppose there were not three, but $100$ circles, and $9900$ points were marked. Prove that the original circles can be recovered uniquely.
1.In triangle $ABC$, point $M$ lies on side $AC$, and point $N$ lies on side $BC$. Can line segments $AN$ and $BM$ split triangle $ABC$ into four parts with equal areas?
2. In triangle $ABC$, points $N$ and $M$ lie on side $BC$, and points $K$ and $L$ lie on side $AC$. Can $9$ parts, into which triangle ABC is divided by segments$ AN, AM, BK, BL$, have equal areas?
3. In triangle $ABC$, point $M$ lies on side $AB$, point $K$ on $BC$, and point $L$ on $AC$. How many parts did the segments $AK, BL, CM$ divide the triangle $ABC$ into if the areas of all the parts obtained are equal?
4. In quadrilateral $ABCD$, point $K$ lies on side $BC, L$ on side $CD, M$ on $AD$, $N$ on $AB$. Can all parts, into which $ABCD$ divided the segments $AK, BL, CM, DN$, have equal areas?
A stencil is cut out of cardboard in the form of the requlat $n$-gon with a side of $1$ cm. We will call it an $n$-stencil. The following geometric constructions are allowed:
a) attach the stencil to a sheet of paper so that several vertices of the stencil coincide with the points previously marked on the paper;
b) completely or partially circle the stencil attached to the sheet, marking all the vertices obtained in this case;
c) mark the intersection points of the segments already drawn;
d) attach the stencil to a sheet of paper so that one of the sides stencil lay on the previously drawn segment.
1. Prove that using operations a) –– d) and a $10$-stencil, you can construct the center of a regular $10$-gon with a side of $1$ cm.
2. Prove that using operations a) –– c) and a $12$-stencil, you can construct the center of a regular $12$-gon with a side of $1$ cm.
3. Prove that using operations a) –– b) and a $10$-stencil, you can construct the center of a regular $10$-gon with a side of $1$ cm.
4. Prove that using operations a) –– b) and a $14$-stencil you can construct the center of a regular $14$-gon with a side of $1$ cm.
5. Prove that using operations a) –– b) and a $2014$-stencil, you can construct the center of a regular $2014$-gon with a side of $1$ cm.
6. Prove that using operations a) –– d) and an $n$-stencil you can construct the center of a regular $n$-gon with a side of $1$ cm for any $n> 4$. In addition to operations a) –– d), it is allowed to connect by a segment any two points, the distance between which is not more than $1$ cm.
Grade 10: 1-3,5 Grade 11: 1-2,4,6
A triangle $ABC$ is considered, on the sides of which points are selected $X, Y$ and $Z$ (one point on each side). Moreover, it is known that the centers of the inscribed circles triangles $ABC$ and $XYZ$ coincide.
1. Let points $X, Y$ and $Z$ be the midpoints of the sides of triangle $ABC$. Prove that triangles $ABC$ and $XYZ$ are equilateral.
2. Let the radius of the incircle of $\vartriangle XYZ$ be half the radius of the incircle of $\vartriangle ABC$. Prove that if triangle $XYZ$ is equilateral, then $ABC$ is equilateral.
3. Let the radius of the incircle of $\vartriangle XYZ$ be half the radius of the incircle of $\vartriangle ABC$. Prove that $\vartriangle XYZ$ is equilateral.
4. Let the ratio of the radius of the incircle of $\vartriangle ABC$ to the radius of the incircle of $\vartriangle XYZ$ be equal to $t$. Prove that$$t^2 \le \frac{xy}{ (p-x) (p-y)},$$where $x = YZ$, $y = XZ$, $p$ is the semiperimeter of $\vartriangle XYZ$.
The perpendicular bisector of the side $AC$ of the triangle $ABC$ intersects the $BC$ segment and the ray $AB$ at points $D$ and $E$, respectively. Points $M$ and $N$ are the midpoints of the segments $AC$ and $DE$, respectively.
1. It turned out that $\angle BAC = 2\angle BCA$. Prove that $DM \le DB$.
2. Prove that $\angle ABC + \angle MBN = 180^o$ if $AB = BE$.
3. The point $J\ne D$ on the segment BD is chosen such that $DE = JE$. It is known that $\angle ABM = 90^o$. Find the ratio of the areas of triangles $ADJ$ and $BEJ$.
4. For what values of $\alpha$ can the equality $\angle MBN = \alpha = \angle ABC$ be satisfied?
In all problems $O$ denotes the center of the circumcircle of triangle $ABC$, and $I$ the center of its incircle.
1. Consider an acute-angled triangle $ABC$ and its orthocenter $H$. It turns out that points $B, O, H$ and $C$ lie on the same circle. Prove that point $I$ lies on the same circle.
2. Points $X$ and $Y$ are the midpoints of arcs $AC$ and $AB$ of the circumscribed circle of triangle $ABC$, respectively. The segment $XY$ and the side of the triangle $AC$ meet at the point $Z$. Prove that$$|IZ|> \frac{|AC|-|IC|}{2}.$$3. Given a non-isosceles triangle $ABC$ with angle $\angle A = 60^o$. Prove that the intersection point of lines $OI$ and $BC$ is equidistant from points $A$ and $I$.
4. An arbitrary acute-angled triangle $ABC$ is given. $X$ is some point inside the triangle. The circumcircles of triangles $AOX, BOX$ and $COX$ intersect the circumcircle of triangle $ABC$ at points $A_1, B_1$ and $C_1$, respectively. Prove that $X$ is the center of the incircle $ABC$ if and only if $X$ is the orthocenter $A_1B_1C_1$.
Let $I$ be the center of the inscribed circle $\omega$ of the triangle $ABC$. The circumscribed circle of the triangle $AIC$ intersects $\omega$ at points $P$ and $Q$ so that $P$ and $A$ lie on one side of the straight line $BI$, and $Q$ and $C$ on the other. We denote by $M$ the midpoint of the smaller arc $AB$ of the circumscribed circle of the triangle $ABC$, and by $N$ the midpoint of the smaller arc $BC$.
1. Prove that if $PQ\parallel AC$, then triangle $ABC$ is isosceles.
2. Given a triangle $DEF$. The circle passing through the vertices $E$ and $F$ intersects the sides $DE$ and $DF$ at points $X$ and $Y$, respectively. The bisector of angle $\angle DEY$ intersects $DF$ at point $Y '$, and the bisector of angle $\angle DFX$ intersects $DE$ at point $X'$. Prove that $XY\parallel X'Y '$.
3. Prove that $MN> PQ$
4. Let $L$ be the point of intersection of lines $AP$ and $CM, S$ be the point of intersection of lines $AN$ and $CQ$. Prove that $LS\parallel PQ$.
5. Prove that $MN\parallel PQ$.
6. Let $T$ be the point of intersection of lines $AP$ and $CQ$, and $K$ be the point of intersection of lines $MP$ and $NQ$. Prove that $T, K$ and $I$ are collinear.
Grade 9: 1,3-5 Grade 10: 1-2,5-6
An arbitrary triangle ABC with orthocenter $H$ is given. The inner and external bisectors of angle $B$ intersect the line $AC$ at points $L$ and $K$, respectively. Two circles are considered: $\omega_1$ the circumcircle of the triangle $AHC$, $\omega_2$ having segment $KL$ as it's diameter.
1. Let point $T$ be such that $TL$ is the angle bisector of the triangle $ATC$ . Prove that $TK$ is the external bisector of the same triangle.
2. Let $X$ be a point of intersection of the circles $\omega_1$ and $\omega_2$ such that $X$ and $B$ lie on opposite sides wrt line $AC$. Prove that the point $X$ lies at the alitude $BH$ of the triangle $ABC$.
3. Let $Y$ be a point of intersection of the circles $\omega_1$ and $\omega_2$ such that the points $Y$ and $B$ lie on the same side wrt line $AC$. Prove that point $Y$ lies on the median $BM$.
4. Prove that the tangent to the circle $\omega_1$ and $\omega_2$ at the point of intersection with the median $BM$ intersects the line $AC$ at the midpoint of the segment $KL$.
A circle $\omega$ with a center at point $I$ is inscribed in the triangle ABC and touches its sides $AB$ and $AC$ at points $D$ and $E$, respectively. The angle bisectors of the triangle $ADE$ intersect at point $J$. The segments $BJ$ and $CJ$ intersect the segment $DE$ at points $P$ and $Q$, respectively.
1. Prove that $PJ> PD$.
2. It is known that $EJ = DE$. Find the angle $BAC$.
3. Prove that the perimeter of the triangle $BJC$ is greater than the perimeter of the perimeter $BDEC$.
4. Points $M$ and $N$ are the midpoints of $DJ$ and $JE$ respectively. Prove that $PM = QN$.
Georgy Konstantinovich has a garden, through which Erich sometimes runs. Erich runs in a straight line, but each time a new one. Georgy Konstantinovich wants to buy and place anti-tank hedgehogs inside or on the border of the garden (in the form of several segments) so that Erich is guaranteed to rest against them. The length of a hedgehog is the sum of the lengths of its constituent segments.
1. Let the garden have the shape of an equilateral triangle with side $1$. Prove that hedgehogs with total length \sqrt3 are enough for Georgy Konstantinovich.
2a. Let the garden have the shape of a square with side $1$. Prove that hedgehog with total length $2.65$ are enough for Georgiy Konstantinovich.
2b. Let the garden have the shape of a square with side $1$. Prove that hedgehog with total length $2.64$ are enough for Georgiy Konstantinovich.
3a. Let the garden have the shape of an equilateral triangle with side $1$. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least $\frac{3\sqrt3}{4}$.
3b. Let the garden have the shape of an equilateral triangle with side 1. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least $1.29$.
4. Let the garden have the shape of a square with side $1$. Prove that Georgy Konstantinovich will have to buy hedgehogs of total length at least $2$.
Grade 9: 1,2a,3a,4 , Grade 10: 1,2a,3b,4 Grade 11: 1,2b,3b,4
Two circles inscribed in an angle with a vertex $R$ meet at points $A$ and $B$. A line is drawn through A that intersects the smaller circle at point $C$, and the larger one at point $D$. It turned out that $AB = AC = AD$.
1. Prove that the tangents to the circles at point $A$ are perpendicular.
2. Let $C$ and $D$ coincide with the points of tangency of the circles and the angle. Prove that the $\angle R$ is right.
3. Let $C$ and $D$ coincide with the points of tangency of the circles and the angle. Find the angle $ADR$.
4. Prove that if $\angle R$ is right, then $C$ and $D$ coincide with the tangency points of the circles and the angle.
5. Let $\angle R = 135^o$. The perpendicular from A to the nearest side of the angle intersects the smaller circle at point $P$, the perpendicular from $A$ to the second side intersects $BP$ at point $Q$. Finally, let $O_1$ and $O_2$ be the centers of the original circles, $O$ be the center of the circle circumscribed around $\vartriangle ABQ$. Prove that $BO$ is the bisector of the angle $O_1BO_2$.
6. What values can the angle $ \angle RAO_1$ take, where $O_1$ is the center of the smaller circle?
Grade 9: 1-4 , Grade 10: 2-5 Grade 11: 2-4,6
There is a square with a side of $2$. Vasya painted a finite number in it polygons so that there are no filled points at a distance of $1$ (the border is considered unpainted). Let $A$ be the filled set, $S (A)$ is its area.
1. Give an example of A such that $S (A)\ge 1$.
2. Prove that $S (A) \le 2$.
3. Give an example of $A$ with $S (A)\ge 2.5 -\sqrt2$
4. Estimate $S (A)$ from above as $1.65$.
5. Give an example of $A$ such that $S (A) \le 1.14$.
Grades 9: 1-4 Grade 11: 1,2,4,5
An equilateral triangle $ABC$ is given. Points $X$ and $Y$ on sides $AB$ and $BC$, respectively. Equilateral triangles $AXD$ and $BY E$ are built in the outside wrt the original triangle $ABC$. Point $F$ is such that triangle $DEF$ is equilateral, and points $F$ and $C$ lie on the same side wrt DE.
1. It is known that $AX = BY$, and G is the intersection point of the side $AB$ and the line passing through $F$ parallel to $BC$. Prove that $GF = AB$.
2. It is known that $AB$ is parallel to $FC$. Prove that $AX = BY$.
3. Let the initial triangle $ABC$ be fixed and its side equal to $a$, and the rest of the points are not fixed. Let $F'$ be a point different from $F$ such that the triangle $DEF'$ is equilateral. Find locus of points F' and its length.
4. Let G be the point of intersection of the side $AB$ and the line passing through $F$ parallel to $BC$. Prove that $BG = AX$.
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