geometry problems from International Schoolchildren Tournament "Mathematical Multiathlon" (Russia) with aops links in the names
Математическое многоборье
collected inside aops: Seniors , Juniors
2008 - 2017 (lasted only these years)
Reggata Juniors
Inside the rectangle $ABCD$, whose sides $AB = CD = 15$ and $BC = AD = 10$, a point $P$ is given such that $AP = 9$, $BP = 12$. Find $CP$.
Given a convex pentagon $ABCDE$ such that $AB = AE = DC = BC + DE = 1$ and $\angle ABC = \angle DEA = 90^o$. What is the area of this pentagon?
An isosceles triangle $ABC$ with a base $AC$ is inscribed in circle $\omega$. It turned out that the radius of the inscribed circle $ABC$ is equal to the radius of the circle tangent to the smaller arc $BC$ of the circle $\omega$ and the side of the $BC$ in its midpoint (see fig.). Find the ratio of the sides of the triangle $ABC$.
In a convex polygon $A_1A_2...A_{2006}$ opposite sides are parallel ($A_1A_2\parallel A_{1004}A_{1005}, ...$). Prove that the diagonals $A_1A_{1004},A_2A_{1005},...,A_{1003}A_{2006}$ intersect at one point if and only if every two opposite sides are equal.
In a triangle $ABC$ with an angle $BAC$ equal to $24^o$, the points $X$ and $Y$ are taken on the sides $AB$ and $AC$, respectively. In this case, a circle centered at $Y$ passing through $A$ also passes through $X$, and a circle centered at $X$ passing through $B$ also passes through $C$ and $Y$. Find $\angle ABC$.
$AL, BM, CN$ are medians of triangle $ABC$, intersecting at point $K$. It is known that the quadrilateral $CLKM$ is cyclic and $AB = 2$. Find the length of the median $CN$.
Can a right isosceles triangle be split into $6$ different right isosceles triangles?
Let $E$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. It is known that the perimeters of triangles $ABE, BCE, CDE, DAE$ are the same, and the radii of the inscribed circles of triangles $ABE, CE, CDE$ are equal to $3, 4, 6$, respectively. Find the radius of the inscribed circle of triangle $DAE$.
An equilateral triangle $ABC$ is given. Point $D$ is such that $\angle BDC = 90^o$ and points $B, A$ lie in different half-planes wrt line $BC$. Point $M$ is the midpoint of side $AB$. Find the angle $BDM$.
In triangle $ABC$, point $M$ is the midpoint of side $AB$ and $BD$ is the angle bisector. Prove that $\angle MDB = 90^o$ if and only if $AB=3BC$.
In the triangle $ABC$ from the vertex $A$, the altitude $AH$ was drawn. It turned out that $CH:HB=CA^2: AB^2 \ne 1$. What values can the angle $A$ take?
In a right-angled triangle $ABC$ with a right angle $A$, bisectors $BB_1$ and $CC_1$ are drawn. From points $B_1$ and $C_1$, perpendiculars $B_1B_2$ and $C_1C_2$ are drawn on the hypotenuse $BC$. What is the angle $B_2AC_2$ ?
Point $P$ lies on side $BC$ of square $ABCD$. A square $APRS$ was built with side the segment $AP$ . Prove that the angle $RCD$ is $45^o$. (The vertices of both squares are labeled clockwise.)
In an isosceles triangle $ABC$ ($AB = AC$), the angle $BAC$ is $40^o$. Points $S$ and $T$ lie on sides $AB$ and $BC$ respectively, such that $\angle BAT = \angle BCS = 10^o$. Segments $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.
The angle bisector $BD$ is drawn in an isosceles triangle $ABC$ with base $BC$. It turned out that $BD + DA = BC$. Find the angles of triangle $ABC$.
The $h_a$ and $h_b$ are drawn on the adjacent sides $a$ and $b$ of parallelogram $ABCD$, respectively. It is known that $a + h_a = b + h_b$. Consider segments $AB$, $AC$, $AD$, $BC$, $BD$, $CD$. What is the largest number of different ones among them?
On the sides $AB$ and $BC$ of triangle $ABC$, there are points $X$ and $Y$, respectively, such that $\angle BYX = \angle AYC$ and $\frac{BY}{Y C} = \frac{2BX}{XA}$ . Prove that triangle $ABC$ is right-angled.
On the lateral side $AB$ of a right trapezoid $ABCD$ ($AB\perp BC$), a semicircle is constructed (having it as diameter) that touches the side $CD$ at point $K$. The diagonals of the trapezoid meet at point $O$. Find the length of the segment $OK$ if the lengths of the bases of the trapezoid $ABCD$ are equal to $2$ and $3$.
In trapezoid $ABCD$ with bases $AB$ and $CD$, it turned out that $AD = DC = CB <AB$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ are respectively such that $\angle ADE = \angle AEF$. Prove that $4CF \le BC$.
The quadrilateral $ABCD$ is inscribed in a circle. It is known that $AB = AC$ and $BC = CD$. The diagonals of the quadrilateral $ABCD$ meet at the point $O$, and the point $X$ is the midpoint of the arc $CD$ not containing the point $A$. Prove that $XO \perp AB$.
Points $M, N, P$ are the midpoints of sides $AB, CD$ and $DA$ of the inscribed quadrilateral $ABCD$. It is known that $\angle MPD=150^o$, $\angle BCD=140^o$. Find the angle $\angle PND$.
(D. Maksimov)
There are $2013$ line segments of unit length on the plane, each intersecting with everyone. Prove that all of them can be covered with a circle of radius $1.5$.
(A. Shapovalov)
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.
In parallelogram $ABCD$, $BD=BC$. A point $M$ on $AC$ is such that $3AM=AC$. Prove that $AM=BM$.
(13th Ural Tournament, Major League)
A circle with center $O$ is inscribed in the triangle $ABC$. The point $L$ lies on the extension of the side $AB$ beyond $A$. The tangent from $L$ intersects the side $AC$ in the point $K$. Find $\angle KOL$ if $\angle BAC = 50^o$.
(revision of the problem of the Soros Olympiad 1995)
From the point $A_0$ black and red rays are drawn with an angle $7^o$ between them. Then a polyline $A_0A_1...A_{20}$ is drawn (possibly self-intersecting, but with all vertices different), in which all segments have length $1$, all vertices with even numbers lie on the black ray and the ones with odd numbers - on the red ray. What is the index of the vertex that is farthest from $A_0$?
(old Mathematical Kangaroo)
In the convex quadrilateral $ABCD$, the bisectors of the angles $A$ and $C$ are parallel and intersect the diagonal $BD$ in two distinct points $P$ and $Q$, so that $BP = DQ$. Prove that the quadrilateral $ABCD$ is a parallelogram.
(A.Chapovalov)
A paper rectangle $ABCD$ ($AB = 3$, $BC = 9$) is folded in such way that vertices $A$ and $C$ coincide. What is the area of the obtained pentagon?
(additional problems from Mathematical Kangaroo 2014)
Different points $A,B$ and $C$ are marked on the straight line in such a way that $AB=AC=1$. On the segments $AB$ and $AC$, a square $ABDE$ and an equilateral triangle $ACF$ are constructed in one half-plane, respectively. Find the angle between lines $BF$ and $CE$.
In a quadrilateral $ABCD$, point $M$ is the midpoint of side $AB$. Prove that if the angle $DMC$ is right , then $AD+BC \ge CD .$
An angle bisector $AD$ is drawn in an acute-angled triangle $ABC$. The perpendicular drawn from point $B$ on line $AD$ intersects the circumscribed circle of the triangle $ABD$ at a point $E$ other than $B$. Prove that points $A,E$ and the center of the circumcircle $O$ of triangle $ABC$ are collinear.
On side $AB$ of triangle $ABC$, points $K$ and $L$ are selected in such a way that $\angle ACK = \angle KCL = \angle LCB$. The point $M$ on the side $BC$ is such that $\angle BKM =\angle MKC$ . It turned out that $ML$ is the bisector of the angle $KMB$. Find $\angle CLM$.
Given a parallelogram $ABCD$. Points $E ,F$ are marked on lines $AB ,BC$ respectively such that $AF=AB$ and $CE=CB$, and the points $E$ and $F$ do not coincide with $B$. Prove that $DE=DF$.
The vertices of a convex polygon are located at the nodes of an lattice points, moreover, none of its sides passes along the lattice lines. Prove that the sum of the lengths of the vertical segments of the lattice lines enclosed within the polygon, equal to the sum of the horizontal lines
In a regular hexagon $ABCDEF$, the point $M$ is the midpoint of the diagonal $AC$, $N$ is the midpoint of side $DE$. Prove that triangle $FMN$ is equilateral.
$P$ and $Q$ are the midpoints of sides $BC$ and $AD$ of rectangle $ABCD$ , respectively. Diagonals of rectangle PQDC meet at point $R$. It turned out that $AP$ is a bisector of the angle $BAR$. Find the length of the side $BC$ if $AB=1$.
Is it possible to mark points $A, B, C, D, E$ on a straight line so that the distances between them in centimeters are equal: $AB = 6$, $BC = 7$, $CD = 10$, $DE = 9$, $AE = 12$ ?
On the sides $BC$ and $AB$ of the triangle $ABC$, there are points $L$ and $K$, respectively, such that $AL$ is the bisector of the angle BAC, ZACK = ZABC, ZCLK = ZBKC. Prove that $AC = KB$
The diagonals of the convex quadrilateral $ABCD$ are perpendicular and intersect at the point $O$, and $BC = AO$. Point $F$ is such that $CF \perp CD$ and $CF = BO$. Prove that triangle $ADF$ is isosceles.
In an acute-angled triangle $ABC$ on the side $AC$, a point P is chosen such that $2AP = BC$. Points $X$ and $Y $are symmetric to point $P$ wrt vertices $A$ and $C$. It turned out that $BX = BY$. What is the angle $C$ of the original triangle?
Team Juniors
Find the locus of the points of intersection of the medians of the triangles, all the vertices of each of which lie on different sides of the given square.
Two unequal circles $\omega_1$ and $\omega_2$ of radii $r_1$ and $r_2$, respectively, intersect at points $A$ and $B$. On the plane, a point $O$ is taken, for which $\angle OAB$ is $90^o$, and a circle $\omega$ with center $O$ is drawn, tangent to the internally to the circles $\omega_1$ and $\omega_2$. Find the radius of the circle $\omega$.
Point $D$ is selected inside the triangle ABC. The circumscribed circles of triangles $CAD$ and $CBD$ intersect the segments $CB$ and $CA$, at points $E$ and $F$ , respectively. It turned out that $BE = AF$. Prove that $CD$ is the bisector of the angle $\angle ACB$.
You are given a parallelogram $ABCD$. On rays $DB$ and $AC$, there were such points $K$ and $L$, respectively, that $KL \parallel BC$, $\angle BCD = 2 \angle KLD$. Prove that $AK \perp DL$.
In an acute-angled triangle $ABC$, a square is inscribed with side $m$ so that its two vertices lie on the side $AB$, and one vertex on the sides $BC$ and $AC$ . Denote $h_c $ the altitude of the triangle $ABC$ drawn from the vertex $C$ , and $c$ the side $AB$. Prove that $\frac{1}{h_c}=\frac{1}{m}+\frac{1}{c}$.
In the non-isosceles triangle $ABC$, let $M$ be the point of intersection of the medians and $I$ be the point of intersection of the angle bisectors . It turned out that line $MI$ is perpendicular to the side $BC$ . Prove that $AB + AC = 3BC$.
The angle bisector $AL$ is drawn in triangle $ABC$. It is known that $AB = 2007$, $BL = AC$. Find the sides of triangle $ABC$ if known to be integers.
Points $P$ and $Q$ on side$AB$ of convex quadrilateral $ABCD$ are such that $AP = QB$. Point $X$ is a non-$D$ intersection point of the circumscribed circles of triangles $APD$ and $DQB$, and point $Y$ is a non-$C$ intersection point of the circumscribed circles of triangles $ACP$ and $QCB$. Prove that points $C, D, X$ and $Y$ lie on the same circle.
A paper triangle with sides $a, b, c$ bent in a straight line so that the vertex opposite side of length $c$, hit this side. It is known that in the resulting quadrilateral two angles are equal, to the fold line. Find the lengths of the segments into which the vertex that gets there divides side $c$.
Given an isosceles triangle $ABC$ ($AB = BC$). On the side $AB$, point $K$ is selected, and on the side $BC$ point $L$ so that $AK + CL = \frac12AB$. Find the locus of the midpoints of the line segments $KL$.
In a regular heptagon $ABCDEFG$, the sides are $1$. The diagonals $AD$ and $CG$ meet at a point $H$. Prove that $FH = \sqrt2$.
Inside the triangle $ABC$ is chosen point $P$ such that $\angle ABP=\angle CPM$, where $M$ is the midpoint of the segment $AC$. The line $MP$ intersects the circumcircle of the triangle $APB$ at points $P$ and $Q$. Prove that $QA=PC$.
Several chords are drawn in a circle so that every pair of them intersects inside the circle. Prove that all the drawn chords can be intersected by the same diameter.
(A.Chapovalov)
The point $I_b$ is the center of an excircle of the triangle $ABC$, that is tangent to the side $AC$. Another excircle is tangent to the side $AB$ in the point $C_1$. Prove that the points $B, C, C_1$ and the midpoint of the segment $BI_b$ lie on the same circle.
(inspired by olympiad of the Faculty of Mathematics and Mechanics of SPBU)
Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Given two points $C, D$ on $\ell$ such that $B$ is between $C$ and $D. E, F$ are the intersections of $\omega$ and $AC, AD$, respectively, and $G, H$ are the intersections of $\omega$ and $CF, DE$, respectively. Prove that $AH = AG$.
Let $ABC$ be an isosceles triangle ($AB = AC$). On the extensions of the sides $BC, AB$ and $AC$, points $P, X, Y$ are selected in such a way that $PX \parallel AC$ and $PY \parallel AB$ and point $P$ lies on the ray $CB$ . Point $T$ is the midpoint of the arc $BC$ of the circumscribed circle of the triangle $ABC$ ($T \ne A$). Prove that $PT \perp XY$.
Construct a right-angled triangle for a given hypotenuse $c$, if it is known that the median drawn to $c$ is the geometric mean of its legs.
A quadrilateral $ABCD$ is inscribed in a circle. Lines$ AB$ and $CD$ meet at point $E$, lines $AD$ and $BC$ met at point $F$. The bisector of angle $AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$, and the bisector of angle $BFD$ intersects side $AB$ at point $P$ and side $CD$ at point $Q$. Prove that the quadrilateral $MNPQ$ is a rhombus.
In a right-angled triangle $ABC$, the altitude $CH$ is drawn to the hypotenuse. The bisector $BD$ of angle $B$ intersects the altitude at point $E$. Let $K$ be the intersection point of line segments $AE$ and $HD$. Prove that the quadrilateral $CDKE$ and the triangle $AHK$ have equal areas.
(Peru Geometrico)
Geometry Round Juniors
Inside the triangle $ABC$, a point $M$ is taken such that $\angle CMB = 100^o $. Perpendicular bisectors of $BM$ and $CM$ intersect the sides $AB$ and $AC$, respectively, at points $P$ and $Q$. Points $P, Q$ and $M$ lie on one straight line. Find the value of $\angle CAB$ .
Given a quadrilateral $ABCD$ inscribed in a semicircle $\omega$ with diameter $AB$. Lines $AC$ and $BD$ intersect at point $E$, lines $AD$ and $BC$ intersect at point $F$. Line $EF$ intersects semicircle $\omega$ at point $G$, and line $AB$ at point $H$. Prove that $E$ is the midpoint of segment $GH$ if and only if $G$ is midpoint of $FH$.
An acute-angled triangle $ABC$ was drawn on the plane, in which the angle is $A\ne 60^o$. The point of intersection of the altitudes $H$ and the center of the circumscribed circle $O$ was marked in it, then straight lines $m = BH$, $n = CH$ were drawn. After that, all but the straight lines were erased from the drawing, those $m$ and $n$, points $O$ (that is, there were two straight lines and one point). Reconstruct triangle $ABC$ using a compass and ruler.
Let $ABCD$ be a convex quadrilateral with $\angle DAC = 30^o$, $\angle BDC = 50^o$, $\angle CBD = 15^o$, and $\angle BAC = 75^o$. The diagonals of the quadrilateral intersect at point $P$. Find the value of $\angle APD$ .
$ABC$ is an acute-angled triangle, $AD$ is its angle bisector, and $BM$ is its altitude . Prove that $\angle DMC> 45^o$.
$ABCD$ is trapezoid with bases $BC$ and $AD$. Equilateral triangles $ADK$ and $BCL$ are built on the bases ouside $ABCD$. Prove that $AC, BD$ and $KL$ meet at one point.
Cut a square into $5$ pieces, from which you can make $3$ pairs of different squares.
In $\vartriangle ABC$, $AB = AC$, $\angle BAC = 100^o$. Inside $\vartriangle ABC$, a point $M$ is taken such that $\angle MCB = 20^o$, $\angle MBC = 30^o$. Find $\angle BAM$.
The median $BD$ is drawn in the triangle $ABC$, and the point of intersection of the medians $G$ is marked on it. A straight line parallel to $BC$ and passing through point $G$ intersects $AB$ at point $E$. It turned out that $\angle AEC = \angle DGC$. Prove that $\angle ACB = 90^o$.
Points $X$ and $Y$ are selected on the sides $AB$ and $CD$ of rectangle $ABCD$, respectively. The segments $AY$ and $DX$ intersect at the point $P$, and the segments $CX$ and $BY$ intersect at the point $Q$. Prove that $PQ\ge \frac12 AB$.
In an acute-angled triangle $ABC$, the altitude $AD$ is drawn. Points $M$ and $N$ are symmetrical to point $D$ wrt lines $AC$ and $AB$, respectively. Ray $AO$ (where $O$ is the center of the circumscribed circle of triangle $ABC$) intersects $BC$ at point $E$. Prove that $\angle CME = \angle BNE$.
The bisector of angle $A$ of triangle $ABC$ intersects the circumscribed circle of triangle $ABC$ at point $M$, and the side $BC$ at point $A_1$. A circle $\omega$ was drawn through points $B$ and $C$ with center at point $M$. Chord $XY$ of circle $\omega$ passes through point $A_1$. Prove that the centers of the inscribed circles of triangles $ABC$ and $AXY$ coincide.
In triangle $ABC$, the median $AA_1$ is drawn and point $M$, the point of intersection of the medians, is marked on it . Point $K$ lies on side $AB$ is such that $MK \parallel AC$. It turned out that $AM = CK$. Find the angle $ACB$ .
$2011$ points are marked on the plane. A pair of marked points $A$ and $B$ is said to be isolated if all other points are strictly outside the circle constructed on $AB$ as the diameter. What is the smallest number of isolated pairs possible?
In triangle $ABC$, points $M$ and $L$ on side $BC$ are the feet of the median and angle bisector, respectively, drawn from vertex $A$. Points $P$ and $Q$ are the feet of perpendiculars drawn from point $L$ on sides $AB$ and $AC$, respectively. Point $X$ lies on the median $AM$ such that $XL\perp BC$. Prove that points $P, X$ and $Q$ are collinear.
In the convex quadrilateral $ABCD$, it turned out that $AB + CD =\sqrt2 AC$ and $BC + DA = \sqrt2 BD$. Prove that $ABCD$ is a parallelogram.
In a parallelogram $ABCD$, the angle bisector at $A$ meets side $BC$ in its midpoint $M$. Assume that $\angle BDC = 90^o$. Find the angles of the parallelogram $ABCD$.
In a trapezoid $ABCD$ with the parallel sides $AD$ and $BC$, the diagonals are orthogonal. The line parallel to $AD$ and passing through the intersection of the diagonals meets the lateral sides $AB$ and $CD$ at points $K$ and $L$ respectively. Point $M$ on side $AB$ is such that $AM = BK$. Prove that $LM = AB$.
The incircle of an isosceles triangle $ABC$ with $AB = BC$ is tangent to $BC$ and $AB$ at $E$ and $F$ respectively. A half-line trough $A$ inside the angle $EAB$ intersects the incircle at points $P$ and $Q$. The lines $EP$ and $EQ$ meet the line $AC$ at $P'$ and $Q'$. Prove that $P'A = Q'C$.
Prove or disprove that any triangle of area $3$ can be covered by an axially symmetric convex polygon of area $5$.
Points $D$ on the side $AC$ and $E$ on the side $BC$ of triangle ABC are such that $\angle ABD=\angle CBD=\angle CAE$ and $\angle ACB=\angle BAE$. Let $F$ be the intersection point of segments $BD$ and $AE$. Prove that $AF=DE$.
(Ф.Ивлев, Ф.Бахарев)
A hexagon $ABCDEF$ is inscribed into a circle. $X$ is the intersection point of the segments $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is theintersection of $BE$ and $CF$. Given $AX=DY$ and $CY=FZ$, prove that $BX=EZ$.
(Д.Максимов, Ф.Петров)
A quadrilateral $ABCD$ is inscribed into a circle, given $AB>CD$ and $BC>AD$. Points $K$ and $M$ are chosen on the rays $AB$ and $CD$ respectively in such a way that $AK=CM=\frac 12 (AB+CD)$. Points $L$ and $N$ are chosen on the rays $BC$ and $DA$ respectively in such a way that $BL=DN=\frac 12 (BC+AD)$. Prove that the $KLMN$ is a rectangle of the same area as $ABCD$.
(Eisso J.Atzema, proposed by В.Дубровский)
Point $O$ is the circumcenter and point $H$ is the orthocenter in an acute non-isosceles triangle $ABC$. Circle $\omega_A$ is symmetric to the circumcircle of $AOH$ with respect to $AO$. Circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that circles $\omega_A$, $\omega_B$ and $\omega_C$ have a common point, which lies on the circumcircle of $ABC$.
(Ф.Бахарев, inspired by Iran TST 2013)
In a right triangle $ABC$ with the right angle $B$, the angle bisector $CL$ is drawn. The point $L$ is equidistant from the points $B$ and the midpoint of the hypotenuse $AC$. Find the angle $BAC$.
(F.Nilov)
In a convex quadrilateral $ABCD$ the equality $\angle BCA +\angle CAD = 180^o$ holds. Prove that $AB + CD \ge AD + BC$.
(A. Smirnov inspired by Serbian regional olympiad 2014)
You are given a circle and its chord $AB$. At the ends of the chord to the circle are drawn tangent and equal segments $AK$ and $BL$, lying on different sides wrt line $AB$. Prove that line $AB$ divides the segment $KL$ in half.
Consider a pentagonal star formed by diagonals of an arbitrary convex pentagon.
Let's circle $10$ segments of its outer contour one by one solid and dotted lines (see figure).
Prove that the product of the lengths of solid segments is equal to the product of the lengths of the dotted segments .
On the bisectors of angles $A, B, C, D$ of the convex quadrilateral $ABCD$ taken points $A ', B', C ', D'$, respectively, so that line $A'B'$ is parallel to $AB$, line $B'C'$ is parallel to $BC$ and line $C'D'$ is parallel to $CD$. Prove that
a) the line $D'A'$ is parallel to $DA$;
b) $B'D '|| AC$, if additionally it is known that $A'C '|| BD$ .
Angle $B$ of triangle $ABC$ is twice the angle $C$. Circle of radius $AB$ with the center $A$ intersects the perpendicular bisector of the segment $BC$ at point $D$, lying inside the angle $BAC$. Prove that $\angle DAC = \frac13 \angle A$.
Is it possible to construct a triangle (with a compass and a unmarked ruler) given two given angles $\alpha$ and $\beta$ and
a) known perimeter $P$
b) any any altitude of the triangle?
If it is possible, give the construction algorithm (sequence of actions) and indicate the number of different triangles in each case; if not ,justify your answer.
Find the sides of a right triangle if it's perimeter $P$ and it's area $S$ are known
In the sea off the coast of Kamchatka, three suspicious fishing vessels are regularly recorded, traditionally located at the tops of one isosceles triangle with the angle $120^o$ and lengths of the sides $20$ in km. Find the locus of points (GMT) in the space where it is possible to place technical means of control (with a range of no more than $30$ km) for illegal catch of Kamchatka crab, so that all points of this GMT are equidistant from vessels - potential violators of environmental legislation.
Two straight-line railways intersect at point $N$ at an angle of $60^o$. Inside this angle there is an airfield (point A) at distances of $10$ km and $20$ km from these roads. Find the locations of points $B$ and $C$ (places of loading and unloading cargo) on these roads so that the cost of cyclic transportation along the highway from $A$ to $B$, then to $C$ with a return to $A$ is minimal. Considering that the transportation of $1$ ton of cargo along the highway $ABCA$ costs $100$ rubles for $1$ km, determine
a) is it possible for a cargo of $10$ t to keep within the amount of $50$ thousand rubles, excluding the costs of loading and unloading cargo?
b) the same question for the amount of 60 thousand rubles taking into account the cost of a full reloading of goods ($2,500$ rubles per $10$ t) at $2$ points out of $4$? in $3$ points out of $4$? at all $4$ points?
The bisector $BD$ of angle $B$ is drawn in an isosceles triangle $ABC$ ($AB = AC$). The perpendicular to $BD$ at point $D$ intersects line $BC$ at point $E$. Find $BE$ if $CD = d$.
(A. A. Egorov)
The billiard table has a parallelogram shape. Two balls, placed in the middle of one of the sides, hit so that they bounced off different adjacent sides, after which both hit the same point on the opposite side. One ball traveled twice the distance before bouncing than after. Find the ratio of the lengths of the path segments before and after the bounce for the other ball.
(I.N.Sergeev)
Equilateral triangle $ABC$ and right-angled triangle $ABD$ are constructed on segment $AB$ on opposite sides of it, in which $\angle ABD = 90^o$, $\angle BAD = 30^o$. The circumcircle of the first triangle intersects the median $DM$ of the second at point $K$. Find the ratio $AK: KB$.
(A variation of the Nguyen Dung Thanh problem from Cut-the-knot)
Points $K, L, M$ and $N$ are taken on the sides $AB, BC, CD$, and $AD$ of the convex quadrilateral $ABCD$, respectively, so that $KLMN$ is a rectangle and $AK <KB$, $BL> LC$ and $CM <MD$. Can the area of this rectangle be more than half the area of the quadrilateral $ABCD$?
(I.N.Sergeev)
Reggata Seniors
Inside the rectangular parallelepiped $ABCDA'B'CD'$ there are two balls, so that the first one touches all three faces containing $A$, and the second three faces containing $C'$. In addition, the balls touch each other. The radii of the balls are $10$ and $11$, respectively. The lengths of the edges $AB$ and $AD$ are $25$ and $26$. Find the length of the edge $AA'$.
$ABC$ is an equilateral triangle. On one side of the plane $ABC$, the perpendicular to it segments were laid $AA'= AB $and $BB' = AB / 2$. Find the angle between the planes $ABC$ and $A_1B_1C$.
A quadrilateral $ABCD$ is inscribed in a circle $\omega$ and such that $AB = AD$ and $BC = CD$ (deltoid). It turned out that the radius of the inscribed circle ABC is equal to the radius of the circle tangent to the smaller arc $BC$ of the circle $\omega$ and the side of the $BC$ in its midpoint (see fig. ). Find the ratio of the side $AB$ to the radius of the inscribed circle of the triangle $ABC$.
Let $AB$ be the diameter of the circle $\omega$. Line $\ell$ touches $\omega$ at point $B$. Points $C$ and $D$ lie on line $\ell$, and point $B$ lies between points $C$ and $D$. Lines $AC$ and $AD$ intersect circle $\omega$ at points E and $F$, respectively. Lines $CF$ and $DE$ intersect the circle $\omega$ at points $G$ and $H$. Prove that $AH = AG$.
$AB$ is the diameter of the unit circle centered at point $O$. $C$ and $D$ are points on the circle such that $AC$ and $BD$ intersect inside the circle at point $Q$ and $\angle AQB = 2\angle COD$. Find the distance from $O$ from line $CD$.
The circumscribed circle of triangle $ABC$ is fixed. Find the locus of the points of intersection of the medians of the triangles $ABC$.
The radius of the circumscribed circle of the triangle is $2$, and the lengths of all altitudes are integers. Find the sides of the triangle.
$B_1$ is the midpoint of the side $AC$ of the triangle $ABC$ , $C_1$ is the midpoint of the side $AB$ of the triangle $ABC$ . The circumscribed circles of triangles $ABB_1$ and $ACC_1$ intersect at point $P$. Line $AP$ intersects the circumscribed circle of the triangle $AB_1C_1$ at point $Q$. Find $\frac{AP}{AQ}$.
The altitude $AH$ and the ange bisector $AL$ were drawn in the triangle $ABC$. It turned out that $BH: HL: LC = 1: 1: 4$ and the side $AB$ is equal to $1$. Find the lengths of the other sides of the triangle $ABC$.
Points $A, B, C, D, X$ are given on the circle, and $\angle AXB = \angle BXC = \angle CXD$. The distances $AX = a$, $BX = b$, $CX = c$ are known. Find $DX$.
A segment $AB$ is given. For any point $C$, we mark a point $B_1$ on the segment$ AC$ such that $AB_1: B_1C = 1: 2$, and a point $A_1$ on the segment $BC$ such that $BA_1: A_1C = 2: 1$. Find the locus of the intersection points $AA_1$ and $BB_1$, provided that the points $A, A_1, B, B_1$ lie on the same circle.
Given a rectangular trapezoid $ABCD$ ($BC\parallel AD$ and $AB \perp AD$), the diagonals of which are perpendicular and intersect at point $M$. On the sides $AB$ and $CD$, points $K$ and $L$ are selected, respectively, that $MK$ is perpendicular to $CD$, and $ML$ is perpendicular to$ AB$. It is known that $AB = 1$. Find the length of the segment $KL$.
In a regular pyramid $ABCDS$ ($S$ vertex), the length of $AS$ is $1$, and the angle $ASB$ is equal to $30^o$. Find the length of the shortest path from $A$ to $A$ that intersects all lateral edges other than $AS$.
In the quadrilateral $ABCD$ with angles $\angle A = 60^o$, $\angle B = 90^o$, $\angle C = 120^o$, $M$ isthe point of intersection of the diagonals . It turned out that $BM = 1$ and $MD = 2$. Find the area of $ABCD$.
Points $A, B, C$ are given on the straight line, and point $B$ lies between $A$ and $C$, $AB = 3$ and $BC = 5$. Let $BMN$ be an equilateral triangle. Find the smallest value of $AM + CN$.
In the inscribed hexagon $ABCDEF$, it turned out that $AB = BC$, $CD = DE$ and $EF = FA$. Prove that $S_{ABCDEF} = 2S_{BDF}$.
Points $K, L$ and $M$ are taken on the sides $AB, BC$ and $AC$ of a triangle $ABC$, respectively. Suppose that the circumradii of the triangles $AKM$, $BKL$, $CLM$ and $KLM$ are equal. Prove that the triangles $ABC$ and $KLM$ are similar.
Through a point $X$ inside a square $ABCD$, segments $PQ$ and $EF$ parallel to the sides $AD$ and $AB$ respectively are constructed, with the endpoints on the sides of the square ($P$ on $AB$, $F$ on $AD$). If $S_{ECQX} = 2S_{PXFA}$, determine $\angle EAQ$.
In a tetrahedron $SABC$, the circumradii of the faces SAB, SBC and $SAC$ are equal to $108$. The radius of the inscribed sphere of the tetrahedron equals $35$, and the distance between its center and $S$ equals $125$. Find the radius of the circumsphere of the tetrahedron, assuming that its center lies inside the tetrahedron.
Points $K, L, M$ and $N$ lie on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$, respectively. If $\angle KLA = \angle LAM = \angle AMN = 45^o$, prove that $KL^2 + AM^2 = LA^2 +MN^2$.
Points $M$ and $N$ are chosen on a hypotenuse $AC$ of right isosceles triangle $ABC$. Given that $\angle MBN=45^o$, prove that it is possible to construct a right triangle from the segments $AM, MN$ and $NC$.
(folklore)
Given a point $Q$ inside a convex polyhedron $M$. A line $\ell$ passes through $Q$ and intersects the surface of $M$ at points $A$ and $B$. Prove that for infinitely many of lines $\ell$ the equality $AQ= BQ$ holds.
(Putnam 1977 B4 )
A quadrilateral $ABCD$ is inscribed in a circle. $K, L, M$ and $N$ are chosen on the segments $AB, BC, CD$ and $DA$ respectively in such a way that $AK=KB=6$, $BL=3$, $LC=12$, $CM=4$, $MD=9$, $DN =18$, $NA=2$. Prove that quadrilateral $KLMN$ can be inscribed into a circle.
(Ф.Бахарев)
All the vertices of a regular polygon lie on the surface of a cube, but its plane does not contain any of the cube’s faces. What is the maximum possible number of vertices of such a polygon?
(A.Chapovalov)
The incircle of the triangle $ABC$ touches the side $AC$ at $D$. The $\angle BDC$ is equal to $60^o$. Prove that the inscribed circles of triangle $ABD$ and $CBD$ touch $BD$ at the same point and find the ratio of the radii of these circles.
(M.Volchkevich)
On the sides $BC, CA$ and $AB$ of an acute triangle$ ABC$ the points $A_1, B_1$ and $C_1$ are chosen respectively. The circumscribed circles of the triangles $AB_1C_1, BC_1A_1$ and $CA_1B_1$ intersect at P. The points $O_1, O_2$ and $O_3$ are the centres of these circles. Prove that $4S_(O_1O_2O_3) \ge S_(ABC)$.
(A. Smirnov inspired by Macedonia 2014)
$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB+BC+CA=2s$. Prove that $s < AP+BP+CP < 2s$.
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$ .
Let $\vartriangle ABC$ be an acute scalene triangle, and let $N$ be the center of the circle which pass through the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\vartriangle ABC$ at $B$ and $C$. Prove that $A, D$ and $N$ are collinear iff $\angle BAC = 45^o$.
$ABC$ is an acute angle triangle such that $AB>AC$ and $\angle BAC=60^o$. Let’s denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ratio $PO:HQ$.
Given triangle $ABC$ with right angle $C$, find all points $X$ on the hypotenuse $AB$ such that $CX^2 =AX \cdot BX.$
Given triangle $ABC$, point $O$ is the center of its excircle that is tangent to the side $BC$ and extensions of the sides $AB$ and $AC$. Prove that points $A,B,C$ and the center of the circumcircle of triangle $ABO$ are concyclic.
Given isosceles triangle $ABC$ ($AB=BC$), $D$ is the midpoint of its base $AC, E$ is the projection of $D$ on the side $BC$, and $F$ is the second intersection of circumcircle of the triangle $ADB$ with the segment $AE$. Prove that the line $BF$ bisects the segment $DE$.
Triangle $ABC$ with angle bisector $BL$ is given such that $BL=AB$. Point $M$ on the extension of $BL$ beyond point $L$ is such that $ \angle BAM+\angle BAL=180^o$. Prove that $BM=BC$.
$AD$ is a diameter of the circumcircle of the quadrilateral $ABCD$. Point $E$ is symmetric to the point $A$ with respect to the midpoint of $BC$. Prove that $DE \perp BC$.
Incenter $I$ of the acute triangle $ABC$ lies on the bisector of an acute angle between the altitudes $AA_1$ and $CC_1$. Bisector of the angle $B$ intersects the opposite side of the triangle at the point $L$. Prove that points $A_1, I, L, C$ lie on one circle.
Given point $P$ inside the triangle $ABC$ such that points symmetrical to $P$ with respect to the midpoint of $BC$ and to the bisector of angle $A$ lie on one line with the point $A$. Prove that projections of the point $P$ on the sides $AB$ and $AC$ are equidistant from the midpoint of $BC$.
Point $L$ on the bisector of the angle $A$ of the triangle $ABC$ is such that $\angle LBC = \angle LCA = \angle LAB$. Prove that the lengths of the sides of the triangle form a geometric progression.
Geometry Round Seniors
On the plane, two disjoint circles of equal radius are given and point $O$ is the midpoint of a segment with ends at the centers of these circles. A straight line $\ell$, parallel to the line of the centers of these circles, intersects them at points $A, B, C$ and $D$. A straight line $m$ passing through $O$ intersects them at points $E, F, G$ and $H$. Find the radius of these circles if it is known that $AB=BC=CD=14$ and $EF=FG=GH=6$.
Two circles were drawn on the plane $\omega_1$ and $\omega_2$ and points $A$ and $B$ on them so that the segment $AB$ has the greatest possible length. $AR$ and $AS$ are tangents from point $A$ to circle $\omega_2$. $BP$ and $BQ$ are tangents from point $B$ to circle $\omega_1$. The circle $\alpha_1$ touches the circle $\omega_1$ internally and the rays $AR$ and $AS$. The circle $\alpha_2$ touches the circle $\omega_2$ internally and the rays $BP$ and $BQ$. Prove that the radii of the circles $\alpha_1$ and $\alpha_2$ are equal.
Given a triangle ABC and points $D$ and $E$ on the sides $AB$ and $AC$ such that $DE \parallel BC$. Let $P$ be an arbitrary point inside $\vartriangle ADE$. Lines $PB$ and $PC$ meet $DE$ at points $F$ and $G$, respectively. Let the circumcircles of triangle $PDG$ and triangle $PFE$ meet for the second time at point $Q$. Prove that points $A, P, Q$ are collinear.
Are there several points and several straight lines on the plane such that each point lies exactly on $2008$ lines and exactly $2008$ points lie on each line?
$ABC$ is an acute-angled triangle, $AD$ is its angle bisector, and $BM$ is its altitude . Prove that $\angle DMC> 45^o$.
A quadrilateral $ABCD$ is inscribed in a circle with a diameter of $BD$. Point $A_1$ is symmetrical to point $A$ wrt straight line $BD$, point $B_1$ is symmetric to point $B$ wrt straight line $AC$. Let $P$ be the point of intersection of lines $CA_1$ and $BD$, $Q$ be the point of intersection of lines $DB_1$ and $AC$. Prove that $AC \perp PQ$.
An arbitrary tetrahedron is given. The segments connecting the vertices with the intersection points of the medians of the opposite faces intersect at the point $M$. Prove that at least two of the projections of the point $M$ on the faces of the tetrahedron fall inside the corresponding faces.
Let $\omega$ be the circumscribed circle of a non-isosceles triangle $ABC$. Circle $\gamma$ touches lines $AB$ and $AC$ at points $P$ and $Q$, respectively, and circle $\omega$ internally at point $T$. Lines $AT$ and $PQ$ intersect at point $S$. Prove that the circumcircle of triangle $IST$, where I is the center of the incircle of triangle $ABC$, touches $\omega$.
Let $H$ be the orthocenter of an isosceles triangle $ABC$ ($AC = AB$). The point $L$ lies on the base $BC$ such that $LH\parallel AC$, and the point $M$ lies on the side $AC$ such that $ML \parallel AB$. Ray $LH$ intersects $AB$ at point $K$. Prove that $\angle LMK = 90^o$.
Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Segments $AC$ and $BD$ are chords of circles $\omega_1$ and $\omega_2$, respectively, and segment $AB$ and ray $CD$ intersect at point $P$. Ray $BD$ and segment $AC$ intersect at point $X$. Point $Y$ lies on circle $\omega_1$ that $PY\parallel BD$. The point $Z$ lies on the circle $\omega_2$ such that $PZ\parallel AC$. Prove that points $Q, X, Y$ and $Z$ are collinear.
The inscribed circle in triangle $ABC$ touches the side $BC$ at point $A_1$. Segment $AA_1$ intersects the inscribed circle for the second time at point $P$. Segments $CP$ and $BP$ intersect the inscribed circle at points $M$ and $N$, respectively. Point $Q$ is the midpoint of the segment $MN$. Prove that $\angle MQA_1= \angle MQP$.
Point $P$ is marked inside the triangle $ABC$. Let $h_a, h_b$ and $h_c$ be the respective altitudes of the triangle $ABC$. Prove that$$\frac{PA}{h_b+h_c}+\frac{PB}{h_a+h_c}+\frac{PC}{h_a+h_b}\ge 1$$
In rectangle $ABCD$, point $P$ is the midpoint of side $AB$, and point $Q$ is the foot of the perpendicular drawn from point $C$ on $PD$. Prove that $BQ = BC$.
Given a tetrahedron $ABCD$. The sphere passing through the vertices $A, B$ and $C$ intersects intersects the side edges $DA, DB$, and $DC$ at points $A_1, B_1$, and $C_1$. Refelct these points wrt the midpoint of the corresponding edges and obtainpoints $A_2, B_2$ and $C_2$. Prove that points $A, B$ and $C$ are equidistant from the center of the circumscribed sphere of the tetrahedron $DA_2B_2C_2$ .
Points $A, B, C, D$ lie on the circle in the indicated order, with $AB$ and $CD$ not parallel. The length of the arc $AB$ containing points $C$ and $D$ is twice the length of the arc $CD$ that does not contain points $A$ and $B$. Point $E$ is specified by the conditions $AC = AE$ and $BD = BE$. It turned out that the perpendicular from point $E$ to line $AB$ passes through the midpoint of the arc $CD$ not containing points $A$ and $B$. Find $\angle ACB$.
$2011$ points are marked on the plane. A pair of marked points $A$ and $B$ is said to be isolated if all other points are strictly outside the circle constructed on $AB$ as on the diameter. What is the largest number of isolated pairs that can be?
Let $ABC$ be a triangle with a right angle at $C$ and let $M$ be the midpoint of the side $AB$. Point $Q$ on side $CB$ is such that $\frac{BQ}{QC }= 2$. Prove that $\angle QAB = \angle QMC$.
The incircle of an isosceles triangle $ABC$ with $AB = BC$ is tangent to $BC$ and $AB$ at $E$ and $F$ respectively. A half-line trough $A$ inside the angle $EAB$ intersects the incircle at points $P$ and $Q$. The lines $EP$ and $EQ$ meet the line $AC$ at $P'$ and $Q'$. Prove that $P'A = Q'C$.
Let $P$ be an arbitrary point inside a tetrahedron $ABCD$. Denote by $R$ the circumradius of the tetrahedron and by $x$ the distance from $P$ to the circumcenter. Prove the inequalities$$(R + x) (R - x)^3 \le PA\cdot PB\cdot PC\cdot PD \le (R + x)^3 (R - x)$$
Let $ABC$ be a scalene triangle. Denote by $I_A$ the center of the excircle at side $BC$ and by $A_1$ its tangency point with the corresponding side. Points $I_B, I_C, B_1, C_1$ are defined analogously. Show that the circumcircles of triangles $AI_AA_1$, $BI_BB_1$ and $CI_CC_1$ have two common points.
$AA_1$ and $BB_1$ are the altitudes in an acute triangle $ABC$, and $O$ is its circumcenter. Prove that the areas of the triangles $AOB_1$and $BOA_1$ are equal.
(folklore)
Two circles $\omega$ and $\gamma$ have the same center, and $\gamma$ lies inside $\omega$. Let $O$ be an arbitrary point on $\omega$. $OA$ and $OB$ are the tangent lines through $O$ to $\gamma$. Circle with center $O$ and radius $OA$ meets $\omega$ at points $C$ and $D$. Prove that the line $CD$ contains a midline of the triangle $OAB$.
(F. Ivlev)
For which $a>1 $ there exists a convex polyhedron such that the ratio of areas of any two of its faces is greater than $a$?
(A. Shapovalov)
Let $H, I, O$ be the orthocenter, incenter and circumcenter of an acute triangle $ABC$ respectively. Prove that $\angle AIH= 90^o$ if and only if $OI \parallel BC$.
(F. Ivlev)
A closed six-segmented polyline in space is given. Each of its segments is parallell to one of the orthogonal coordinate axes. Prove that its vertices lie on one sphere or in one plane.
(Iran 2014)
A set $M$ of points in the $3$-dimensional space is called interesting, if for any plane there exist at least $100$ points in $M$ outside this plane. For which minimal $d$ any interesting set contains an interesting subset with at most $d$ points?
(IMC 2013.9)
Two congruent circles $b$ and $c$ touching each other are inscribed in the angles $B$ and $C$ of a square $ABCD$. A tangent to circle $b$ is drawn from vertex $A$ and a tangent to $c$ is drawn from $D$ (see fig. ). Prove that the circle inscribed in the triangle bounded by these tangents and the side $AD$ is congruent to the given circles.
(a) Let $A_1A_2A_3A_4A_5$ be a pentagram, i.e., a closed self-intersecting five-edge polyline (see fig. ). Denote by $B_1, B_2, ..., B_5$ the points of intersection of its nonadjacent edges: $B_n$ is the intersection point of $A_{i}A_{i+1}$ and $A_kA_{k+1}$, where the indices of all points are different and we assume $A_6=A_1$. Prove that $A_1B_2\cdot A_2B_3\cdot A_3B_4\cdot A_4B_5\cdot A_5B_1 = B_1A_2\cdot B_2A_3\cdot B_3A_4\cdot B_4A_5\cdot B_5A_1$.
(b) Prove that this equation holds for any polyline $A_1A_2A_3A_4A_5$ that has no parallel edges if we define $B_n$ as the intersection points of the same edges as in (a) or their extensions.
Let $n + 1$ points, $A_1, A_2, ..., A_n$ and $B$, be given in the plane such that no three of them are collinear. It is known that for any two points $A_i$ and $A_j$ one can find a third point $A_k$ that completes a triangle $A_iA_jA_k$ containing the point $B$ in its interior. Prove that $n$ is odd.
On the sides $BC$ and $AC$ of a triangle $ABC$ points $B'$ and $A'$, respectively, are taken. The circumcircle of the triangle $ABC$ meets the line through $C$ parallel to $A'B'$ in a point $D$ (other than $C$). The circumcircle of the triangle $A'B'C$ meets the line through $C$ parallel to $AB$ in a point $E$ (other than $C$). Prove that the lines $AB, A'B'$, and $DE$ are concurrent.
A circle inscribed in the right angle of an isosceles right triangle divides the hypotenuse into three equal parts. The leg of the triangle is of length $1$. Find the radius of the circle.
The length of each bisectors of a triangle is at least one unit. Prove that its area is at least $\sqrt3 / 3$.
A chain of three equilateral triangles $ABC, CDE$, and $EFG$, whose vertices are written counterclockwise, is arranged so that $D$ is the midpoint of $AG$. Prove that triangle $BFD$ is also equilateral.
For any four vertices of a convex polyhedron, the tetrahedron with these vertices lies inside the polyhedron. Does there exist a polyhedron such that none of such tetrahedrons lies inside it?
Find the angles of a triangle $ABC$, knowing that its altitude $CD$ and angle bisector $BE$ meet at a point $M$ such that $CM = 2MD$ and $BM = ME$.
(A. A. Egorov)
Is it possible to construct a $27$-gon that has an inscribed circle and side lengths $1, 2, ..., 27$?
You can arrange its sides in any desired order.
(I. A. Sheipak)
Eight wooden balls are placed in a cubic box measuring $1\times 1\times 1$. Can the sum of their radii be greater than $2$?
(V. N. Dubrovsky, K. A. Knop)
Let $KMLN$ be a square and let $CML$ be a right triangle constructed externally on the side $ML$ as hypotenuse. The sides $CM$ and $CL$ are extended to meet the line $KN$ at $A$ and $B$, respectively. Denote by $P$ the intersection point of segments $AL$ and $KM$, and by $Q$, the intersection point of $BM$ and $NL$. Prove that $CPQ$ is an isosceles triangle.
(Stan Fulger)
Team Seniors
Two unequal circles $\omega_1$ and $\omega_2$ of radii $r_1$ and $r_2$, respectively, intersect at points $A$ and $B$. On the plane, a point $O$ is taken, for which $\angle OAB$ is $90^o$, and a circle $\omega$ with center $O$ is drawn, tangent to the internally to the circles $\omega_1$ and $\omega_2$. Find the radius of the circle $\omega$.
A point on a side of a equilateral triangle is projected onto the other two sides. Prove that the line connecting the starting point with the center of the triangle bisects the segment between the projections.
Triangle $ABC$ with an obtuse angle $C$ is inscribed in a circle with center $O$. It turned out that $AC + BC = 2OC$. The segment $OC$ intersects the side $AB$ at point $D$. Prove that the inscribed circles of triangles $ADC$ and $BDC$ are equal.
Points $P$ and $Q$ are taken inside the triangle $ABC$ such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on the segment $BC$. Prove that $\angle APB + \angle DPC = 180^o$ if and only if $\angle AQC + \angle DQB = 180^o$.
A circle $\Omega$ is given. Circles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$, $\omega_5$ and $\omega_6$ touch $\Omega$ internally. In addition, the circles $\omega_1$, $\omega_3$ and $\omega_5$ touch each other externally, the circle $\omega_2$ touches the circles $\omega_1$ and $\omega_3$ externally, the circle $\omega_4$ touches the circles $\omega_3$ and $\omega_5$ externally, the circle $\omega_6$ touches the circles $\omega_5$ and $\omega_1$ externally (see fig.). Prove that there is a circle tangent to the common internal tangent pairs of circles $\omega_i$ and $\omega_{i+1}$, $i=1,2,3,4,5,6$ (we assume $\omega_7= \omega_1$).
Line segment $AL$ is the angle bisector of triangle $ABC$, $I_1$ and $I_2$ are centers of the circles, inscribed in triangles $ABL$ and $ACL$, respectively. Line $I_1I_2$ crosses the sides$ AB$ and $AC$ at points $C_1$ and $B_1$, respectively. Prove that lines $BB_1$, $CC_1$ and $AL$ intersect at one point.
Given a triangle $ABC$ and concentric circles $\omega_b,\omega_c$ centered at $A$. An arbitrary ray starting from $A$ intersects these circles at points $B'$ and $C'$, respectively. The perpendicular bisector of the segments $BB'$ and $CC'$ meet at the point $X$. Prove that the points $X$, constructed in this way for all rays emerging from $A$, lie on one straight line.
In a regular heptagon $ABCDEFG$, the sides are $1$. The diagonals $AD$ and $CG$ meet at a point $H$. Prove that $FH = \sqrt2$.
Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$. Straight lines $a$ and $b$, intersect the circle $\gamma_1$ for the second time at points$ A$ and $B$, respectively, and the circle $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let $X$ be an arbitrary point of the plane that does not lie on the given lines $a, b$ and circles $\gamma_1$ , $\gamma_2$ . Circles circumscribed around triangles $ATX$ and $BTX$ intersect the circle $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that lines $TX, A_1B_2$ and $A_2B_1$ meet in one point.
Two medians divide a triangle into a quadrilateral and three isosceles triangles. Prove that initial triangle is also isosceles.
(A. Shapovalov)
Prove that in acute-angled triangle the sum of its medians not greater than the sum of radii of its excircles.
(F. Ivlev)
The point $I_b$ is the center of an excircle of the triangle $ABC$, that is tangent to the side $AC$. Another excircle is tangent to the side $AB$ in the point $C_1$. Prove that the points $B, C, C_1$ and the midpoint of the segment $BI_b$ lie on the same circle.
(inspired by olympiad of the Faculty of Mathematics and Mechanics of SPBU)
The spheres $S_1, S_2$ and $S_3$ are externally tangent to each other and all are tangent to some plane at the points $A, B$ and $C$. The sphere $S$ is externally tangent to the spheres $S_1, S_2$ and $S_3$ and is tangent to the same plane at the point $D$. Prove that the projections of the point $D$ onto the sides of the triangle $ABC$ are vertices of an equilateral triangle.
(folklore)
Through the center of an equilateral triangle $ABC$ an arbitrary line $\ell$ is drawn that intersects the sides $AB$ and $BC$ in points $D$ and $E$. A point $F$ is constructed, so that $AE = FE$ and $CD = FD$. Prove that the distance from F to the line $\ell$ does not depend on the choice of $\ell$ .
(M.Volchkevich)
Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Given two points $C, D$ on $\ell$ such that $B$ is between $C$ and $D. E, F$ are the intersections of $\omega$ and $AC, AD$, respectively, and $G, H$ are the intersections of $\omega$ and $CF, DE$, respectively. Prove that $AH = AG$.
Let $ABC$ be a triangle such that it’s circumcircle radius is equal to the radius of $A$-excircle. Suppose that the $A$-excircle touches $BC, AC, AB$ at $M, N, L$. Prove that $O$ (center of circumcircle) is the orthocenter of $MNL$
In acute triangle $ABC$ angle$ B$ is greater than $C$. Let $M$ be the midpoint of $BC$. $D$ and $E$ are the feet of the altitudes from $C$ and $B$ respectively. $K$ and $L$ are midpoints of $ME$ and $MD$ respectively. If the line $KL$ intersects the line through $A$ parallel to $BC$ at point $T$, prove that $TA = TM$.
In a triangle $ABC$ with angle $A= 15^o$ the sides denoted by $a,b,c$ and satisfy the equation $b =\sqrt {a(a+c)}$ . Find the area of $ABC$ if the radius of inscribed circle $r=1$.
(Write the answer as a sum of numbers.)
A tin sheet with area $2a^2$ is used to make a closed box with the shape of a cuboid with maximum volume $V_m=\max V$.
1) Find the sides values of such box. The sheet can be deformated only with straight lines and cutted. But it is impossible to connect (glue or welding) the different parts of the sheet.
2) The same question about $\max V$ and the sides values of opened box (without lid) that can be made from a tin sheet with sizes $b\cdot c$ . Find $\max V$ if the perimeter is known.
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