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Champions Tournament 2001-19 (Ukraine) 47p (-14)

geometry problems from the Ukrainian  Champions Tournament  (Турнір чемпіонів) with aops links

collected inside aops :

2001 - 2019 (-2014)


Juniors 2004-2019 

Is it possible to cut a rectangle into $9$ rectangles so that no two adjacent rectangles form a rectangle together? The figure shows how you can cut a rectangle into five parts in this way.

In the triangle $ABC$, the side $AC$ is the smallest. On the sides $AB$ and $CB$ we took the points $K$ and $L$, respectively, such that $KA = AC = CL$. Let $M$ be the point of intersection of $AL$ and $KC$. Let $I$ be the center of the circle inscribed in the triangle $ABC$, points $M$ and $I$ do not coincide. Prove that the line $MI$ is perpendicular to the line $AC$.

The sides of the pentagon, taken consecutively, are $4$ cm,$ 6$ cm, $8$ cm, $7$ cm and $9$ cm. Is it possible to inscribe a circle in this pentagon?

Two circles $s_1$ and $s_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$. Let $M$ be an arbitrary point of the circle $s_1$, $MA$ intersect with $s_2$ at the point $P$, and $MB$ intersects with $s_2$ at point $Q$. Prove that if the quadrilateral $AO_1BO_2$ can be inscribed in a circle, then $AQ$ and $BP$ intersect at circle $s_1$.

The bisector $AK$, the median $BL$ and the altitude $CM$ are drawn in the triangle $ABC$. The triangle $KML$ is equilateral. Prove that the triangle $ABC$ is equilateral.

Using a compass and a ruler, restore the $ABC$ triangle given three points: $D, E, M$, where the points $D$ and $E$ are the midpoints of the altitudes $AH$ and $CP$ of the $ABC$ triangle, and the point $M$ is the midpoint of the side $AC$.

Let $A_1$ and $C_1$ be projections of the vertices $A$ and $C$ of the triangle $ABC$ on the bisector of the outer corner at the vertex $B$, respectively. Prove that the segments $AC_1$ and $CA_1$ intersect at the bisector of the angle $ABC$ of the triangle.

The bisector $BL$ is drawn in the triangle $ABC$. A tangent is drawn through the point $L$ to the circle circumscribed around the triangle $BLC$, which intersects the side $AB$ at the point $P$. Prove that the line $AC$ touches the circle circumscribed around the triangle $BPL$.

Given a triangle $ABC$, in which $AB<BC$. Point $D$ lies on the side $BC$ is such that $AB=DC$. Prove that the line passing through the midpoints of the segments $AC$ and $BD$ is parallel to the bisector of the angle $ABC$.

Triangles are considered, all vertices of each of which lie on three different ones sides of a square. Find the locus of points of centroids of such triangles.

Given a parallelogram $ABCD$ with an angle $A$ equal to $60^o$. Point $O$ is the center of the circle circumscribed around the triangle $ABD$. The line $AO$ intersects the bisector of the outer angle $C$ at the point $K$. Find the ratio $AO:OK$.

Inside the segment $AC$, an arbitrary point $B$ was chosen and circles with diameters $AB$ and $BC$ were constructed. On the circles (in same half-plane wrt $AC$) the points $M$ and $L$ were chosen, respectively, so that $\angle MBA = \angle LBC$. Points $K$ and $F$ are marked on the rays $BM$ and $BL$, respectively, so that $BK = BC$,$ BF = AB$. Prove that the points $M, K, F$ and $L$ lie on the same circle.

Diagonals of a convex $n$-gon that do not intersect divide it into triangles such that each vertex of the $n$-gon is a vertex of an odd number of triangles. Prove that $n$ is divisible by $3$.

Consider all possible parabolas $y = x^2 + px + q$ intersecting the positive coordinate semiaxes at three different points. For each such triple of points, a circle passing through them is constructed. Prove that all these circles have a common point.

In the triangle $ABC$, the point $D$ lies on the side AC, the angles $ABD$ and $BCD$ are equal, $AB = CD$, and $AE$ is the bisector of the triangle $ABC$. Prove that the line $ED$ is parallel to the line $AB$.

2014 missing

In the isosceles triangle $ABC$ ($AB=AC$) the bisector $BD$ is drawn (point $D$ belongs to $AC$). It is known that $BC = BD + AD$. Find the angle $BAC$.

An equilateral triangle $ABC$ is given. On the extension of the side $AB$ beyond the point $A$, the point $D$ is chosen, on the extension of the side $BC$ beyond the point $C$, the point $E$, on the extension of the side $AC$ beyond the point $C$, the point $F$ so that $CF=AD$ and $AC + EF = DE $. Find the angle $BDE$.

2017 Champions Tournament Juniors p2
In a convex hexagon $ABCDEF$ the angles at the vertices $B, C, E$ and $F$ are equal, and the lines $BC$ and $EF$ are parallel. Prove that $AB + AF=CD+ DE$.

2018 Champions Tournament Juniors p3
In a convex quadrilateral $ABCD$, the angles $A$ and $C$ are equal to $100^o$. Points $X$ and $Y$ are selected on sides $AB$ and $BC$ respectively so that $AX =CY$. It turned out that the straight line $YD$ is parallel to the bisector of angle $ABC$. Find the angle $AXY$ .

Given a circle of length $90$. Is it possible to mark $10$ points on it so that among the arcs with ends at these points there are arcs with all integer lengths from $1$ to $89$?

2019 Champions Tournament Juniors p3
On the extension of the median $AM$ of an isosceles triangle $ABC$ with base $AC$ beyond point $M$ is taken point $P$ so that the angle $CBP$ is equal to the angle $BAP$. Find the angle $ACP$ .


Seniors  2001-19 



Let $G$ be the point of intersection of the medians in the triangle $ABC$. Let us denote $A_1, B_1, C_1$ the second points of intersection of lines $AG, BG, CG$ with the circle circumscribed around the triangle. Prove that $AG + BG + CG \le A_1C + B_1C + C_1C$.

 (Yasinsky V.A.)
Three cars move uniformly along three different straight roads (each on its own road). It is known that at some three points in time the cars were on the same straight line. Prove that then these cars will be on the same straight line at any time (each time has its own straight line).

(Yasinsky V.A.)
Given a convex pentagon $ABCDE$ in which $\angle ABC = \angle AED = 90^o$, $\angle BAC= \angle DAE$. Let $K$ be the midpoint of the side $CD$, and $P$ the intersection point of lines $AD$ and $BK$, $Q$ be the intersection point of lines $AC$ and $EK$. Prove that $BQ = PE$.

The point $P$ is outside the circle \omega with center $O$. Lines $\ell_1$ and $\ell_2$ pass through a point $P$, $\ell_1$ touches the circle $\omega$ at the point $A$ and $\ell_2$ intersects $\omega$ at the points $B$ and $C$. Tangent to the circle $\omega$ at points $B$ and $C$ intersect at point $Q$. Let $K$ be the point of intersection of the lines $BC$ and $AQ$. Prove that $(OK) \perp (PQ)$.

Three sets of parallel lines are drawn on the plane, ten lines in each set. What is the largest number of triangles they will cut this plane?

Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.

Two different circles $\omega_1$ ,$\omega_2$, with centers $O_1, O_2$ respectively intersect at the points $A, B$. The line $O_1B$ intersects $\omega_2$ at the point $F (F \ne B)$, and the line $O_2B$ intersects $\omega_1$ at the point $E (E\ne B)$. A line was drawn through the point $B$, parallel to the $EF$, which intersects $\omega_1$ at the point $M (M \ne B)$, and $\omega_2$ at the point $N (N\ne B)$. Prove that the lines $ME, AB$ and $NF$ intersect at one point.

Given a triangle $ABC$, the line passing through the vertex $A$ symmetric to the median $AM$ wrt the line containing the bisector of the angle $\angle BAC$ intersects the circle circumscribed around the triangle $ABC$ at points $A$ and $K$. Let $L$ be the midpoint of the segment $AK$. Prove that $\angle BLC=2\angle BAC$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on the base $BC$ such that $BD:DC = 2: 1$. Note on the segment $AD$ a point $P$ such that $\angle BAC= \angle BPD $. Prove that $\angle BPD = 2 \angle CPD$.

Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.

The polyhedron $PABCDQ$ has the form shown in the figure. It is known that $ABCD$ is parallelogram, the planes of the triangles of the $PAC$ and $PBD$ mutually perpendicular, and also mutually perpendicular are the planes of triangles $QAC$ and $QBC$.  Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces.
Given a right triangle ABC, in which \angle C=90^o. On its hypotenuse AB is arbitrary marked the point P. The point Q is symmetric to the point P wrt AC. Let the lines PQ and BQ intersect AC at points O and R respectively. Denote by S the base of the perpendicular from the point R on the line AB (S \ne P), and let T be the point of intersection of lines OS and BR. Prove that R is the center of the circle inscribed in the triangle CST.

Given a quadrangular pyramid SABCD, the basis of which is a convex quadrilateral ABCD. It is known that the pyramid can sphere a ball. Let P be the point of contact of this sphere with the base ABCD. Prove that  \angle APB + \angle CPD = 180^o.

a) Are there two equal hexagons, all vertices of which coincide, but no two sides coincide?
b) Are there three equal hexagons, all vertices of which coincide, but no two sides coincide?
(Recall that a polygon on a plane is called a simple (without self-intersections) closed polyline, the adjacent links of which do not lie on one line.)


On the sides $AB$ and $BC$  arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such  that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and E the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.

The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).

About the triangle $ABC$ it is known that $AM$ is its median, and $\angle AMC = \angle BAC$. On the ray $AM$ lies the point $K$ such that $\angle ACK = \angle  BAC$. Prove that the centers of the circumcircles of the triangles $ABC, ABM$ and $KCM$ lie on the same line.

On the base of the $ABC$ of the triangular pyramid $SABC$ mark the point $M$ and through it were drawn lines parallel to the edges $SA, SB$ and $SC$, which intersect the side faces at the points $A1_, B_1$ and $C_1$, respectively. Prove that $\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le  \sqrt{SA+SB+SC}$

2014 missing

Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.

Let $t$ be a line passing through the vertex $A$ of the equilateral $ABC$, parallel to the side $BC$. On the side $AC$ arbitrarily mark the point $D$. Bisector of the angle $ABD$ intersects the line $t$at the point $E$. Prove that $BD=CD+AE$

2017 Champions Tournament Seniors p4
Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$

2018 Champions Tournament Seniors p3
The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.

2019 Champions Tournament Seniors p2
The quadrilateral $ABCD$ is inscribed in the circle and the lengths of the sides $BC$ and $DC$ are equal, and the length of the side $AB$ is equal to the length of the diagonal $AC$. Let the point $P$ be the midpoint of the arc $CD$, which does not contain point $A$, and $Q$ is the point of intersection of diagonals $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.


source: https://complex.edu.vn.ua/

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