Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

There is a square $ ABCD. $ $ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $ They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $ Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $ If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.

Let $Oxy$ be a fixed rectangular coordinate system in the plane. Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied: (a) If $OA_1 = OB_1$, $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$. (b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$. (c) There exists such a number $\phi \in (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$. (d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then$ f(A_1,A_2) = \frac12$. Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$.

In a convex quadrilateral $ABCD$ the angles $\angle A$ and $\angle C$ are equal and the bisector of $\angle B$ passes through the midpoint of the side $CD$. If it is known that $CD = 3AD$, calculate $\frac{AB}{BC}$.

Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.

Let $A$ be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of $A$.

Given a triangle $ABC$, and let $ E $ and $ F $ be the feet of altitudes from vertices $ B $ and $ C $ to the opposite sides. Denote $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ ABC $. Given that $ FA=FC $, prove that $ OEHF $ is a parallelogram.

Let $\Delta ABC$ be a triangle with orthocenter $H$ and $\Gamma$ be the circumcircle of $\Delta ABC$ with center $O$. Consider $N$ the center of the circle that passes through the feet of the heights of $\Delta ABC$ and $P$ the intersection of the line $AN$ with the circle $\Gamma$. Suppose that the line $AP$ is perpendicular to the line $OH$. Prove that $P$ belongs to the reflection of the line $OH$ by the line $BC$.

The $M$ set consists of $k$ non-intersecting segments on the line. It is possible to put an arbitrary segment shorter than $1$ cm on the line in such a way, that his ends will belong to $M$. Prove that the total sum of the segment lengths is not less than $1/k$ cm.

Let $ABCD$ be a parallelogram and let $AX$ and $AY$ be the altitudes from $A$ to $CB, CD$, respectively. A line $\ell \perp XY$ bisects $AX$ and meets $AB, BC$ at $K, L$. Similarly, a line $d \perp XY$ bisects $AY$ and meets $DA, DC$ at $P, Q$. Show that the circumcircles of $\triangle BKL$ and $\triangle DPQ$ are tangent to each other.

Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that \[ p(p \minus{} c) \equal{} (p \minus{} a)(p \minus{} b) \equal{} S, \] where $ S$ is the area of the triangle.

Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.

The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.

Chords $A_1A_2, A_3A_4, A_5A_6$ of a circle $\Omega$ concur at point $O$. Let $B_i$ be the second common point of $\Omega$ and the circle with diameter $OA_i$ . Prove that chords $B_1B_2, B_3B_4, B_5B_6$ concur.

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be.

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$ [i]Proposed by Kyle Lee[/i]

Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

In the triangle $ABC$ the angle bisector $AD$ is drawn, $E$ is the point of tangency of the inscribed circle to the side $BC$, $I$ is the center of the inscribed circle $\Delta ABC$. The point ${{A} _ {1}}$ on the circumscribed circle $\Delta ABC$ is such that $A {{A} _ {1}} || BC$. Denote by $T$ - the second point of intersection of the line $E {{A} _ {1}}$ and the circumscribed circle $\Delta AED$. Prove that $IT = IA$.

The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]