$X$ and $Y$ are two points lying on or on the extensions of side $BC$ of $\triangle{ABC}$ such that $\widehat{XAY} = 90$. Let $H$ be the orthocenter of $\triangle{ABC}$. Take $X'$ and $Y'$ as the intersection points of $(BH,AX)$ and $(CH,AY)$ respectively. Prove that circumcircle of $\triangle{CYY'}$,circumcircle of $\triangle{BXX'}$ and $X'Y'$ are concurrent.

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

Let $ABCD$ be a quadrilateral such that all sides have equal length and $\angle{ABC} =60^o$. Let $l$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$). Let $E$ and $F$ be the points of intersection of $l$ with $AB$ and $BC$ respectively. Let $M$ be the point of intersection of $CE$ and $AF$. Prove that $CA^2 = CM \times CE$.

Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.

$ABC$ is a right triangle with $\angle C = 90^o$. Let $N$ be the middle of arc $BAC$ of the circumcircle and $K$ be the intersection point of $CN$ and $AB$. Assume $T$ is a point on a line $AK$ such that $TK=KA$. Prove that the circle with center $T$ and radius $TK$ is tangent to $BC$. (Mykhailo Sydorenko)

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?

Let $ABCD$ be a convex quadrilateral, with $\measuredangle ABC > 90^\circ$, $\measuredangle CDA > 90^\circ$ and $\measuredangle DAB = \measuredangle BCD$. Let $E$, $F$ and $G$ be the reflections of $A$ with respect to the lines $BC$, $CD$ and $DB$. Finally, let the line $BD$ meet the line segment $AE$ at a point $K$, and the line segment $AF$ at a point $L$. Prove that the circumcircles of the triangles $BEK$ and $DFL$ are tangent to each other at $G$.

Let $AD$ be a median of $ABC$ and $S$ its midpoint. Let $E$ be a intersection point of $AB$ and $CS$. Prove that $BE=2AE$

In a triangle $ABC, D$ lies on $AB, E$ lies on $AC$ and $ \angle ABE = 30^o, \angle EBC = 50^o, \angle ACD = 20^o$, $\angle DCB = 60^o$. Find $\angle EDC$.

A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments as shown below. This cup has height $h$ and volume $V$. A second cup that is exactly the same shape as the first is held upright inside the first cup so that its base is a distance of $\tfrac{h}2$ from the base of the first cup. The volume of liquid that will t inside the first cup and outside the second cup can be written $\tfrac{m}{n}\cdot V$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] pair s = (10,1); draw(ellipse((0,0),4,1)^^ellipse((0,-6),2,.5)); fill((3,-6)--(-3,-6)--(0,-2.1)--cycle,white); draw((4,0)--(2,-6)^^(-4,0)--(-2,-6)); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-2.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); pair s = (10,-2); draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5)); fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-4.1)--cycle,white); draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6)); //darn :([/asy]

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.

A convex quadrilateral $ABCD$ is given. There rays $BA$ and $CD$ meet in $P$, and the rays $BC$ and $AD$ meet in $Q$. Let $H$ be the projection of $D$ on $PQ$. Prove that $ABCD$ is cyclic if and only if the angle between the rays beginning at $H$ and tangent to the incircle of triangle $ADP$ is equal to the angle between the rays beginning at $H$ and tangent to the incircle of triangle $CDQ$. Also find out whether $ABCD$ is inscribable or circumscribable and justify.

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

The triangle $ABC$ is given. Consider the point $C'{}$ on the side $AB$ such that the segment $CC'$ divides the triangle into two triangles with equal radii of inscribed circles. Denote by $t_c$ the length of the segment $CC'$. Similarly, we define $t_a$ and $t_b$. Express the area of triangle $ABC$ in terms of $t_a,t_b$ and $t_c$. [i]Proposed by K. Mosevich[/i]

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.

A point $D$ lie on the lateral side $BC$ of an isosceles triangle $ABC$. The ray $AD$ meets the line passing through $B$ and parallel to the base $AC$ at point $E$. Prove that the tangent to the circumcircle of triangle $ABD$ at $B$ bisects $EC$.

Let $ABCD$ be an isosceles trapezoid with bases $AB = 92$ and $CD = 19$. Suppose $AD = BC = x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2 = $ $ \textbf{(A)}\ 1369\qquad\textbf{(B)}\ 1679\qquad\textbf{(C)}\ 1748\qquad\textbf{(D)}\ 2109\qquad\textbf{(E)}\ 8825 $

Let $\vartriangle ABC$ be a triangle with $G$ as its centroid, which is the intersection of the three medians of the triangle, as shown in the diagram. If $\overline{GA} \perp \overline{GB}$ and $AB = 7$, compute $AC^2 + BC^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/1/240be132c6adcfde0334a000e1f916a6292907.png[/img]

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

On the side $AB$ of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed circle, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.