The hypotenuse $AB$ of a right-angled triangle $ABC$ touches the corresponding excircle $\omega$ at point $T$. Point $S$ is symmetrical $T$ relative to the bisector of angle $C$, $CH$ is the height of the triangle. Prove that the circumcircle of triangle $CSH$ touches the circle $\omega$.

The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Let $ABC$ be a triangle with circumcenter $O$. Points $P$ and $Q$ are interior to sides $CA$ and $AB$, respectively. Circle $\omega$ passes through the midpoints of segments $BP$, $CQ$, $PQ$. Prove that if line $PQ$ is tangent to circle $\omega$, then $OP = OQ$.

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$

For a triangle $ \triangle ABC (\angle B > \angle C) $, $ D $ is a point on $ AC $ satisfying $ \angle ABD = \angle C $. Let $ I $ be the incenter of $ \triangle ABC $, and circumcircle of $ \triangle CDI $ meets $ AI $ at $ E ( \ne I )$. The line passing $ E $ and parallel to $ AB $ meets the line $ BD $ at $ P $. Let $ J $ be the incenter of $ \triangle ABD $, and $ A' $ be the point such that $ AI = IA' $. Let $ Q $ be the intersection point of $ JP $ and $ A'C $. Prove that $ QJ = QA' $.

A right circular cone stands on plane $P$. The radius of the cone’s base is $r$, its height is $h$. A source of light is placed at distance $H$ from the plane, and distance $1$ from the axis of the cone. What is the illuminated part of the disc of radius $R$, that belongs to $P$ and is concentric with the disc forming the base of the cone?

Let $ABCD$ be a convex quadrilateral such that your diagonals $AC$ and $BD$ are perpendiculars. Let $P$ be the intersection of $AC$ and $BD$, let $M$ a midpoint of $AB$. Prove that the quadrilateral $ABCD$ is cyclic, if and only if, the lines $PM$ and $DC$ are perpendiculars.

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

[b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts: (a) their ages are integers, (b) the product of their ages is $36$, (c) the sum of their ages is equal to the number of windows in the high school of the chemist and the mathematician. The mathematician considered this problem and noted that there was not enough information to obtain a unique solution. The chemist then noted that his oldest son had red hair. The mathematician then announced that he had determined the ages of the three sons. Please (aspiring mathematicians) determine the ages of the chemists three sons and explain your solution. [b]p2.[/b] A square is inscribed in a triangle. Two vertices of this square are on the base of the triangle and two others are on the lateral sides. Prove that the length of the side of the square is greater than and less than $2r$, where $r$ is a radius of the circle inscribed in the triangle. [b]p3.[/b] You are given any set of $100$ integers in which none of the integers is divisible by $100$. Prove that it is possible to select a subset of this set of $100$ integers such that their sum is a multiple of $100$. [b]p4.[/b] Find all prime numbers $a$ and $b$ such that $a^b + b^a$ is a prime number. [b]p5.[/b] $N$ airports are connected by airlines. Some airports are directly connected and some are not. It is always possible to travel from one airport to another by changing planes as needed. The board of directors decided to close one of the airports. Prove that it is possible to select an airport to close so that the remaining airports remain connected. [b]p6.[/b] (A simplified version of the Fermat’s Last Theorem). Prove that there are no positive integers $x, y, z$ and $z \le n$ satisfying the following equation: $x^n + y^n = z^n$. PS. You should use hide for answers.

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt n$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$. [asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("$A$", A, SW); B=(6,15); label("$B$", B, NW); C=(30,15); label("$C$", C, NE); D=(24,0); label("$D$", D, SE); P=(5.2,2.6); label("$P$", (5.8,2.6), N); Q=(18.3,9.1); label("$Q$", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy]

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

Let $M$ be a point on the side $BC$ of triangle $ABC$ and let $P$ and $Q$ denote the circumcentres of triangles $ABM$ and $ACM$ respectively. Let $L$ denote the point of intersection of the extended lines $BP$ and $CQ$ and let $K$ denote the reflection of $L$ through the line $PQ$. Prove that $M, P, Q$ and $K$ all lie on the same circle.

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

In a village there are only $10$ houses, arranged in a circle of a radius $r$ meters. Each has is the same distance from each of the two closest houses. Every year on Sunday of Pascoa, the village priest makes the Easter visit, leaving the parish house (point $A$) and following the path described in Figure 1. This year the priest decided to take the path represented in the Figure 2. Prove that this year the priest will walk another $10r$ meters. [img]https://cdn.artofproblemsolving.com/attachments/a/9/a6315f4a63f28741ca6fbc75c19a421eb1da06.png[/img]

Two fixed circles $\omega$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $C$ and $D$ be two fixed points on the circle $\omega$. Let $P$ be a moving point on $\omega$. Line $PA$ meets circle $\Omega$ again at $Q$. Prove that the second intersection $R$ of two circumcircles of triangles $QPC$ and $QBD$ always lies on a fixed circle. [i]Proposed by buratinogigle[/i]

Inside the parallelogram $ABCD$ are the circles $\gamma_1$ and $\gamma_2$, which are externally tangent at the point $K$. The circle $\gamma_1$ touches the sides $AD$ and $AB$ of the parallelogram, and the circle $\gamma_2$ touches the sides $CD$ and $CB$. Prove that the point $K$ lies on the diagonal $AC$ of the paralelogram.

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.