Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.

Let $ABC$ be a acute triangle in which $O$ and $H$ are the center of the circumscribed circle, respectively the orthocenter. Let $M$ be a point on the small arc $BC$ of the circumscribed circle (different from $B$ and $C$) and be $D, E, F$ be the symmetrical of the point $M$ to the lines $OA, OB, OC$. We note with $K$ the intersection of $BF$ and $CE$ and $I$ is the center of the circle inscribed in the triangle $DEF$. a) Show that the segment bisectors of the segments $EF$ and $IK$ intersect on the circle circumscribed to triangle $ABC$. a) Prove that points $H, K, I$ are collinear.

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

Point $M$ is the midpoint of side $BC$ of an acute—angled triangle $ABC$. The point $U$ is symmetric to the orthocenter $ABC$ relative to its circumcenter. The point $S$ inside triangle $ABC$ is such that $US = UM$. Prove that $SA + SB + SC + AM < AB + BC + CA$.

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is: $\textbf{(A)}\ \sqrt{3}:1 \qquad \textbf{(B)}\ \sqrt{3}:\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{3}:2 \qquad \textbf{(D)}\ 3:\sqrt{2} \qquad \textbf{(E)}\ 3:2\sqrt{2}$

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

Let $M=\{A_{1},A_{2},\ldots,A_{5}\}$ be a set of five points in the plane such that the area of each triangle $A_{i}A_{j}A_{k}$, is greater than 3. Prove that there exists a triangle with vertices in $M$ and having the area greater than 4. [i]Laurentiu Panaitopol[/i]

A circle inscribed in the square $ABCD$, with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ into $MN$. Find the value of segment $PI$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)

A tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. If the volume of the tetrahedron is $V$ , compute $V^2$ .

Construct a triangle by its altitude , median and angle bisector originating from one vertex.

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.

Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. [asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression: $$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$

[i]25 problems for 30 minutes.[/i] [b]p1.[/b] Chad, Ravi, Kevin, and Meena are four of the $551$ residents of Chadwick, Illinois. Expressing your answer to the nearest percent, how much of the population do they represent? [b]p2.[/b] Points $A$, $B$, and $C$ are on a line for which $AB = 625$ and $BC = 256$. What is the sum of all possible values of the length $AC$? [b]p3.[/b] An increasing arithmetic sequence has first term $2014$ and common difference $1337$. What is the least odd term of this sequence? [b]p4.[/b] How many non-congruent scalene triangles with integer side lengths have two sides with lengths $3$ and $4$? [b]p5.[/b] Let $a$ and $b$ be real numbers for which the function $f(x) = ax^2+bx+3$ satisfies $f(0)+2^0 = f(1)+2^1 = f(2) + 2^2$. What is $f(0)$? [b]p6.[/b] A pentomino is a set of five planar unit squares that are joined edge to edge. Two pentominoes are considered the same if and only if one can be rotated and translated to be identical to the other. We say that a pentomino is compact if it can fit within a $2$ by $3$ rectangle. How many distinct compact pentominoes exist? [b]p7.[/b] Consider a hexagon with interior angle measurements of $91$, $101$, $107$, $116$, $152$, and $153$ degrees. What is the average of the interior angles of this hexagon, in degrees? [b]p8.[/b] What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa? [b]p9.[/b] What is the first time after $4:56$ (a.m.) when the $24$-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with difference $1$? (For example, $23:41$ is one of those moments, while $23:12$ is not.) [b]p10.[/b] Chad has trouble counting. He wants to count from $1$ to $100$, but cannot pronounce the word "three," so he skips every number containing the digit three. If he tries to count up to $100$ anyway, how many numbers will he count? [b]p11.[/b] In square $ABCD$, point $E$ lies on side $BC$ and point $F$ lies on side $CD$ so that triangle $AEF$ is equilateral and inside the square. Point $M$ is the midpoint of segment $EF$, and $P$ is the point other than $E$ on $AE$ for which $PM = FM$. The extension of segment $PM$ meets segment $CD$ at $Q$. What is the measure of $\angle CQP$, in degrees? [b]p12.[/b] One apple is five cents cheaper than two bananas, and one banana is seven cents cheaper than three peaches. How much cheaper is one apple than six peaches, in cents? [b]p13.[/b] How many ordered pairs of integers $(a, b)$ exist for which |a| and |b| are at most $3$, and $a^3-a = b^3-b$? [b]p14.[/b] Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist? [b]p15.[/b] Jordan is swimming laps in a pool. For each lap after the first, the time it takes her to complete is five seconds more than that of the previous lap. Given that she spends 10 minutes on the first six laps, how long does she spend on the next six laps, in minutes? [b]p16.[/b] Chad decides to go to trade school to ascertain his potential in carpentry. Chad is assigned to cut away all the vertices of a wooden regular tetrahedron with sides measuring four inches. Each vertex is cut away by a plane which passes through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches? Note: A tetrahedron is a solid with four triangular faces. In a regular tetrahedron, these faces are all equilateral triangles. [b]p17.[/b] Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero? [b]p18.[/b] For art class, Jordan needs to cut a circle out of the coordinate grid. She would like to find a circle passing through at least $16$ lattice points so that her cut is accurate. What is the smallest possible radius of her circle? Note: A lattice point is defined as one whose coordinates are both integers. For example, $(5, 8)$ is a lattice point whereas $(3.5, 5)$ is not. [b]p19.[/b] Chad's ant Arctica is on one of the eight corners of Chad's toolbox, which measures two decimeters in width, three decimeters in length, and four decimeters in height. One day, Arctica wanted to go to the opposite corner of this box. Assuming she can only crawl on the surface of the toolbox, what is the shortest distance she has to crawl to accomplish this task, in decimeters? (You may assume that the toolbox is oating in the Exeter Space Station, so that Arctica can crawl on all six faces.) [b]p20.[/b] Jordan is counting numbers for fun. She starts with the number $1$, and then counts onward, skipping any number that is a divisor of the product of all previous numbers she has said. For example, she starts by counting $1$, $2$, $3$, $4$, $5$, but skips 6, a divisor of $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$. What is the $20^{th}$ number she counts? [b]p21.[/b] Chad and Jordan are having a race in the lake shown below. The lake has a diameter of four kilometers and there is a circular island in the middle of the lake with a diameter of two kilometers. They start at one point on the edge of the lake and finish at the diametrically opposite point. Jordan makes the trip only by swimming in the water, while Chad swims to the island, runs across it, and then continues swimming. They both take the fastest possible route and, amazingly, they tie! Chad swims at two kilometers an hour and runs at five kilometers an hour. At what speed does Jordan swim? [img]https://cdn.artofproblemsolving.com/attachments/f/6/22b3b0bba97d25ab7aabc67d30821d0b12efc0.png[/img] [b]p22.[/b] Cameron has stolen Chad's barrel of oil and is driving it around on a truck on the coordinate grid on his truck. Cameron is a bad truck driver, so he can only move the truck forward one kilometer at a $4$ $EMC^2$ $2014$ Problems time along one of the gridlines. In fact, Cameron is so bad at driving the truck that between every two one-kilometer movements, he has to turn exactly $90$ degrees. After $50$ one-kilometer movements, given that Cameron's first one-kilometer movement was westward, how many points he could be on? [b]p23.[/b] Let $a$, $b$, and $c$ be distinct nonzero base ten digits. Assume there exist integers $x$ and $y$ for which $\overline{abc} \cdot \overline{cb} = 100x^2 + 1$ and $\overline{acb} \cdot \overline{bc} = 100y^2 + 1$. What is the minimum value of the number $\overline{abbc}$? Note: The notation $\overline{pqr}$ designates the number whose hundreds digit is $p$, tens digit is $q$, and units digit is $r$, not the product $p \cdot q \cdot r$. [b]p24.[/b] Let $r_1, r_2, r_3, r_4$ and $r_5$ be the five roots of the equation $x^5-4x^4+3x^2-2x+1 = 0$. What is the product of $(r_1 +r_2 +r_3 +r_4)$, $(r_1 +r_2 +r_3 +r_5)$, $(r_1 +r_2 +r_4 +r_5)$, $(r_1 +r_3 +r_4 +r_5)$, and $(r_2 +r_3 +r_4 +r_5)$? [b]p25.[/b] Chad needs seven apples to make an apple strudel for Jordan. He is currently at 0 on the metric number line. Every minute, he randomly moves one meter in either the positive or the negative direction with equal probability. Arctica's parents are located at $+4$ and $-4$ on the number line. They will bite Chad for kidnapping Arctica if he walks onto those numbers. Also, there is one apple located at each integer between $-3$ and $3$, inclusive. Whenever Chad lands on an integer with an unpicked apple, he picks it. What is the probability that Chad picks all the apples without getting bitten by Arctica's parents? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]

Let $AB\varGamma\varDelta$ be a rhombus. (a) Prove that you can construct a circle $(c)$ which is inscribed in the rhombus and is tangent to its sides. (b) The points $\varTheta,H,K,I$ are on the sides $\varDelta\varGamma,B\varGamma,AB,A\varDelta$ of the rhombus respectively, such that the line segments $KH$ and $I\varTheta$ are tangent on the circle $(c)$. Prove that the quadrilateral defined from the points $\varTheta,H,K,I$ is a trapezium.

Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$. a) Prove that $DE$ is parallel to $BC$. b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.

Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?

Let $\vartriangle ABC$ be a triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\omega$. $AA'$, $BB'$ and $CC'$ are altitudes of $\vartriangle ABC$ with $A'$ in $BC$, $B'$ in $AC$ and $C'$ in $AB$. $P$ is a point on the segment $AA'$. The perpenicular line to $B'C'$ from $P$ intersects $BC$ at $K$. $AA'$ intersects $\omega$ at $M \ne A$. The lines $MK$ and $AO$ intersect at $Q$. Prove that $\angle CBQ = \angle PBA$.

Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.