Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. At what angle $\theta_g$ during the swing is the tension in the rod the greatest? $\textbf{(A) } \text{The tension is greatest at } \theta_g = \theta_0.\\ \textbf{(B) } \text{The tension is greatest at }\theta_g = 0.\\ \textbf{(C) } \text{The tension is greatest at an angle } 0 < \theta_g < \theta_0.\\ \textbf{(D) } \text{The tension is constant.}\\ \textbf{(E) } \text{None of the above is true for all values of } \theta_0 \text{ with } 0 < \theta_{0} < \frac{\pi}{2}$

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

The triangle $ABC$ is given. Let $A'$ be the midpoint of the side $BC$, $B_c{}$ be the projection of $B{}$ onto the bisector of the angle $ACB{}$ and $C_b$ be the projection of the point $C{}$ onto the bisector of the angle $ABC$. Let $A_0$ be the center of the circle passing through $A', B_c, C_b$. The points $B_0$ and $C_0$ are defined similarly. Prove that the incenter of the triangle $ABC$ coincides with the orthocenter of the triangle $A_0B_0C_0$.

Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a,b,c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt {2p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^n+b^n+c^n$.

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

The inscribed circle $\omega$ of an equilateral $\Delta ABC$ is tangent to its sides $AB$,$BC$ and $CA$ in points $D$,$E$, and $F$, respectively. Point $H$ is the foot of the altitude from $D$ to $EF$. Let $AH\cap BC=X,BH\cap CA=Y$. It is known that $XY\cap AB=T$. Let $D$ be the center of the circumscribed circle of $\Delta BYX$. Prove that $OH\perp CT$.

Let $a$, $b$, and $c$ be real angles such that \newline \[3\sin a + 4\sin b + 5\sin c = 0\] \[3\cos a + 4\cos b + 5\cos c = 0.\] \newline The maximum value of the expression $\frac{\sin b \sin c}{\sin^2 a}$ can be expressed as $\frac{p}{q}$ for relatively prime $p,q$. Compute $p+q$.

Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.

A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c\plus{}s$? [asy]unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label("$M$",(-1,0),W); label("$C$",(-0.1,-0.3)); label("$A$",(-0.4,0.7)); label("$B$",(0.7,0.4));[/asy]$ \textbf{(A)}\ \frac{24}{5} \qquad \textbf{(B)}\ \frac{32}{5} \qquad \textbf{(C)}\ 8\plus{}\sqrt5 \qquad \textbf{(D)}\ 5\plus{}\frac{16\sqrt5}{5} \qquad \textbf{(E)}\ 10\plus{}5\sqrt5$

We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$. We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!

Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.

Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$? $\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

Problems 7, 8 and 9 are about these kites. To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid. [asy] for (int a = 0; a < 7; ++a) { for (int b = 0; b < 8; ++b) { dot((a,b)); } } draw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy] What is the number of square inches in the area of the small kite? $ \text{(A)}\ 21\qquad\text{(B)}\ 22\qquad\text{(C)}\ 23\qquad\text{(D)}\ 24\qquad\text{(E)}\ 25 $

Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.

A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is $141$. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.

Prove that there is a unique $1000$-digit number $N$ in base $2022$ with the following properties: [list=1] [*] All of the digits of $N$ (in base $2022$) are $1$’s or $2$’s, and [/*] [*] $N$ is a multiple of the base-$10$ number $2^{1000}$. [/*] [/list] (Note that you must prove both that such a number exists and that there is not more than one such number. You do not have to write down the number! In fact, please don’t!)

Let be the polynomial $ f=X^4+X^2\in\mathbb{Z}_2[X] $ Find: a) its degree.. b) the splitting field of $ f $ c) the Galois group of $ f $ (Galois group of its splitting field)

A copsychus and a sparrow, each initially located at one of the vertex of a regular polygon with $103$ edges, fly clockwise to another vertex each. The copsychus moves across $\ell$ edges each time while the sparrow moves through$ d$ edges of the polygon, where $\ell \ne d$ are both integers less than $103$. Assume that, during their journeys, the copsychus has stopped at $m$ vertices while sparrow has stopped at $n$ vertices of the polygon, for $m \ge n \ge 3$. Determine the value of $m, n$ given that there is only one common single vertex of the polygon that both of birds have stopped at, and there is only one vertex that neither of the birds have reached. Vu Thi Khoi, Topo University, Hanoi Mathematics Institute, Vietnam, Hoang Qu6c Vietnam, Hanoi.

Prove that if an integer $n$ is of the form $4m+3$, where $m$ is another integer, then $n$ is not a sum of two perfect squares (a perfect square is an integer which is the square of some integer).

On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$. Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?

Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.