Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.

In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub)

Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ 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 $AE$ intersects the circle $\omega$ through $AO_1O_2$ at $F$. Prove that the length of segment $EF$ is equal to the diameter of $\omega$.

Let $ABCD$ be a cyclic quadrilateral with $3AB = 2AD$ and $BC = CD$. The diagonals $AC$ and $BD$ intersect at point $X$. Let $E$ be a point on $AD$ such that $DE = AB$ and $Y$ be the point of intersection of lines $AC$ and $BE$. If the area of triangle $ABY$ is $5$, then what is the area of quadrilateral $DEY X$?

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? $\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$. (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.

Let $ABC$ be a triangle inscribed in the circle $(O)$. Tangents at $B,C$ of the circles $(O)$ meet at $T$ . Let $M,N$ be the points on the rays $BT,CT$ respectively such that $BM = BC = CN$. The line through $M$ and $N$ intersects $CA,AB$ at $E, F$ respectively; $BE$ meets $CT$ at $P, CF$ intersects $BT$ at $Q$. Prove that $AP = AQ$. Trần Quang Hùng

The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$

Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.

There are \(n\) numbers arranged in a circle, and each number equals the absolute value of the difference between its two neighbors. Is it necessarily true that all numbers are equal to zero if: a) \(n=2025\); b) \(n=2024\)? [i]Proposed by Anton Trygub[/i]

Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number

Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$. Answer the following questions. (1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$, prove that $GQ>1$. (2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$. (ii) Express the volume $V(n)$ of $K_n$ in terms of $n$. (3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$. (4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$

Evaluate $\sum_{n=1}^\infty \frac{n}{(2^n-2^{-n})^2}+\frac{(-1)^nn}{(2^n-2^{-n})^2}$

The incircle of triangle $ABC$ touches its sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. The point $B_2$ is symmetric to $B_1$ with respect to line $A_1C_1$, lines $BB_2$ and $AC$ meet in point $B_3$. points $A_3$ and $C_3$ may be defined analogously. Prove that points $A_3, B_3$ and $C_3$ lie on a line, which passes through the circumcentre of a triangle $ABC$. [b] proposed by L. Emelyanov[/b]

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$.

How many pairs of positive integers $(a,b)$ are there such that the roots of polynomial $x^2-ax-b$ are not greater than $5$? $ \textbf{(A)}\ 40 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 65 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ \text{None of the preceding} $

A policeman is trying to catch a robber on a board of $2001\times2001$ squares. They play alternately, and the player whose trun it is moves to a space in one of the following directions: $\downarrow,\rightarrow,\nwarrow$. If the policeman is on the square in the bottom-right corner, he can go directly to the square in the upper-left corner (the robber can not do this). Initially the policeman is in the central square, and the robber is in the upper-left adjacent square. Show that: $a)$ The robber may move at least $10000$ times before the being captured. $b)$ The policeman has an strategy such that he will eventually catch the robber. Note: The policeman can catch the robber if he reaches the square where the robber is, but not if the robber enters the square occupied by the policeman.

let $f(x) = \sum_{n=0}^{\infty} 2^{-n} ||2^n x||$ , where ||x|| is the distance between x and the closest integer to x. Are the level sets $\{ x \in [0,1] : f(x)=y \}$ Lebesgue measurable for almost all $y \in f(R)$?

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

Let $A_1$, $A_2$, $\ldots\,$, $A_n$ be finite sets. Prove that \[ \Bigl| \bigcup_{1 \le i \le n} A_i \Bigr| \ge \frac{1}{2} \sum_{1 \le i \le n} \left| A_i \right| - \frac{1}{6} \sum_{1 \le i < j \le n} \left| A_i \cap A_j \right| \, . \] Recall that if $S$ is a finite set, then its cardinality $|S|$ is the number of elements of $S$.

On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?

At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$. Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S=\sum_{1\le i<j\le 5}x_ix_j$. $ \textbf{(A)}$ What value of $x_1, x_2, x_3, x_4, x_5$ maximizes $S$? $ \textbf{(A)}$ Find, with proof, the value of $x_1, x_2, x_3, x_4, x_5$ which minimizes of $S$ if $|x_i-x_j|\le 2$ for any $1\le i$, $j\le 5$.