The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.

An equilateral triangle has inside it a point with distances 5,12,13 from the vertices . Find its side.

In triangle $ABC$ medians of triangle $BE$ and $AD$ are perpendicular to each other. Find the length of $\overline{AB}$, if $\overline{BC}=6$ and $\overline{AC}=8$

Let \(\mathbb{R}\) denote the set of real numbers. A subset \(S\subseteq\mathbb{R}\) is called [i]dense[/i] if any non-empty open interval of \(\mathbb{R}\) contains at least one element in \(S\). For a function \(f:\mathbb{R}\to\mathbb{R}\), let \(\mathcal{O}_f(x)\) denote the set \(\left\{x,f(x),f(f(x)),\ldots\right\}\). (a) Is there a function \(g:\mathbb{R}\to\mathbb{R}\), continuous everywhere in \(\mathbb{R}\) such that \(\mathcal{O}_g(x)\) is dense for all \(x\in\mathbb{R}\) for all \(x\in\mathbb{R}\)? (b) Is there a function \(h:\mathbb{R}\to\mathbb{R}\), continuous at all but a single \(x_0\in\mathbb{R}\), such that \(\mathcal{O}_h(x)\) is dense for all \(x\in\mathbb{R}\)? [i]Proposed by the ICMC Problem Committee[/i]

The product $ 8\times .25\times 2\times .125 = $ $ \text{(A)}\ \frac{1}8\qquad\text{(B)}\ \frac{1}4\qquad\text{(C)}\ \frac{1}2\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2 $

Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?

Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$, participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$, player $i$ has a higher skill level than player $j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$. During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$, the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$. Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$th round. [i]Proposed by Ray Li[/i]

In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$? $ \begin{tabular}{cccc} &a&b&c \\ &d&e&f \\ + & g&h&i \\ \hline 1&6&6&5 \end{tabular}$ [i]Proposed by Eugene Chen[/i]

In how many ways can a $3\times 3$ grid be filled with integers from $1$ to $12$ such that all three of the following conditions are satisfied: (a) both $1$ and $2$ appear in the grid, (b) the grid contains at most $8$ distinct values, and (c) the sums of the numbers in each row, each column, and both main diagonals are all the same? Rotations and reflections are considered the same.

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions. (a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $. (b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

Let $S$ be the set containing all positive integers whose decimal representations contain only 3’s and 7’s, have at most 1998 digits, and have at least one digit appear exactly 999 times. If $N$ denotes the number of elements in $S$, find the remainder when $N$ is divided by 1000.

There are 2023 dice on the table. For 1 dollar, one can pick any dice and put it back on any of its four (other than top or bottom) side faces. How many dollars at a minimum will guarantee that all the dice have been repositioned to show equal number of dots on top faces? [i]Egor Bakaev[/i]

Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If \[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\] then \[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.

The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

The numbers from 1 to $n$ are arranged in some way on a circle. What’s the smallest value of $n$, for which no matter how the numbers are arranged there exist ten consecutively increasing numbers $A_1<A_2<A_3…<A_{10}$ such that $A_1 A_2…A_{10}$ is a convex decagon?

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Let $ABC$ be a scalene acute triangle whose orthocenter is $H$. $M$ is the midpoint of segment $BC$. $N$ is the point where the segment $AM$ intersects the circle determined by $B, C$, and $H$. Show that lines $HN$ and $AM$ are perpendicular.

What is the measure of the acute angle formed by the hands of the clock at $4:20$ PM? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.

Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]