Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.

Prove that we can give a color to each of the numbers $1,2,3,...,2013$ with seven distinct colors (all colors are necessarily used) such that for any distinct numbers $a,b,c$ of the same color, then $2014\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a,b,c$.

What is the $288$th term of the sequence $a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f,...?$

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

A lightbulb is given that emits red, green or blue light and an infinite set $S$ of switches, each with three positions labeled red, green and blue. We know the following: [list=1] [*]For every combination of the switches the lighbulb emits a given color. [*]If all switches are in a position with a given color, the lightbulb emits the same color. [*]If there are two combinations of the switches where each switch is in a different position, the lightbulb emits a different color for the two combinations. [/list] We create the following set $U$ containing some of the subsets of $S$: for each combination of the switches let us observe the color of the lightbulb, and put the set of those switches in $U$ which are in the same position as the color of the lightbulb. Prove that $U$ is an ultrafilter on $S$. In other words, prove that $U$ satisfies the following conditions: [list=1] [*]The empty set is not in $U.$ [*]If two sets are in $U,$ their intersection is also in $U.$ [*]If a set is in $U,$ every subset of $S$ containing it is also in $U.$ [*]Considering a set and its complement in $S,$ exactly one of these sets is contained in $U.$ [/list]

What is the sum of real roots of the equation \[ \left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}? \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ -\dfrac {3}{2} \qquad\textbf{(D)}\ -\dfrac {5}{4} \qquad\textbf{(E)}\ -\dfrac {7}{12} $

Let $\triangle ABC$ be acute. The feet of altitudes from the corners $A, B, C$ are $ D, E, F$, respectively. If $|DF|=3, |FE|=4,$ and $|DE|=5$, then what is the radius of the circle with center $C$ and tangent to $DE$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$

Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.

Given: point $ A $, line $ p $, and circle $ k $. Construct a triangle $ ABC $ with angles $ A = 60^\circ $, $ B = 90^\circ $, whose vertex $ B $ lies on line $ p $, and vertex $ C $ - on circle $ k $.

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.

Define $ n_a!$ for $ n$ and $ a$ positive to be \[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\] where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to $ \textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

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]