Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?

In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called [i]ascending [/i] if $a_i < a_j$ and [i]descending[/i] if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k$ non-overlapping ascending pairs or at least $k$ non-overlapping descending pairs.

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

In acute triangles $ABC$, $AD, BE ,CF$ are altitudes, with $D, E, F$ on the sides $BC, CA, AB$, respectively. Prove that $$DE + DF \le BC$$

Find all positive integers $k$ such that the product of the first $k$ primes increased by $1$ is a power of an integer (with an exponent greater than $1$).

$n\ge 2$ positive integers are written on the blackboard. A move consists of three steps: 1) choose an arbitrary number $a$ on the blackboard, 2) calculate the least common multiple $N$ of all numbers written on the blackboard, and 3) replace $a$ by $N/a$. Prove that using such moves it is always possible to make all the numbers on the blackboard equal to $1$. [i](A. Naradzetski)[/i]

Determine all pairs of $(m, n)$ such that is possible to tile the table $ m \times n$ with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except $m \times n$) regularly covered with figures.

For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is \[ (-1)^{\alpha(i-1, j-1)}. \] Compute the remainder when $\det M$ is divided by $1000$. [i] Proposed by Evan Chen [/i]

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

Finite number of $2017$ units long sticks are fixed on a plate. Each stick has a bead that can slide up and down on it. Beads can only stand on integer heights $( 1, 2, 3,..., 2017 )$. Some of the bead pairs are connected with elastic bands. $The$ $young$ $ant$ can go to every bead, starting from any bead by using the elastic bands. $The$ $old$ $ant$ can use an elastic band if the difference in height of the beads which are connected by the band, is smaller than or equal to $1$. If the heights of the beads which are connected to each other are different, we call it $valid$ $situation$. If there exists at least one $valid$ $situation$, prove that we can create a $valid$ $situation$, by arranging the heights of the beads, in which $the$ $old$ $ant$ can go to every bead, starting from any bead.

The sequence $1,2,3,4,0,9,6,9,4,8,7,\ldots$ is formed so that each term, starting from the fifth, is the units digit of the sum of the previous four. (a) Do the digits $2,0,0,4$ occur in the sequence in this order? (b) Will the initial digits $1,2,3,4$ ever occur again in this order?

The vertices of 100-gon (i.e., polygon with 100 sides) are colored alternately white or black. One of the vertices contains a checker. Two players in turn do two things: move the checker into other vertice along the side of 100-gon and then erase some side. The game ends when it is impossible to move the checker. At the end of the game if the checker is in the white vertice then the first player wins. Otherwise the second player wins. Does any of the players have winning strategy? If yes, then who? [i]Remark.[/i] The answer may depend on initial position of the checker.

Let $ABCD$ be a parallelogram, $E$ a point on the line $AB$, beyond $B, F$ a point on the line $AD$, beyond $D$, and $K$ the point of intersection of the lines $ED$ and $BF$. Prove that quadrilaterals $ABKD$ and $CEKF$ have the same area.

On the sides $AB$ and $AC$ of the triangle $ABC$, the points $P$ and $Q$ are chosen, respectively, so that $PQ\parallel BC$. Segments $BQ$ and $CP$ intersect at point $O$. Point $A'$ is symmetric to point $A$ relative to line $BC$. The segment $A'O$ intersects the circumcircle $w$ of the triangle $APQ$ at the point $S$. Prove that circumcircle of $BSC$ is tangent to the circle $w$.

Let $S$ be the set of all 3-tuples $(a,b,c)$ that satisfy $a+b+c=3000$ and $a,b,c>0$. If one of these 3-tuples is chosen at random, what's the probability that $a,b$ or $c$ is greater than or equal to 2,500?

At a math contest there are $2n$ students participating. Each of them submits a problem to the jury, which thereafter gives each students one of the $2n$ problems submitted. One says that the contest is [i]fair[/i] is there are $n$ participants which receive their problems from the other $n$ participants. Prove that the number of distributions of the problems in order to obtain a fair contest is a perfect square.

How many ways can we pick four $3$-element subsets of $\{1, 2, ..., 6\}$ so that each pair of subsets share exactly one element?

The $X{}$ pentomino consists of five $1\times1$ squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an $8\times 8$ chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such $X{}$ pentominoes.) [i]Alexandr Gribalko[/i]

Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties (a) $\displaystyle f(A) \neq 0$, for $\displaystyle A \neq \phi$; (b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$, for all $\displaystyle A,B \in \mathcal P (M)$, where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.

Let $P_1$, $P_2$, $P_3$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $P_i$. In other words, find the maximum number of points that can lie on two or more of the parabolas $P_1$, $P_2$, $P_3$ .

For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)? $\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 98 \qquad\textbf{(D)}\ 99 \qquad\textbf{(E)}\ 100$

Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

Let $n$ be a positive integer. Each cell on an $n \times n$ board will be filled with a positive integer less than or equal to $2n-1$ such that for each index $i$ with $1 \leq i \leq n$, the $2n-1$ cells in the $i^{\text{th}}$ row or $i^{\text{th}}$ collumn contain distinct integers. (a) Is this filling possible for $n=4$? (b) Is this filling possible for $n=5$?

Liam writes the number $0$ on a board, then performs a series of turns. Each turn, he chooses a nonzero integer so that for every nonzero integer $N,$ he chooses $N$ with $3^{- |N|}$ probability. He adds his chosen integer $N$ to the last number written on the board, yielding a new number. He writes the new number on the board and uses it for the next turn. Liam repeats the process until either $8$ or $9$ is written on the board, at which point he stops. Given that Liam eventually stopped, find the probability the last number he wrote on the board was $9.$