Perimeter of a convex quadrilateral is $2004$ and one of its diagonals is $1001$. Can another diagonal be $1$ ? $2$ ? $1001$ ?

Show that there exists a degree $58$ monic polynomial $$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$ such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.

Simplify: $ \sqrt [3]{\frac {17\sqrt7 \plus{} 45}{4}}$

In an exhibition where $2015$ paintings are shown, every participant picks a pair of paintings and writes it on the board. Then, Fake Artist (F.A.) chooses some of the pairs on the board, and marks one of the paintings in all of these pairs as "better". And then, Artist's Assistant (A.A.) comes and in his every move, he can mark $A$ better then $C$ in the pair $(A,C)$ on the board if for a painting $B$, $A$ is marked as better than $B$ and $B$ is marked as better than $C$ on the board. Find the minimum possible value of $k$ such that, for any pairs of paintings on the board, F.A can compare $k$ pairs of paintings making it possible for A.A to compare all of the remaining pairs of paintings. [b]P.S:[/b] A.A can decide $A_1>A_n$ if there is a sequence $ A_1 > A_2 > A_3 > \dots > A_{n-1} > A_n$ where $X>Y$ means painting $X$ is better than painting $Y$.

Let $O$ be the circumcenter of triangle $ABC$. Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$, respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$, and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$. [i]Proposed by Medeubek Kungozhin, Kazakhstan[/i]

Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$ Determine all integers $m$ such that every term of the sequence is a square.

What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$? A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$

Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.

A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both?

A $3\times2\times2$ right rectangular prism has one of its edges with length $3$ replaced with an edge of length $5$ parallel to the original edge. The other $11$ edges remain the same length, and the $6$ vertices that are not endpoints of the replaced edge remain in place. The resulting convex solid has $8$ faces, as shown below. Find the volume of the solid. [i]Proposed by Justin Hsieh[/i]

Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? $\textbf{(A)}\ \frac{2}{7} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{4}{7} \qquad \textbf{(E)}\ \frac{5}{8}$

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.

Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.