Prove that there exist $16$ subsets of set $M = \{1,2,...,10000\}$ with the following property: For every $z \in M$ there are eight of these subsets whose intersection is $\{z\}$.

Given real numbers $a_i,b_i~(i=1,2,\ldots,n)$ such that \begin{align*} &a_1\ge a_2\ge\ldots\ge a_n>0,\\ &b_1\ge a_1,\\ &b_1b_2\ge a_1a_2,\\ &\vdots\\ &b_1b_2\cdots b_n\ge a_1a_2\cdots a_n, \end{align*}prove that $b_1+b_2+\ldots+b_n\ge a_1+a_2+\ldots+a_n$.

[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label("D",(0,1),NW);label("E",(.4,1),N);label("C",(1,1),NE); label("P",(0,.6),W);label("M",(.25,.55),E);label("Q",(1,.2),E); label("A",(0,0),SW);label("B",(1,0),SE); //Credit to Zimbalono for the diagram[/asy] Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is $\textbf{(A) }5:12\qquad\textbf{(B) }5:13\qquad\textbf{(C) }5:19\qquad\textbf{(D) }1:4\qquad \textbf{(E) }5:21$

Team $ A$ and team $ B$ play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team $ B$ wins the second game and team $ A$ wins the series, what is the probability that team $ B$ wins the first game? $ \textbf{(A)}\ \frac{1}{5}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{1}{2}\qquad \textbf{(E)}\ \frac{2}{3}$

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.

In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.

Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values. (Paolo Leonetti)

For real numbers $a, b$ and $c$ we have \[(2b-a)^2 + (2b-c)^2 = 2(2b^2-ac).\] Prove that the numbers $a, b$ and $c$ are three consecutive terms in some arithmetic sequence.

Does there exist a solution to the equation \[x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65\] in integers with $x, y, z, u, v$ greater than $1998$?

Consider the quadratic equation $x^2-2mx-1=0$, where $m$ is an arbitrary real number. For what values ​​of $m$ does the equation have two real solutions, such that the sum of their cubes is equal to eight times their sum.

Let $AL$ be the bisector of a triangle $ABC$, $X$ be an arbitrary point on the external bisector of angle $A$, the lines $BX$, $CX$ meet the perpendicular bisector to $AL$ at points $P, Q$ respectively. Prove that $A$, $X$, $P$, $Q$ are concyclic. Proposed by: Y.Shcherbatov

For $\alpha \in (0,1)$ we consider the equation $\{x\{x\}\}= \alpha$. a) Prove that the equation has rational solutions if and only if there exist $m,p,q\in\mathbb{Z}$, $0<p<q$, $\gcd(p,q)=1$, such that $\alpha = \left( \frac pq\right)^2 + \frac mq$. b) Find a solution for $\alpha = \frac {2004}{2005^2}$.

Compute $3^{3^{\ldots^3}} \mod{333},$ where there are $3^{3^3}$ $3$'s in the exponent.

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free. [i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]

Calculate with at most $10\%$ relative error \[\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx\]

Let $A$ be a $n\times n$ matrix such that $A_{ij} = i+j$. Find the rank of $A$. [hide="Remark"]Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.[/hide]

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.

Let $ABCD $ be cyclic quadrilateral with circumcircle $\omega $ and $M $ be any point on $\omega $.\\ Let $E $ and $F $ be the intersection of $AB,CD $ and $AD,BC $ respectively.\\ $ME $ intersects lines $AD,BC $ at $P,Q $ and similarly $MF$ intersects lines $AB,CD $ at $R,S $.\\ Let the lines $PS $ and $RQ $ meet at $X $.\\ Prove that as $M $ varies over $\omega $\\ $MX $ passes through fixed point.\\ [i]Proposed by Mehdi Etesami Fard [/i]

Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$

If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.

Prove that for any reals $a> b> c$, the inequality $a^2(b - c) + b^2(c - a) + c^2(a - b)> 0$.

Evaluate $2000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3$

For a natural number $n$, let $D (n)$ be the set of (positive integers) divisors of $n$. Furthermore let $d (n)$ be the number of divisors of $n,$ that is, the cardinality of $D (n)$. For each such $n$, prove the equality $$\sum_{k\in D(n)} d(k)^3=\left( \sum_{k\in D(n)} d(k)\right) ^2.$$