Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Determine the sum of the positive integers $n$ such that there exist primes $p,q,r$ satisfying $p^{n} + q^{2} = r^{2}$.

Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?

Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.

What is the smallest number of $5\times 1$ tiles which must be placed on a $31\times 5$ rectangle (each covering exactly $5$ unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a $52\times 5$ rectangle?

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

Each of the $9$ cells in a $3\times 3$ grid is colored either blue or white with equal probability. The expected value of the area of the largest square of blue cells contained within the grid is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.

A finite set $S$ consists of primes, and $3$ is not in $S$. Prove that there exists a positive integer $M$ such that for every $p \in S$ one can shuffle the digits of $M$ to get a number divisible by $p$ and not divisible by all other numbers in $S$. (Note: the first digit of a positive integer can not be zero). [i]A. Voidelevich[/i]

Find all $n\in \mathbb Z^+$, so that \[ z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} } \] is prime.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Two congruent squares are packed into an isoceles right triangle as shown below. Each of the squares has area 10. What is the area of the triangle? [asy] draw((0,0) -- (3*sqrt(10), 0) -- (0, 3*sqrt(10)) -- cycle); draw((0,0) -- (2*sqrt(10), 0) -- (2*sqrt(10), sqrt(10)) -- (0, sqrt(10))); draw((sqrt(10), sqrt(10)) -- (sqrt(10), 0)); [/asy] $\mathrm{(A) \ } 40 \qquad \mathrm{(B) \ } 90 \qquad \mathrm {(C) \ } \frac{85}{2} \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 45$

Let $ABCD$ be a parallelogram. The line $\Delta$ meets the lines $AB, BC, CD$ and $DA$ at $M, N, P$ and $Q,$ respectively. Let $R$ be the intersection point of the lines $AB,DN$ and let $S$ be intersection point of the lines $AD, BP.$ Prove that $RS \parallel \Delta.$ [asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); pen qqzzcc = rgb(0,0.6,0.8); pen wwwwff = rgb(0.4,0.4,1); draw((2,2)--(6,2),qqzzcc+linewidth(1.6pt)); draw((6,2)--(4,0),qqzzcc+linewidth(1.6pt)); draw((-1.95,(+12-2*-1.95)/2)--(12.24,(+12-2*12.24)/2),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0+3*-1.95)/3)--(12.24,(-0+3*12.24)/3),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0-0*-1.95)/6)--(12.24,(-0-0*12.24)/6),qqzzcc+linewidth(1.6pt)); draw((4,0)--(4,4),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((2,2)--(8.14,0),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((-1.95,(+32.56-4*-1.95)/4.14)--(12.24,(+32.56-4*12.24)/4.14),qqzzcc+linewidth(1.6pt)); dot((0,0),ds); label("$A$", (0,-0.3),NE*lsf); dot((4,0),ds); label("$B$", (4.02,-0.33),NE*lsf); dot((2,2),ds); label("$D$", (1.81,2.07),NE*lsf); dot((6,2),ds); label("$C$", (6.16,2.08),NE*lsf); dot((3,3),ds); label("$Q$", (2.97,3.22),NE*lsf); dot((5,1),ds); label("$N$", (4.99,1.19),NE*lsf); label("$\Delta$", (1.7,3.76),NE*lsf); dot((6,0),ds); label("$M$", (5.9,-0.33),NE*lsf); dot((4,2),ds); label("$P$", (4.02,2.08),NE*lsf); dot((4,4),ds); label("$S$", (3.94,4.12),NE*lsf); dot((8.14,0),ds); label("$E$", (8.2,0.09),NE*lsf); clip((-1.95,-6.96)--(-1.95,4.99)--(12.24,4.99)--(12.24,-6.96)--cycle); [/asy]

Suppose that all subspaces of cardinality at most $ \aleph_1$ of a topological space are second-countable. Prove that the whole space is second-countable. [i]A. Hajnal, I. Juhasz[/i]

Given a positive integer $n$. An inversion of a permutation is the amount of pairs $(i,j)$ such that $i<j$ and the $i$-th number is smaller than $j$-th number in the permutation. Prove that for every positive integer $k \leq n$ there exist exactly $\frac{n!}{k}$ permutations in which the inversion is divisible by $k$.

Let $n\geq1$ be an integer. We have two sequences, each of $n$ positive real numbers $a_1,a_2,\ldots ,a_n$ and $b_1,b_2,\ldots ,b_n$ such that $a_1+a_2+\ldots +a_n=1$ and $ b_1+b_2+\ldots +b_n=1$. Find the smallest possible value that the sum can take $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.$$

Let $ a,b,c$ positive reals such that $ ab \plus{} bc \plus{} ca \equal{} 3$, show that: $ \displaystyle a^2 \plus{} b^2 \plus{} c^2 \plus{} 3 \ge \frac {a(3 \plus{} bc)^2}{(c \plus{} b)(b^2 \plus{} 3)} \plus{} \frac {b(3 \plus{} ca)^2}{(a \plus{} c)(c^2 \plus{} 3)} \plus{} \frac {c(3 \plus{} ab)^2}{(b \plus{} a)(a^2 \plus{} 3)}$ ([i]Anass BenTaleb, Ali Ben Bari High School - Taza,Morocco[/i])

Let $O$ be a point inside a triangle $ABC$ and let the lines $AO,BO$, and $CO$ meet sides $BC,CA$, and $AB$ at points $A_1,B_1$, and $C_1$, respectively. If $AA_1$ is the longest among the segments $AA_1,BB_1,CC_1$, prove that $$OA_1+OB_1+OC_1\le AA_1.$$

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Prove that for every $\varepsilon >0$ there exists a positive integer $n$ and there are positive numbers $a_1, \ldots, a_n$ such that for every $\varepsilon < x < 2\pi - \varepsilon$ we have $$\sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right|$$.

The non-negative real numbers $x, y, z$ satisfy the equality $x + y + z = 1$. Determine the highest possible value of the expression $E (x, y, z) = (x + 2y + 3z) (6x +3y + 2z)$.

Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle.

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

A pile of cards, numbered with $1$, $2$, ..., $n$, is being shuffled. Afterwards, the following operation is repeatedly performed: If the uppermost card of the pile has the number $k$, then we reverse the order of the $k$ uppermost cards. Prove that, after finitely many executions of this operation, the card with the number $1$ will become the uppermost card of the pile.