Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: [b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$; [b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ [b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

Evaluate $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \min (n, k) \left( \frac12 \right)^n\left( \frac13 \right)^k$$

Alice and Bob are playing the following game: They have an $8\times8$ chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black. Or he does nothing. After that, Alice may decide to continue the game or not. The goal of Alice is to maximize the number of connected components of black grids, on the other hand, Bob wants to minimize that number. If both of them are extremely smart, how many connected components will be in the end of the game?

Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

Given $a,b,c,d>0$, prove that: \[\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),\] where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.

At a certain orphanage, every pair of orphans are either friends or enemies. For every three of an orphan's friends, an even number of pairs of them are enemies. Prove that it's possible to assign each orphan two parents such that every pair of friends shares exactly one parent, but no pair of enemies does, and no three parents are in a love triangle (where each pair of them has a child).

As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called [i]banners[/i]. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 [i]banners[/i].

For every positive integer $n$ we take the greatest divisor $d$ of $n$ such that $d\leq \sqrt{n}$ and we define $a_n=\frac{n}{d}-d$. Prove that in the sequence $a_1,a_2,a_3,...$, any non negative integer $k$ its in the sequence infinitely many times.

Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.) [asy] defaultpen(linewidth(0.8)); pair O=origin; fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray); fill(O--Arc(O, 1, 20, 160)--cycle, white); fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray); fill(O--Arc(O, 1, 200, 340)--cycle, white); fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 2, 160, 200)--cycle, white); fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray); fill(O--Arc(O, 2, -20, 20)--cycle, white); fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray); draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3)); draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy] $ \textbf{(A)} \frac{\pi}8\qquad \textbf{(B)}\frac{\pi}7\qquad \textbf{(C)}\frac{\pi}6\qquad \textbf{(D)}\frac{\pi}5\qquad \textbf{(E)}\frac{\pi}4 $

Consider $ABC$ with $AC > AB$ and incenter $I$. The midpoints of $\overline{BC}$ and $\overline{AC}$ are $M$ and $N$, respectively. If $\overline{AI}$ is perpendicular to $\overline{IN}$, then prove that $\overline{AI}$ is tangent to the circumscribed circle of $\vartriangle BMI$.

Show that there are infinitely many positive real numbers a which are not integers such that a(a-3{a}) is an integer.

Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix. Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.

a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$ |a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $ a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$ |a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$

For each of the following, provide proof or a counterexample: a) Every $2\times2$ matrix with real entries can we written as the sum of the squares of two $2\times2$ matrices with real entries. b) Every $3\times3$ matrix with real entries can we written as the sum of the squares of two $3\times3$ matrices with real entries.

Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{max}$ and $P_{min}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{max}$ and $P_{min}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$. [i]Proposed by [b]FedeX333X [/b][/i]

A natural number is written on the blackboard. In each step, we erase the units digit and add four times the erased digit to the remaining number, and write the result on the blackboard instead of the initial number. Starting with the number $13^{2006}$, is it possible to obtain the number $2006^{13}$ by repeating this step finitely many times?

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]

Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$). We define function $f:S\rightarrow S$ as follow: $\forall z\in S$, $ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$ $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$. Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.

For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1$ prove that $ a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32,$ and determine the cases of equality.