Find the number of subsets of $\{ 1,2,3,4,5,6,7,8 \}$ that are subsets of neither $\{1,2,3,4,5\}$ nor $\{4,5,6,7,8\}$.

Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.

Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other? $\textbf{(A) }A \text{ east}, B \text{ west} \qquad \textbf{(B) } A\text{ north}, B\text{ south} \qquad \textbf{(C) } A\text{ north}, B\text{ west} \qquad \textbf{(D) } A\text{ up}, B\text{ south} \qquad \textbf{(E) } A\text{ up}, B\text{ west}$

Carson the farmer has a plot of land full of crops in the shape of a $6 \times 6$ grid of squares. Each day, he uniformly at random chooses a row or a column of the plot that he hasn’t chosen before and harvests all of the remaining crops in the row or column. Compute the expected number of connected components that the remaining crops form after $6$ days. If all crops have been harvested, we say there are $0$ connected components.

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap): a) $20$ pieces of dumplings; b) $24$ pieces of dumplings; . (The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.) [i]Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly. [/i]

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. Determine all polynomials $Q(x) \in \mathbb{Z}[x]$, such that for every positive integer $n$, there exists a polynomial $R_n(x) \in \mathbb{Z}[x]$ satisfies $$Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).$$

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour? (Mebius, Sharygin)

Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that \[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\] Prove that $f(x) = f^{-1}(x).$

How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Denote the foot of perpendicular from $A,B,C$ to opposing sides as $D,E,F$. Let $P$ a foot of perpendicular from $F$ to $DE$, and $Q(\neq F)$ a intersection point of line $FP$ and circumcircle of $BDF$. Prove that $\angle PBQ=\angle PAD$.

Let us call a set of positive integers [i]nice[/i], if its number of elements is equal to the average of all its elements. Call a number $n$ [i]amazing[/i], if one can partition the set $\{1,2,\ldots,n\}$ into nice subsets. a) Prove that any perfect square is amazing. b) Prove that there exist infinitely many positive integers which are not amazing.

Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?

There are $ 3n, n \in \mathbb{Z}^\plus{}$ girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $ 3n$ students had just one time to be on duty on the same day. (1) When $ n\equal{}3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion. (2) Prove that $ n$ is an odd number.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time.

Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

Find all $4$-tuples $(a,b,c,d)$ of natural numbers with $a \le b \le c$ and $a!+b!+c!=3^d$

Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities: \[\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\ \dfrac{1}{6-z}+2\le x \\ \end{array}\right.\]

Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

Find all pairs $(m, n)$ of positive integers such that $m^2 + n^2 =(m + 1)(n + 1).$