Compute the sum of all positive integers $n$ for which there exists a real number $x$ satisfying $$\left(x +\frac{n}{x} \right)^n= 2^{20}.$$

Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles.

Draw on the plane a set of points whose coordinates $(x,y)$ satisfy the equation $x^3 + y^3 = x^2y^2 + xy$.

The integers from 1 to 49 are written in a $7\times 7$ table, such that for any $k\in\{1,2,\ldots,7\}$ the product of the numbers in the $k$-th row equals the product of the numbers in the $(8-k)$-th row. [list=a] [*]Prove that there exists a row such that the sum of the numbers written on it is a prime number. [*]Give an example of such a table. [/list] [i]Cristi Săvescu[/i]

Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$ and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely many numbers divisible by $ 2009$.

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied: $ (1)$ $ n$ is not a perfect square; $ (2)$ $ a^{3}$ divides $ n^{2}$.

5-Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB \equal{} CD$.

Let be given a triangle $ ABC$. Points $ P$ on side $ AC$ and $ Y$ on the production of $ CB$ beyond $ B$ are chosen so that $ Y$ subtends equal angles with $ AP$ and $ PC$. Similarly, $ Q$ on side $ BC$ and $ X$ on the production of $ AC$ beyond $ C$ are such that $ X$ subtends equal angles with $ BQ$ and $ QC$. Lines $ YP$ and $ XB$ meet at $ R$, $ XQ$ and $ YA$ meet at $ S$, and $ XB$ and $ YA$ meet at $ D$. Prove that $ PQRS$ is a parallelogram if and only if $ ACBD$ is a cyclic quadrilateral.

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is [asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy] $\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

You are given a piece of paper. You can cut the paper into $8$ or $12$ pieces. Then you can do so for any of the new pieces or let them uncut and so on. Can you get exactly $60$ pieces¿ Show that you can get every number of pieces greater than $60$.

What is the largest value for $m$ for which I can find nonnegative integers $a_1, a_2, \ldots, a_m < 2024$ such that for all indices $i>j,$ $17$ divides $\tbinom{a_i}{a_j}$?

It is now between $10:00$ and $11:00$ o'clock, and six minutes form now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? $\textbf{(A) }10:05\dfrac{5}{11}\qquad\textbf{(B) }10:07\dfrac{1}{2}\qquad\textbf{(C) }10:10\qquad\textbf{(D) }10:15\qquad$ $\textbf{(E) }10:17\dfrac{1}{2}$

Two cars travel around a track at equal and constant speeds, each completing a lap every hour. From a common starting point, the first starts at time $t=0$ and the second at an arbitrary later time $t=T>0.$ Prove that there is a total period of exactly one hour during the motion in which the first has completed twice as many laps as the second.

Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?

For a pair $ A \equal{} (x_1, y_1)$ and $ B \equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \equal{} |x_1 \minus{} x_2| \plus{} |y_1 \minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane.

Let $O$ be the circumcenter of $\triangle ABC.$ The circle $k$ passing through $A$ and $B$ cuts $AC$ and $BC$ at $P$ and $Q,$ respectively. Prove that $PQ$ and $OC$ are perpendicular.

The times between $ 7$ and $ 8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $ 84$ degrees are: $ \textbf{(A)}\ \text{7: 23 and 7: 53} \qquad \textbf{(B)}\ \text{7: 20 and 7: 50} \qquad \textbf{(C)}\ \text{7: 22 and 7: 53} \\ \textbf{(D)}\ \text{7: 23 and 7: 52} \qquad \textbf{(E)}\ \text{7: 21 and 7: 49}$

Cookies should be placed on a $7 \times 7$ chess board, so that never four cookies can form a rectangle whose sides are parallel to those of the chessboard. Find the maximum number of biscuits that can be positioned in this way. Note: If you do the same job for a $13 \times 13$ chessboard, you get a biscuit. If you solve it for an infinite number of squares of chessboards, you get two biscuits. If you solve them for all sidelengths, , you even get two three biscuits (We cannot distribute more cookies, otherwise we run the risk of them to form a rectangle).

Let $a, b$ be the roots of the quadratic polynomial $Q(x) = x^2 + x + 1$, and let $u, v$ be the roots of the quadratic polynomial $R(x) = 2x^2 + 7x + 1$. Suppose $P$ is a cubic polynomial which satises the equations $$\begin{cases} P(au) = Q(u)R(a) \\ P(bu) = Q(u)R(b) \\ P(av) = Q(v)R(a) \\ P(bv) = Q(v)R(b) \end{cases}$$ If $M$ and$ N$ are the coeffcients of $x^2$ and $x$ respectively in $P(x)$, what is the value of $M+ N$?

In convex quadrilateral $ABCD$, only $D$ is an obtuse angle. Use some line segments to divide it into $n$ obtuse triangles. But on its sides (except $A,B,C,D$ ), there is no vertex of triangles we divided into. Prove that if and only if $n\geq4$, we can divide the convex quadrilateral into such $n$ triangles.

$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that $$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$

Let $A,B$ be $n\times n$ matrices with complex entries. Show that there exists a matrix $T$ and an invertible matrix $S$ such that \[ B=S(A+T)S^{-1}\ -T \iff \operatorname{tr}(A) = \operatorname{tr}(B) \]

In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$. [i]Proposed by Andrew Wen[/i]