Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.

Let $n>1$ be an integer and $x_1,x_2,\ldots,x_n$ be $n{}$ arbitrary real numbers. Determine the minimum value of \[\sum_{i<j}|\cos(x_i-x_j)|.\]

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

For any $n$ consecutive integers $a_1, \cdots, a_n$, prove that $$(a_1+\cdots+a_n)\cdot\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\leqslant \frac{n(n+1)\ln(\text{e}n)}{2}.$$

Jar $A$ and Jar $B$ each contain $10$ beans. The number of beans in jar $A$ is doubled, and the number of beans in jar $B$ is halved. How many beans are now in jars $A$ and $B$? $\textbf{(A) }15\qquad\textbf{(B) }20\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40$

Find the shortest distance between the graphs of $y=x^2+5$ and $x=y^2+5$. [i]Proposed by Nathan Ramesh

Let \( ABC \) be a scalene triangle. Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \), respectively, and let \( D \) be any point on segment \( BC \). The circumcircles of triangles \( BDF \) and \( CDE \) intersect line \( EF \) at points \( K \neq F \), and \( L \neq E \), respectively, and intersect at points \( X \neq D \). The point \( Y \) is on line \( DX \) such that \( AY \) is parallel to \( BC \). Prove that points \( K \), \( L \), \( X \), and \( Y \) lie on the same circle.

Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.

The average (mean) of $ 20$ numbers is $ 30$, and the average of $ 30$ other numbers is $ 20$. What is the average of all $ 50$ numbers? $ \textbf{(A)}\ 23 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 27$

Consider a countinuous function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that verifies the following conditions: $ \text{(1)} x f(f(x))=(f(x))^2,\quad\forall x\in\mathbb{R}_{>0} $ $ \text{(2)} \lim_{\stackrel{x\to 0}{x>0}} \frac{f(x)}{x}\in\mathbb{R}\cup\{ \pm\infty \} $ [b]a)[/b] Show that $ f $ is bijective. [b]b)[/b] Prove that the sequences $ \left( (\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}} ) (x) \right)_{n\ge 1} ,\left( (\underbrace{f^{-1}\circ f^{-1}\circ\cdots \circ f^{-1}}_{\text{n times}} ) (x) \right)_{n\ge 1} $ are both arithmetic progressions, for any fixed $ x\in\mathbb{R}_{>0} . $ [b]c)[/b] Determine the function $ f. $ [i]Nelu Chichirim[/i]

Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks? [i]Radu Gologan, Dan Schwarz[/i]

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.

Let $O = (0,0), A = (0,a), and B = (0,b)$, where $0<b<a$ are reals. Let $\Gamma$ be a circle with diameter $\overline{AB}$ and let $P$ be any other point on $\Gamma$. Line $PA$ meets the x-axis again at $Q$. Prove that angle $\angle BQP = \angle BOP$.

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square? $ \textbf{(A)}\ 1-\frac{\sqrt2}2+\frac\pi4\qquad\textbf{(B)}\ \frac12+\frac\pi4\qquad\textbf{(C)}\ 2-\sqrt2+\frac\pi4\qquad\textbf{(D)}\ \frac{\sqrt2}2+\frac\pi4\qquad\textbf{(E)}\ 1+\frac{\sqrt2}4+\frac\pi8 $

A walk consists of a sequence of steps of length 1 taken in the directions north, south, east, or west. A walk is self-avoiding if it never passes through the same point twice. Let $f(n)$ be the number of $n$-step self-avoiding walks which begin at the origin. Compute $f(1)$, $f(2)$, $f(3)$, $f(4)$, and show that \[2^n < f(n) \le 4 \cdot 3^{n - 1}.\]

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.

In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is: ${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $

The student locker numbers at Olympic High are numbered consecutively beginning with locker number $ 1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $ 9$ and four centers to label locker number $ 10$. If it costs $ \$137.94$ to label all the lockers, how many lockers are there at the school? $ \textbf{(A)}\ 2001 \qquad \textbf{(B)}\ 2010 \qquad \textbf{(C)}\ 2100 \qquad \textbf{(D)}\ 2726 \qquad \textbf{(E)}\ 6897$

There are $N$ pupils in a school class, and there are several communities among them. Sociability of a pupil will mean the number of pupils in the largest community to which the pupil belongs (if the pupil belongs to none then the sociability equals $1$). It occurred that all girls in the class have different sociabilities. What is the maximum possible number of girls in the class?