There are $100$ stones in a pile. A partition of the heap in $k $ piles is called [i]special [/i] if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the $k + 1$ piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number $k$, such that there is a special partition of the $100$ stones into $k $ piles. b) Find the minimum number $k $, such that there is a special partition of the $100$ stones in $k $ piles.

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$ [i]Proposed by Ali Zamani [/i]

We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations: [b]1. [/b]Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between. [b]2. [/b]Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right. [b]3.[/b] Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells. The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player. [img]https://i.imgur.com/IjcIDOa.png[/img] [i]Proposed by Luka Tsulaia, Georgia[/i]

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ($1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?

For each $k$ , find the least $n$ in terms of $k$ st the following holds: There exists $n$ real numbers $a_1 , a_2 ,\cdot \cdot \cdot , a_n$ st for each $i$ : $$0 < a_{i+1} - a_{i} < a_i - a_{i-1}$$ And , there exists $k$ pairs $(i,j)$ st $a_i - a_j = 1$.

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

What is the smallest integer $n$ for which any subset of $\{1,2,3,\ldots,20\}$ of size $n$ must contain two numbers that differ by $8$? $\textbf{(A) }2\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15$

Find the smallest positive $\alpha$ (in degrees) for which all the numbers \[\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots\] are negative.

A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.

a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$ b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

Consider a table with one real number in each cell. In one step, one may switch the sign of the numbers in one row or one column simultaneously. Prove that one can obtain a table with non-negative sums in each row and each column.

The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

Given that $2x + 7y = 3$, find $2^{6x + 21y - 4}$. [i]Proposed by Deyuan Li[/i]

Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.