Show that any odd number can be written as the difference between two perfect squares.

Let's call a checkered polygon a [i]strip[/i], which can be traversed entirely, starting from some of its cells and then moving only in two directions - up or to the right. Several such strips can be inserted into each other by shifting by a vector $(-1.1)$. Prove that for any strip consisting of an even number of cells, there is such an odd $k$ that if you combine $k$ of the same strips by inserting them sequentially into each other, then the resulting polygon can be divided along the grid lines into two equal parts. [i]Proposed by I. Markelov, S. Markelov[/i]

Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$. [i]Andrei Bâra[/i]

Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

Let $(G, *)$ a group of $n > 1$ elements, and let $g \in G$ be an element distinct from the identity. Ana and Bob play with the group $G$ on the following way: Starting with Ana and playing alternately, each player selects an element of $G$ that has not been selected before, until each element of $G$ have been selected or a player have selected the elements $a$ and $a * g$ for some $a \in G$. In that case it is said that the player loses and his opponent wins. $a)$ If $n$ is odd, show that, independent of element $g$, one of the two players has a winning strategy and determines which player possesses such a strategy. $b)$ If $n$ is even, show that there exists an element $g \in G$ for which none of the players has a winning strategy. Note: A group $(G, *)$ es a set $G$ together with a binary operation $* : G\times G \to G$ that satisfy the following properties $(i)$ $*$ is asociative: $\forall a, b, c \in G (a * b) * c = a * (b * c)$; $(ii)$ there exists an identity element $e \in G$ such that $\forall a \in G, a *e = e * a = a;$ $(iii)$ there exists inverse elements: $\forall a \in G \exists a^{-1} \in G$ such that $a*a^{-1} = a^{-1} *a = e.$

Let $x,y,z$ be positive numbers such that $x^2+y^2+z^2=1$. Prove that $$\frac x{1-x^2}+\frac y{1-y^2}+\frac z{1-z^2}\ge\frac{3\sqrt3}2$$

Find the complex numbers $ z_1,z_2,z_3 $ of same absolute value having the property that: $$ 1=z_1z_2z_3=z_1+z_2+z_3. $$

Oriol has a finite collection of cards, each one with a positive integer written on it. We say the collection is $n$-[i]complete[/i] if for any integer $k$ from $1$ to $n$ (inclusive), he can choose some cards such that the sum of the numbers on them is exactly $k$. Suppose that Oriol's collection is $n$-complete, but it stops being $n$-complete if any card is removed from it. What is the maximum possible sum of the numbers on all the cards? [i]Proposed by Ariel García[/i]

It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that $i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements $ii)$ Upper bound can be reached for $n=3$

A triangle $ABC$ is considered, and let $D$ be the intersection point of the angle bisector corresponding to angle $A$ with side $BC$. Prove that the circumcircle that passes through $A$ and is tangent to line $BC$ at $D$, it is also tangent to the circle circumscribed around triangle $ABC$.

Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

Altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$ meet at $H$. $A_0$, $B_0$, $C_0$ are the midpoints of $BC$, $CA$, $AB$ respectively. Points $A_2$, $B_2$, $C_2$ on the segments $AH$, $BH$, $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$. Prove that the lines $AC_2$, $BA_2$, $CB_2$ are concurrent.

A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.

Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine? $ \textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16} $

Let $S$ be the set of all subsets of $\left\{2,3,\ldots,2016\right\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, we say that $b^{-1}$ is the unique positive integer less than $2017$ for which $2017$ divides $bb^{-1} -1$. [i]Proposed by Tristan Shin[/i]

Is there positive integer $n$, such that $n+2$ divides $S=1^{2019}+2^{2019}+...+n^{2019}$

Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10: \[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]

Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$

For each integer $i=0,1,2, \dots$, there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$? $ \textbf{a)}\ 8 \qquad\textbf{b)}\ 10 \qquad\textbf{c)}\ 12 \qquad\textbf{d)}\ 15 \qquad\textbf{e)}\ 16 $

Let $a,b,c$ be positive real numbers that satisfy the condition $a + b + c = 1$. Prove the inequality $$\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123$$

Find all solutions of $\frac{a}{x}=\frac{x-a}{a}$ for $x$.

For each integer $ n\geq 4$, let $ a_n$ denote the base-$ n$ number $ 0.\overline{133}_n$. The product $ a_4a_5 \dotsm a_{99}$ can be expressed as $ \frac {m}{n!}$, where $ m$ and $ n$ are positive integers and $ n$ is as small as possible. What is the value of $ m$? $ \textbf{(A)}\ 98 \qquad \textbf{(B)}\ 101 \qquad \textbf{(C)}\ 132\qquad \textbf{(D)}\ 798\qquad \textbf{(E)}\ 962$