An arithmetic sequence of five terms is considered $good$ if it contains 19 and 20. For example, $18.5,19.0,19.5,20.0,20.5$ is a $good$ sequence. For every $good$ sequence, the sum of its terms is totalled. What is the total sum of all $good$ sequences?

In a sport competition, $m$ teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team $x$ such that for each team $y,$ either $x$ wins $y$ or there exists a team $z$ for which $x$ wins $z$ and $z$ wins $y.$ [i][i.e. prove that in every tournament there exists a king.][/i]

Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.

Determine all pairs of polynomials $f,g\in\mathbb{C}[x]$ with complex coefficients such that the following equalities hold for all $x\in\mathbb{C}$: $f(f(x))-g(g(x))=1+i$ $f(g(x))-g(f(x))=1-i$

Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$. [i]Proposed by A. Antropov[/i]

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$ [asy] size(160); defaultpen(linewidth(1.1)); path square = (1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle; filldraw(square,white); filldraw(rotate(30)*square,white); filldraw(rotate(60)*square,white); dot((0,0),linewidth(7)); [/asy] $\textbf{(A)}\: 75\qquad\textbf{(B)} \: 93\qquad\textbf{(C)} \: 96\qquad\textbf{(D)} \: 129\qquad\textbf{(E)} \: 147$

Nico picks $13$ pairwise distinct $3-$digit positive integers. Ian then selects several of these 13 numbers, the ones he wants, and using only once each selected number and some of the operations addition, subtraction, multiplication and division ($+,-,\times ,:$) must get an expression whose value is greater than $3$ and less than $4$. If he succeeds, Ian wins; otherwise, Nico wins. Which of the two has a winning strategy?

1- Find a pair $(m,n)$ of positive integers such that $K = |2^m-3^n|$ in all of this cases : $a) K=5$ $b) K=11$ $c) K=19$ 2-Is there a pair $(m,n)$ of positive integers such that : $$|2^m-3^n| = 2017$$ 3-Every prime number less than $41$ can be represented in the form $|2^m-3^n|$ by taking an Appropriate pair $(m,n)$ of positive integers. Prove that the number $41$ cannot be represented in the form $|2^m-3^n|$ where $m$ and $n$ are positive integers 4-Note that $2^5+3^2=41$ . The number $53$ is the least prime number that cannot be represented as a sum or an difference of a power of $2$ and a power of $3$ . Prove that the number $53$ cannot be represented in any of the forms $2^m-3^n$ , $3^n-2^m$ , $2^m-3^n$ where $m$ and $n$ are positive integers

For each positive integer $k$, let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$. Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*} f(d(n+1)) &= d(f(n)+1)\quad \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*} for all positive integers $n$.

Polygon $A_0A_1...A_{n-1}$ satisfies the following: $\bullet$ $A_0A_1 \le A_1A_2 \le ...\le A_{n-1}A_0$ and $\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 = ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles). Prove that this polygon is regular.

Let $AB$ be the diameter of a circle $\Gamma$ and let $C$ be a point on $\Gamma$ different from $A$ and $B$. Let $D$ be the foot of perpendicular from $C$ on to $AB$.Let $K$ be a point on the segment $CD$ such that $AC$ is equal to the semi perimeter of $ADK$.Show that the excircle of $ADK$ opposite $A$ is tangent to $\Gamma$.

$S$ is a subset of $\{1,2, . . . ,100\}$. What is the maximum number of elements in $S$ such that the product of any two of them is not a perfect square?

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]

For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.

a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.

Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions. (1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.) (2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$? (3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$. Point $M$ is the midpoint of $PE$. Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$. [i]Matsvei Zorka[/i]

A finite set $S$ of positive integers is given. Show that there is a positive integer $N$ dependent only on $S$, such that any $x_1, \dots, x_m \in S$ whose sum is a multiple of $N$, can be partitioned into groups each of whose sum is exactly $N$. (The numbers $x_1, \dots, x_m$ need not be distinct.)

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.

Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy: $2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$ Where in both sides $2$ appeared $1397$ times

Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$. [i]Proposed by Andrew Wu, Deyuan Li, and Andrew Milas[/i]

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Let $x$,$y$ be positive real numbers such that $\frac{1}{1+x+x^2}+\frac{1}{1+y+y^2}+\frac{1}{1+x+y}=1$.Prove that $xy=1.$