Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying \[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

Isosceles triangle $ABC$ has angle $\angle BAC = 135^o$ and $AB = 2$. What is its area?

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that $$1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace 241 \le a_{15}\le 250.$$ What is the sum of digits of $a_{14}$? $\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.

Zack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than $\frac35$ . The expected number of cuts that Zack makes can be written as $\frac{p}{q}$ for $p$ and $q$ relatively prime positive integers. Find $p + q$.

There is a point set on a plane, and seven circles $C_1,C_2,\cdots,C_7$, where $C_7$ passes exactly 7 points in $M$, $C_6$ passes exactly 6 points in $M$, ..., $C_1$ passes exactly 1 point in $M$. Then how many points do set $M$ have at least? $\text{(A)}11\qquad\text{(B)}12\qquad\text{(C)}21\qquad\text{(D)}28$

Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into $100$ pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.

In the $200\times 200$ table in some cells lays red or blue chip. Every chip "see" other chip, if they lay in same row or column. Every chip "see" exactly $5$ chips of other color. Find maximum number of chips in the table.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.

In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)); pair A = (0,0); pair C = (2,0); pair B = dir(57.5)*2; pair E = waypoint(C--A,0.25); pair D = waypoint(C--B,0.25); pair T = intersectionpoint(D--A,E--B); label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N); draw(A--B--C--cycle); draw(A--D); draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$

There are 20 towns on the bay of a circular island. Each town has 20 teams for a mathematical duel. No two of these teams are of equal strength. When two teams meet in a duel, the stronger one wins. For a given number $n\in \mathbb{N}$ one town $A$ can be called [i]“n-stronger”[/i] than $B$, if there exist $n$ different duels between a team from $A$ and team from $B$, for which the team from $A$ wins. Find the maximum value of $n$, for which it is possible for each town to be [i]n-stronger[/i] by its neighboring one clockwise.

Four mathematicians, four physicists, and four programmers gather in a classroom. The $12$ people organize themselves into four teams, with each team having one mathematician, one physicist, and one programmer. How many possible arrangements of teams can exist?