Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Find all triplets of positive integers $a$, $b$ and $c$ such that $a \geq b \geq c$ and $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2$

Let $n$ be an integer, $n \ge 2$. For each number $k = 1, 2, ....., n,$ denote by $a _ k$ the number of multiples of $k$ in the set $\{1, 2,. .., n \}$ and let $x _ k = \frac {1} {1} + \frac {1} {2} + \frac {1} {3} _... + \frac {1} {a _ k}$ . Show that: $$\frac {x _ 1 + x _ 2 + ... + x _ n} {n} \le \frac {1} {1 ^ 2} + \frac {1} {2 ^ 2} + ... + \frac {1} {n ^ 2} $$.

Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$. Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false? (A)$ab^2$ is irrational (B)$a^2b$ is rational (C)$\sqrt{ab}$ is rational (D)$a+b$ is irrational

Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.

In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \angle{EAB}$ and $ \angle{ABC}$ are right angles, $ AB \equal{} 4$, $ BC \equal{} 6$, $ AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle{ADE}$ and $ \triangle{BDC}$? [asy] defaultpen(linewidth(0.8)+fontsize(10));size(200); unitsize(5mm) ; pair A=(0,0), B=(4,0), C=(4,6), D=(0,8), H=intersectionpoint(C--A, D--B); draw(A--B--C--cycle) ; draw(A--B--D--cycle) ; label("E",(0,8), N) ; label("8",(0,4),W) ; label("A",A,S) ; label("B",B,SE) ; label("C",C,NE) ; label("6",(4,3),E) ; label("4",(2,0),S) ; label("D",H,2*dir(85)) ;[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l} 1 \le {u_1} \le 3\\ {u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}. \end{array} \right.$$ Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Let $ABC$ be an acute scalene triangle with incenter $I$, circumcircle $w_1$, and denote the circumcircle of $BIC$ as $w_2$. Suppose point $P$ lies on $w_2$ and is inside $w_1$. Let $X,Y$ lie on $BC$ with $XP \perp BP, YP \perp PC$. Circles $O_1, O_2$ are drawn tangent to $w_1$ at points on the same side of $BC$ as $A$ and tangent to $BC$ at $X,Y$ respectively. Let the centers of those two circles be $Z_1, Z_2$. Let $D$ be the point on $w_2$ opposite to $P$ and let $E$ be the foot of the altitude from $P$ to $BC$. Show that $DE \perp Z_1Z_2$

Determine the least integer $k$ for which the following story could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.

Let ABC be a triangle whose vertices are inside a circle $\Omega$. Prove that we can choose two of the vertices of ABC such that there are infinitely many circles $\omega$ that satisfy the following properties: 1. $\omega$ is inside of $\Omega$, 2. $\omega$ passes through the two chosen vertices, and 3. the third vertex is in the interior of $\omega$ .