In each line and column of a table $ (2n \plus{} 1)\times (2n \plus{} 1)$ are written arbitrarly the numbers $ 1,2,...,2n \plus{} 1$. It was constated that the repartition of the numbers is symmetric to the main diagonal of this table. Prove that all the numbers on the main diagonal are distinct.

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room? [img]https://cdn.artofproblemsolving.com/attachments/a/9/5548d7b34551369d1b69eae682855bcc406f9e.jpg[/img]

Let $M$ be a set with $n$ elements. How many pairs $(A, B)$ of subsets of $M$ are there such that $A$ is a subset of $B?$

A positive integer $b$ is called good if there exist positive integers $1=a_1, a_2, \ldots, a_{2023}=b$ such that $|a_{i+1}-a_i|=2^i$. Find the number of the good integers.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Determine all values of $n$ satisfied the following condition: there's exist a cyclic $(a_1,a_2,a_3,...,a_n)$ of $(1,2,3,...,n)$ such that $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$ is a complete residue systems modulo $n$.

A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. [asy] size(150); defaultpen(linewidth(0.65)+fontsize(11)); real r=10; pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; path P=circle(O,r); C=intersectionpoint(B--(B.x+r,B.y),P); draw(Arc(O, r, 45, 360-17.0312)); draw(A--B--C);dot(A); dot(B); dot(C); label("$A$",A,NE); label("$B$",B,SW); label("$C$",C,SE); [/asy]

Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

Consider a $m\times n$ rectangular grid with $m$ rows and $n$ columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that a) $m=4$; b) $m=3$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(25,32)$, is it possible to obtain the pair $(82,98)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(34,60)$, is it possible to obtain the pair $(2000, 2008)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(145,220)$, is it possible to obtain the pair $(363,498)$ after a series of transformations?

[b]a)[/b] Prove that $ 4040100 $ divides $ 2009\cdot 2011^{2011}+1. $ [i]Gabriel Iorgulescu[/i] [b]b)[/b] Let be three natural numbers $ x,y,z $ with the property that $ (1+\sqrt 2)^x=y^2+2z^2+2yz\sqrt 2. $ Show that $ x $ is even. [i]Marius Cavachi[/i]

Find the sum of all positive integers $n$ such that $\frac{2020}{n^3 + n}$ is an integer.

In how many ways can $10$ distinct books be placed onto $3$-shelf bookcase in such a way that no shelf is empty? $ \textbf{(A)}\ 36\cdot 10! \qquad\textbf{(B)}\ 50 \cdot 10! \qquad\textbf{(C)}\ 55 \cdot 10! \qquad\textbf{(D)}\ 81 \cdot 10! \qquad\textbf{(E)}\ \text{None of the above} $

Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.

We say three sets $A_1, A_2, A_3$ form a triangle if for each $1 \leq i,j \leq 3$ we have $A_i \cap A_j \neq \emptyset$, and $A_1 \cap A_2 \cap A_3 = \emptyset$. Let $f(n)$ be the smallest positive integer such that any subset of $\{1,2,3,\ldots, n\}$ of the size $f(n)$ has at least one triangle. Find a formula for $f(n)$.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

Tao plays the following game:given a constant $v>1$;for any positive integer $m$,the time between the $m^{th}$ round and the $(m+1)^{th}$ round of the game is $2^{-m}$ seconds;Tao chooses a circular safe area whose radius is $2^{-m+1}$ (with the border,and the choosing time won't be calculated) on the plane in the $m^{th}$ round;the chosen circular safe area in each round will keep its center fixed,and its radius will decrease at the speed $v$ in the rest of the time(if the radius decreases to $0$,erase the circular safe area);if it's possible to choose a circular safe area inside the union of the rest safe areas sometime before the $100^{th}$ round(including the $100^{th}$ round),then Tao wins the game.If Tao has a winning strategy,find the minimum value of $\biggl\lfloor\frac{1}{v-1}\biggr\rfloor$.

Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.

Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic. [i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

Find the number of distinguishable $2\times 2\times 2$ cubes that can be formed by gluing together two blue, two green, two red, and two yellow $1\times 1\times 1$ cubes. Two cubes are indistinguishable if one can be rotated so that the two cubes have identical coloring patterns.