Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that \[A^2B-2ABA+BA^2=0.\] Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.

Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and $$n^q-q$$ is an integer.

Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.

The following game is played with a group of $n$ people and $n+1$ hats are numbered from $1$ to $n+1.$ The people are blindfolded and each of them puts one of the $n+1$ hats on his head (the remaining hat is hidden). Now, a line is formed with the $n$ people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now, starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players). What is the highest number of guaranteed correct guesses, if the $n$ people can discuss a common strategy? [i]Proposed by Viktor Kiss, Budapest[/i]

The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$. The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$, respectively, prove that $A, E$ and $F$ are collinear.

Consider the set $I=\{ 1,2, \cdots, 2020 \}$. Let $W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I$, $Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I$ be its two subsets. Further, let $X= W \cap Y$. [b](1)[/b] Find the sum of maximal and minimal elements in $X$. [b](2)[/b] An element $n=y(a,b) (a \le b)$ in $Y$ is called [i]excellent[/i], if its representation is not unique (for instance, $20=y(1,5)=y(2,3)$). Find the number of [i]excellent[/i] elements in $Y$. [hide=Note][b](2)[/b] is only for Grade 11.[/hide]

A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\] Show that $k>0$ exists with $x_k=1$.

Let $ ABC$ be an equilateral triangle. Let $ D,E, F,M,N,$ and $ P$ be the mid-points of $ BC, CA, AB, FD, FB,$ and $ DC$ respectively. [b](a)[/b] Show that the line segments $ AM,EN,$ and $ FP$ are concurrent. [b](b)[/b] Let $ O$ be the point of intersection of $ AM,EN,$ and $ FP.$ Find $ OM : OF : ON : OE : OP : OA.$

Prove that for all primes $p$ true is equality $$\sum_{k=1}^{p-1}\left\lfloor\frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}{4}$$

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

Solve the equation $x^4 + 4x^3 - 8x + 4 = 0$.

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$  

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

How many figures can be obtained by intersecting the infinite-dimensional cube $|x_k| \le 1$, $k = 1,2,\ldots$ with a two-dimensional plane?

For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.

Determine all $x$, $y$, $k$ and $n$ positive integers such that: $10^x$ + $10^y$ + $n!$ = $2024^k$

On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point. [i](5 points)[/i]

Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and \[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\] If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.

In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?

Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.

$f(x)$ is a decreasing function defined on $(0,+\infty)$, if $f(2a^2+a+1)<f(3a^2-4a+1)$, then the range value of $a$ is________.

Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shylock any two coins of different colors in exchange for one coin of the other color; for example, he may give Shylock one red coin and one blue coin, and receive one yellow coin in return. Bassanio wishes to end with coins that are all the same color, and he wishes to do this while having as many coins as possible. How many coins will he end up with, and what color will they be?

Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.