In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?

Find the smallest three-digit divisor of the number \[1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\]

Let $0\leq a,b,c,d\leq 10$. For how many ordered quadruples $(a,b,c,d)$ is $ad-bc$ a multiple of $11?$

The midpoint of each side of a convex pentagon is connected by a segment with the centroid of the triangle formed by the remaining three vertices of the pentagon. Prove that these five segments have a common point.

What is the sum of all $k\leq25$ such that one can completely cover a $k\times k$ square with $T$ tetrominos (shown in the diagram below) without any overlap? [asy] size(2cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((1,2)--(2,2)); draw((0,0)--(0,1)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,1)); [/asy] $\text{(A) }20\qquad\text{(B) }24\qquad\text{(C) }84\qquad\text{(D) }108\qquad\text{(E) }154$

There is an $n \times n$ grid which has rows and columns numbered from $1$ to $n$; the cell at row $i$ and column $j$ is denoted as the cell at $(i, j)$. A subset $A$ of the cells is called [i]good[/i] if for any two cells at $(x_1, y), (x_2, y)$ in $A$, the cells $(u, v)$ satisfying $x_1 < u \leq x_2, v<y$ or $x_1 \leq u < x_2, v>y$ are not in $A$. Determine the minimal number of good sets such that they are pairwise disjoint and every cell of the board belongs to exactly one good set.

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

Let $1,7,19,\ldots$ be the sequence of numbers such that for all integers $n\ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence. [i]Proposed by Nathan Xiong[/i]

Solve for real $x$ : \[ (x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3. \]

Let $a$ and $b$ be perfect squares whose product exceeds their sum by $4844$. Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\]

Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\] Prove that $f$ and $g$ have a common non-constant factor.

Suppose that the $101$ positive integers \[2024, 2025, 2026, \ldots , 2124 \]are concatenated in some order to form a $404$-digit number. Can this number be prime?

Suppose $ H$ and $ O$ are the orthocenter and the circumcenter of acute triangle $ ABC$; $ AA_1$, $ BB_1$ and $ CC_1$ are the altitudes of the triangle. Point $ C_2$ is the reflection of $ C$ in $ A_1B_1$. Prove that $ H$, $ O$, $ C_1$ and $ C_2$ are concyclic.

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?

Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.

Call a positive integer $m$ $\textit{good}$ if there exist integers $a, b, c$ satisfying $m=a^3+2b^3+4c^3-6abc$. Show that there exists a positive integer $n<2024$, such that for infinitely many primes $p$, the number $np$ is $\textit{good}$.

For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that $$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$

Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$.

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

Five numbers from a list of nine integers are $7,8,3,5,$ and $9$. The largest possible value of the median of all nine numbers in this list is $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$

Consider all 1000-element subsets of the set $\{1,2,3,\dots,2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? $\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64 $

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets $d$ different integers. For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben. [i]a)[/i] Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ [i]b)[/i] Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ [i]Proposed by Besfort Shala, Kosovo[/i]