The volume of a certain rectangular solid is $ 8 \text{ cm}^3$, its total surface area is $ 32 \text{ cm}^3$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is $ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 44$

Determine all nonzero natural numbers $n$, for which the number $\sqrt{n! + 5}$ is a natural number.

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once. We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant. Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant. Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant. Prove that, there are exactly $40$ winning triangles in this tournament.

For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$

Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\ $\textbf{(a)}$ Prove that $G$ is cyclic. \\ $\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?

Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns – let $S$ be the largest of these sums. What is the minimal value of $S$?

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

A railway line is divided into ten sections by stations $E_1, E_2,..., E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $ 12$ km, and a trip through three consecutive stations is at least $17$ Km. Calculate the distance between $E_2$ and $E_7$.

Let $\Pi$ be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped $P_0$ can be decomposed into parallelepipeds $P_1,P_2, . . . ,P_N\in \Pi$, prove that $P_0\in \Pi$.

A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number. a) What is the minimum possible number of odd squares? b) What is the smallest value that the sum of the perimeters of the odd squares can take?

Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.

Let $P$ and $Q$ be points taken inside of triangle $ABC$ such that $\angle APB=\angle AQC$ and $\angle APC=\angle AQB$. Circumcircle of $APQ$ intersects $AB$ and $AC$ second time at $K$ and $L$ respectively. Prove that $B,C,L,K$ are concyclic.

Prove that $\frac 12 \cdot \sqrt{4\sin^2 36^{\circ} - 1}=\cos 72^\circ$.

Let $ ABC $ be a triangle and let $ AD, BE, CF $ be cevians of the triangle which are concurrent at $ G $. Prove that if $ CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE $ then $ AG \le GD $.

Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).

A minus sign is placed on one square of a $5 \times 5$ board and plus signs are placed on the remaining squares. A move is to select a $2 \times 2, 3 \times 3, 4 \times 4$ or $5 \times 5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.

Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

Find all triples of positive integers $a, b, c$ such that they are pairwise relatively prime and $a^2+3b^2c^2=7^c$.

A projectile in flight is observed simultaneously from four radio stations which are situated at the corners of a square of side $b$. The distances of the projectile from the four stations, taken in order around the square, are found to be $R_1 , R_2 , R_3 $ and $R_4$. Show that $$R_{1}^{2}+ R_{3}^{2}= R_{2}^{2}+ R_{4}^{2}.$$ Show also that the height $h$ of the projectile above the ground is given by $$h^{2}=- \frac{1}{2} b^2 +\frac{1}{4}(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}) -\frac{1}{8 b^{2}}(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}- 2 R_{1}^{2}R_{3}^{2} -2 R_{2}^{2} R_{4}^{2}).$$