There are $5$ ingots weighing $1$, $2$, $3$, $4$ and $5$ kg with an unknown copper content that varies in different ingots. Each ingot must be divided into $5$ parts and $5$ new ingots of the same mass of $1$, $2$, $3$, $4$ and $5$ kg must be made. This requires that the percentage of copper in all pieces be the same, regardless of what it was in the original pieces. What parts should each piece be divided into?

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.

Find all real numbers for which $3^x+4^x=5^x$.

For a positive integer $n>1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$. For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$. Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$. Find the largest integer not exceeding $\sqrt{N}$

Alpha and Beta are playing a game on a $10\times 100$ grid of squares. At each turn, they can fold the grid along any of the interior horizontal or vertical gridlines, which creates a smaller (folded) grid of squares (on the first move, they can choose one of $9$ horizontal or $99$ vertical gridlines). The person who makes the last fold wins. If both players play optimally and Alpha starts, determine with proof who wins.

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove $$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$ For which $n$ does equality hold?

Find all prime numbers $p$ for which $p^3- p + 1$ is a perfect square .

Let $\sigma (n)$ shows the number of positive divisors of $n$. Let $s(n)$ be the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n)- \sigma (n) $.

Let $M=\{(x,y)||xy|=1,x>0\},N=\{(x,y)|\arctan x+\arctan y=\pi\}$. Which one is right? $\text{(A)}M\cup N=\{(x,y)||xy|=1\}\qquad\text{(B)}M\cup N=M$ $\text{(C)}M\cup N=N\qquad\text{(D)}M\cup N=\{(x,y)||xy|=1,x,y\text{ cannot be negative the same time}\}$

Prove that the following inequality holds for each natural number $ n$. \[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]

Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$. $\textit{Proposed by Pablo Serrano, Ecuador}$

It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

Let $n$ be a positive integer, and let $M_n$ be the set of invertible matrices with integer entries and size $n \times n$. (a) Find the largest possible value of $n$ such that there exists a symmetric matrix $A \in M_n$ satisfying \[ \det(A^{20} + A^{24}) < 2024. \] (b) Prove that for every $n$, there exists a matrix $B \in M_n$ such that \[ \det(B^{20} + B^{24}) < 2024. \]

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

We divide a natural number $N$, that has $k$ digits, by $19$ and get a residue of $10^{k-2} -Q$, where $Q$ is the quotient and $Q < 101$. Also, $10^{k-2}-Q$ is larger than $0$. How many possible values of $N$ are there?

Let $G$ be a finite group and $a\in G$ a fixed element. Define the set $$S_a=\{g\in G\mid ga\neq ag, \,ga^2=a^2g\}.$$ Show that: [list=a] [*] if $g\in S_a$, then $ag^{-1}\in S_a$; [*] $|S_a|$ is divisible by $4$.

Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\). [i]Proposed by Raymond Feng[/i]

Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

Characterise all numbers that cannot be written as a sum of $1$ or more consecutive odd numbers.