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.

Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.

$n$ students, numbered from $1$ to $n$ are sitting next to each other in a class. In the beginning the $1$st student has $n$ pieces of paper in one pile. The goal of the students is to distribute the $n$ pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): $\bullet$ They divide one of their piles of paper into two smaller piles. $\bullet$ They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?

With an unmarked ruler only, reconstruct the trapezoid $ABCD$ ($AD \parallel BC$) given the vertices $A$ and $B$, the intersection point $O$ of the diagonals of the trapezoid and the midpoint $M$ of the base $AD$. (Yukhim Rabinovych)

The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$. Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$

A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers $a$, $b$, and $t$. The random triangle has two sides of lengths $a$ and $b$ with the angle between them measuring $15(t - 1)$ degrees. What is the probability that the triangle is a right triangle?

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.

Let $ABC$ an isosceles triangle with $CA=CB$ and $\Gamma$ it´s circumcircle. The perpendicular to $CB$ through $B$ intersect $\Gamma$ in points $B$ and $E$. The parallel to $BC$ through $A$ intersect $\Gamma$ in points $A$ and $D$. Let $F$ the intersection of $ED$ and $BC, I$ the intersection of $BD$ and $EC, \Omega$ the cricumcircle of the triangle $ADI$ and $\Phi$ the circumcircle of $BEF$.If $O$ and $P$ are the centers of $\Gamma$ and $\Phi$, respectively, prove that $OP$ is tangent to $\Omega$

An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?

If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is: $\textbf{(A)}\ 16:9 \qquad \textbf{(B)}\ 9:16 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:4 \qquad \textbf{(E)}\ \text{None of these} $

Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells. The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list] Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\). [i]Proposed by Linus Tang[/i]

Evaluate \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] [i]Proposed by Nathan Xiong[/i]

A sequence $(a_n)$ of positive integers satisfies$(a_m,a_n) = a_{(m,n)}$ for all $m,n$. Prove that there is a unique sequence $(b_n)$ of positive integers such that $a_n = \prod_{d|n} b_d$

Positive integer $q$ is the $k{}$-successor of positive integer $n{}$ if there exists a positive integer $p{}$ such that $n+p^2=q^2$. Let $A{}$ be the set of all positive integers $n{}$ that have at least a $k{}$-successor, but every $k{}$-successor does not have $k{}$-successors of its own. Prove that $$A=\{7,12\}\cup\{8m+3\mid m\in\mathbb{N}\}\cup\{16m+4\mid m\in\mathbb{N}\}.$$