Let $E$ be the set of all continuously differentiable real valued functions $f$ on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Define $$J(f)=\int^1_0(1+x^2)f'(x)^2\text dx.$$ a) Show that $J$ achieves its minimum value at some element of $E$. b) Calculate $\min_{f\in E}J(f)$.

On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$. [i]S. Berlov[/i]

Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)

There is an $n\times n$ chessboard where $n\geq 4$ is a positive even number. The cells of the chessboard are coloured black and white such that adjacent cells sharing a common side have different colours. Let $A$ and $B$ be two interior cells (which means cells not lying on an edge of the chessboard) of distinct colours. Prove that a chess piece can move from $A$ to $B$ by moving across adjacent cells such that every cell of the chessboard is passed through exactly once.

Let $a$ be a positive real number. Collinear points $Z_1, Z_2, Z_3, Z_4$ (in that order) are plotted on the $(x, y)$ Cartesian plane. Suppose that the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 + \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_1$ and $Z_4$, and the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 - \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_2$ and $Z_3$. If $Z_1Z_2 = 5$, $Z_2Z_3 = 1$, and $Z_3Z_4 = 3$, then $a^2$ can be written as $\frac{m + n\sqrt{p}}{q}$, where $m$, $n$, $p$, and $q$ are positive integers, $m$, $n$, and $q$ are relatively prime, and $p$ is squarefree. Find $m + n + p + q$.

How many natural numbers (ie. 1,2,3,...) equal three times the sum of the digits? A. None B. 1 C. 2 D. 3 E. 4 or more

Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.

Let $f : N \to [0, \infty)$ be a function satisfying the following conditions: a) $f(4)=2$ b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$. Find $f(n)$ in closed form.

Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.

Let $a,b,c>0$ satisfy $a\geq b\geq c$. Prove that $$\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).$$

Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

Let $a, b, c, d$, and $e$ be positive integers such that $a\le b\le c\le d\le e$ and that $a+b+c+d+e=1002$. a) Determine the largest possible value of $a+c+e$. b) Determine the lowest possible value of $a+c+e$.

Five numbers 1,2,3,4,5 are written on a blackboard. A student may erase any two of the numbers a and b on the board and write the numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?

For any positive integer $n$, let $f(n)$ denote the sum of the positive integers $k \le n$ such that $k$ and $n$ are relatively prime. Let $S$ be the sum of $\frac{1}{f(m)}$ over all positive integers $m$ that are divisible by at least one of $2$, $3$, or $5$, and whose prime factors are only $2$, $3$, or $5$. Then $S = \frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is a) an integer, b) a prime.

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

Each point in a $12 \times 12$ array is colored red, white or blue. Show that it is always possible to find $4$ points of the same color forming a rectangle with sides parallel to the sides of the array.

A set of $n^2$ counters are labeled with $1,2,\ldots, n$, each label appearing $n$ times. Can one arrange the counters on a line in such a way that for all $x \in \{1,2,\ldots, n\}$, between any two successive counters with the label $x$ there are exactly $x$ counters (with labels different from $x$)?

Suppose that an isosceles right-angled triangle is divided into $m$ acute-angled triangles. Find the smallest possible $m$ for which this is possible.

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)