Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions. (Proposed by Fedor Petrov, St. Petersburg State University)

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

Let $n>1$ be an integer and $x_1,x_2,\ldots,x_n$ be $n{}$ arbitrary real numbers. Determine the minimum value of \[\sum_{i<j}|\cos(x_i-x_j)|.\]

Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?

We consider triangles $ABC$, in which the point $M$ lies on the side $AB$, $AM = a$, $BM = b$, $CM = c$ ($c <a, c <b$). Find the smallest radius of the circumcircle of such triangles.

A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$ Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

Let $x,y,z$ be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}

We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$ Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.

Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$

Let $x, y$ and $z$ be nonnegative real numbers satisfying the following conditions: (1) $x + cy \le 36$,(2) $2x+ 3z \le 72$, where $c$ is a given positive number. Prove that if $c \ge 3$ then the maximum of the sum $x + y + z$ is $36$, while if $c < 3$, the maximum of the sum is $24 + \frac{36}{c}$ .

Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$. Determine the minimum value of the red number at the end of this procedure.

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$ Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$

Determine the minimum value of the expression $$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$ for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.