Let $f\in C^1(a,b)$, $\lim_{x\to a^+}f(x)=\infty$, $\lim_{x\to b^-}f(x)=-\infty$ and $f'(x)+f^2(x)\geq -1$ for $x\in (a,b)$. Prove that $b-a\geq\pi$ and give an example where $b-a=\pi$.

A natural number $n$ is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it $a_0$, $a_1$, $a_2$, $...$, $a_{2n-1}$ the polynomial $x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0$ has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)

Let $x$, $y$, $z > 0$ such that $x^2 + y^2 + z^2 = 3$. Then $$x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}$$

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

A square \( K = 2025 \times 2025 \) is given. We define a [i]stick[/i] as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: [list] [*] If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

A table consisting of $9$ rows and $2001$ columns is filfed with integers $1,2,..., 2001$ in such a way that each of these integers occurs in the table exactly $9$ times and the integers in any column differ by no more than $3$. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).

Assume that $a$ is a given irrational number. (a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$. (b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.

At a gala banquet, $12n + 6$ chairs, where $n \in N$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank n find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)

Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called [i]simple[/i] if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$. Given a simple operation $\times$ on $S$, any string of elements in $S$ can be reduced to a single element, such as $xyz \to x \times (y \times z)$. A string of variables in $S$ is called[i] full [/i]if it contains each variable in $S$ at least once, and two strings are [i]equivalent[/i] if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n=1$. Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$. Determine the number of elements of $T$.

Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is (A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.

Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point

[color=darkred]Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that: [list] [b]a)[/b] $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$ [b]b)[/b] $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$ [/list][/color]

Given three non-collinear points in the plane, find a line which is equally distant from each of the points. How many such lines are there?

Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After choosing $P_k$ (but before choosing $P_{k+1}$) for $k \geq 1$, Bob tells "warmer'' if $P_k$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder''. After Alice has chosen $P_N$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $AB$ is at most $\frac 1 {2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.

Prove that every positive integer which is not a member of the infinite set below is equal to the sum of two or more distinct numbers of the set \[\{ 3,-2, 2^{2}3,-2^{3}, \cdots, 2^{2k}3,-2^{2k+1}, \cdots \}=\{3,-2, 12,-8, 48,-32, 192, \cdots \}.\]

Let be given an open interval $(\alpha; \beta)$ with $\alpha - \beta = \frac{1}{27}$. Determine the maximum number of irreducible fractions $\frac{a}{b}$ in $(\alpha; \beta)$ with $1 \leq b \leq 2007$?

For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\] [i]Proposed by Daniel Zhu.[/i]

Let $\alpha\in(\frac{\pi}{4},\frac{\pi}{2})$, then the order of $(\cos\alpha)^{\cos\alpha},(\sin\alpha)^{\cos\alpha},(\cos\alpha)^{\sin\alpha}$ is $\text{(A)}(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(B)}(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}<(\sin\alpha)^{\cos\alpha}$ $\text{(C)}(\sin\alpha)^{\cos\alpha}<(\cos\alpha)^{\cos\alpha}<(\cos\alpha)^{\sin\alpha}$ $\text{(D)}(\cos\alpha)^{\sin\alpha}<(\cos\alpha)^{\cos\alpha}<(\sin\alpha)^{\cos\alpha}$

Let $n$ be a natural number with $120$ positive divisors (including $1$ and $n$). For each divisor $d$ of $n$, let $q$ be the quotient and $r$ the remainder when dividing $4n - 3$ by $d$. Let $Q$ be the sum of all the quotients $q$, and $R$ the sum of all the remainders $r$ for the $120$ divisions of $4n - 3$ by $d$. Determine all posible values of $Q - 4R$

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$

For all $n\in\mathbb{N}$, we denote by $s(n)$ the sum of its digits. Find all integers $k\geq 2$ such that there exist $a,b\in\mathbb{N}$ with $$s(n^3+an+b)\equiv s(n)\pmod k,$$ for all $n\in\mathbb{N}^*$.