Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]

Find the smallest real number $C$, such that for any positive integers $x \neq y$ holds the following: $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$ Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$. [i]Proposed by Anton Trygub[/i]

Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$. Let $K$ be the area of rhombus $\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$. [asy] size(150);defaultpen(linewidth(0.7)+fontsize(10)); draw(rotate(45)*polygon(4)); pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180); draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$\mathcal{P}$", intersectionpoint( A--(-1,0), F--(0,1) )); label("$\mathcal{S}$", intersectionpoint( E--(-1,0), F--(0,-1) )); label("$\mathcal{R}$", intersectionpoint( D--(1,0), C--(0,-1) )); label("$\mathcal{Q}$", intersectionpoint( B--(1,0), C--(0,1) )); label("$\mathcal{T}$", point); dot(A^^B^^C^^D^^E^^F);[/asy]

Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]

Let $a_1 \ge a_2 \ge ... \ge a_n > 0$ be real numbers. Prove that $$a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)$$

Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$? A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$

Let $n$ be a positive integer and let $x_1,\ldots,x_n$ be $n$ nonzero points in $\mathbb R^2$. Suppose $\langle x_i,x_j\rangle$ (scalar or dot product) is a rational number for all $i,j$ ($1\le i,j\le n$). Let $S$ denote all points of $\mathbb R^2$ of the form $\sum_{i=1}^na_ix_i$ where the $a_i$ are integers. A closed disk of radius $R$ and center $P$ is the set of points at distance at most $R$ from $P$ (includes the points distance $R$ from $P$). Prove that there exists a positive number $R$ and closed disks $D_1,D_2,\ldots$ of radius $R$ such that (a) Each disk contains exactly two points of $S$; (b) Every point of $S$ lies in at least one disk; (c) Two distinct disks intersect in at most one point.

For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$.

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$; ($ii$)$2017|x_1+...+x_{100}$; ($iii$)$2017|x_1^2+...+x_{100}^2$.

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.

Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have \[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]

Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have $$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$ [i]Allen Wang[/i]

Find all real solutions $(x,y,z)$ of the system of equations $$\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}$$

In an acute triangle $ABC$, $AB<AC$. $AD$ is the altitude dropped onto $BC$ and $P$ is a point on $AD$. Let $PE\perp AC$ at $E$, $PF\perp AB$ at $F$ and let $J,K$ be the circumcentres of triangles $BDF, CDE$ respectively. Prove that $J,K,E,F$ are concyclic if and only if $P$ is the orthocentre of triangle $ABC$.

Find the largest value of the expression \[xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}\]

Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$ Prove that $x_1x_2...x_n \ge (n -1)^n$.

Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.

Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.

Find the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.

Lason Jiu gives a problem to Sick Nong and Ayush Agrawal. Sick takes 6 minutes to solve the problem, while Ayush takes 9 minutes. Sick has a 1/3 chance of solving correctly and Ayush has a 2/3 chance of solving correctly. If they solved it incorrectly, they resume solving with the same time and accuracy. Lason gives a rubber chicken to the first person who solves it correctly. If Sick and Ayush solve the question at the same time, Lason checks Sick's work first. The probability that Ayush wins the rubber chicken can be expressed as $\frac{p}{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #8[/i]

What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$ $\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$

The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.

On the sides $AB, BC, CD$ and $DA$ of the parallelogram $ABCD$ marked the points $M, N, K$ and $F$. respectively. Is it possible to determine, using only compass, whether the area of ​​the quadrilateral $MNKF$ is equal to half the area of ​​the parallelogram $ABCD$?

Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.