There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let $$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$ Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions: i) $\deg {{Q}_{k}}(x)=2020.$ ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

Consider all line segments of length $4$ with one end-point on the line $y = x$ and the other end-point on the line $y = 2x$. Find the equation of the locus of the midpoints of these line segments.

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

A board $1$x$k$ is called [i]guayaco[/i] if: -Each unit square is painted with exactly one of $k$ available colors. -If $gcd(i,k)>1$, the $i$th unit square is painted with the same color as $(i-1)$th unit square. -If $gcd(i, k)=1$, the $i$th unit square is painted with the same color as $(k-i)$th unit square. Sebastian chooses a positive integer $a$ and calculates the number of boards $1$x$a$ that are guayacos. After that, David chooses a positive integer $b$ and calculates the number of boards $1$x$b$ that are guayacos. David wins if the number of boards $1$x$a$ that are guayacos is the same as the number of boards $1$x$b$ that are guayacos, otherwise, Sebastian wins. Find all the pairs $(a,b) $ such that, with those numbers, David wins.

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? $ \textbf{(A)}\ 26 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 34$

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

Juli has a deck of $54$ cards and proposes the following game to Bruno. Juli places the cards in a row, some face-up and others face-down. Bruno can repeatedly perform the following move: select a card and flip it along with its two neighbors (turning face-up cards face-down, and vice versa for face-down cards). Bruno wins if, through this process, he manages to turn all the cards face up. Otherwise, Juli wins. Determine which player has a winning strategy and explain it. [b]Note:[/b] When Bruno selects the first or the last card in the row, he flips only two cards. In all other cases, he flips three cards.

Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.

Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.

How many of the rearrangements of the digits $123456$ have the property that for each digit, no more than two digits smaller than that digit appear to the right of that digit? For example, the rearrangement $315426$ has this property because digits $1$ and $2$ are the only digits smaller than $3$ which follow $3,$ digits $2$ and $4$ are the only digits smaller than $5$ which follow $5,$ and digit $2$ is the only digit smaller than $4$ which follows $4.$

Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$

The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common. (a) If $n = 14$, determine the largest possible number of variants the problem committee can prepare. (b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.

For each $ x$ in $ [0,1]$, define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$

Find all positive integers $n$ such that one can place checkers on a $n\times n$ checkerboard such that any square chosen from the checkerboard has exactly $2$ adjacent squares with checkers on it. Two squares are considered adjacent if they both share a common side

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions: [list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$ [*] For all real number $x$, $f(g(x))=g(f(x))=0$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$ [/list]

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

Line $p$ intersects sides $AB$ and $BC$ of triangle $\triangle ABC$ at points $M$ and $K.$ If area of triangle $\triangle MBK$ is equal to area of quadrilateral $AMKC,$ prove that \[\frac{\left|MB\right|+\left|BK\right|}{\left|AM\right|+\left|CA\right|+\left|KC\right|}\geq\frac{1}{3}\]

Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that: \[ f(x^2 + y) = xf(x) + \frac{f(y^2)}{y} \] for any positive real numbers \( x \) and \( y \).

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]

In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.