Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold: $i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$; $ii)$ $\sum_{j=1}^{k} a_j \leq 1$. Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

Consider equilateral triangle $ABC$ with side length $1$. Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$. Compute the sum of all possible values of $\kappa$. [i]2018 CCA Math Bonanza Lightning Round #3.4[/i]

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $. a) Prove that $ I$ lies between $ M $ and $ H $. b) Calculate $ AB ^ 2-AC ^ 2 $. c) Determine $ \dfrac {AB} {AC} $. d) Find the measure of all the sides and angles of the triangle.

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$

Find all triples $(x,y,z)$ of positive integers such that $(x+1)^{y+1}+1=(x+2)^{z+1}$.

let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$. let $CE$ be the perpendicular from $C$ on $AB$ prove that $ CE^2 = AB. CD $

Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum \[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\] Determine all possible values of $S_f$.

Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

Determine the number of pairs of positive integers \( (p, k) \) such that \( p \) is a prime number and \( p^2 + 2^k \) is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.

There are $n$ snails on the plane where the $i$ snail has $a_i$ attack and $d_i$ defense, where $a_i, d_i\in \mathbb{R}$ and each snail has a distinct attack and a distinct defense. We said a 4-tuple of subsets of snails $(S_1, S_2, S_3, S_4)$ is [b]balanced[/b] if $|S_1|+|S_3|$ is either $\lceil n/2\rceil$ or $\lfloor n/2\rfloor$ and there exist real numbers $A, D$ such that \begin{align*} S_1=\{i\ |\ a_i\geq A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_2=\{i\ |\ a_i<A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_3=\{i\ |\ a_i< A\text{ and } d_i< D, 1\leq i\leq n\}\\ S_4=\{i\ |\ a_i\geq A\text{ and } d_i< D, 1\leq i\leq n\} \end{align*} Find the largest integer $k$ such that there is always at least $k$ [b]balanced[/b] 4-tuples of subsets. [i]Proposed by redshrimp[/i]

The pieces in a game are squares of side $1$ with their sides colored with $4$ colors: blue, red, yellow and green, so that each piece has one side of each color. There are pieces in every possible color arrangement, and the game has a million pieces of each kind. With the pieces, rectangular puzzles are assembled, without gaps or overlaps, so that two pieces that share a side have that side of the same color. Determine if with this procedure you can make a rectangle of $99\times 2007$ with one side of each color. And $100\times 2008$? And $99\times 2008$?

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

Let $A, B$ and $C$ three points on a line $l$, in that order .Let $D$ a point outside $l$ and $\Gamma$ the circumcircle of $\triangle BCD$, the tangents from $A$ to $\Gamma$ touch $\Gamma$ on $S$ and $T$. Let $P$ the intersection of $ST$ and $AC$. Prove that $P$ does not depend of the choice of $D$.

Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.

A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.

Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, $\ldots$). What is the largest integer $j$ such that $P$ is divisible by $101^j$?

A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.

For a sequence $a_1,a_2,...,a_m$ of real numbers, define the following sets \[A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\] Let $n$ be a given integer, and $n>2$. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set $A\triangle B$, where $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$

At Zoom University, people’s faces appear as circles on a rectangular screen. The radius of one’s face is directly proportional to the square root of the area of the screen it is displayed on. Haydn’s face has a radius of $2$ on a computer screen with area $36$. What is the radius of his face on a $16 \times 9$ computer screen?

What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle