let $a\le b\le c \le d$ show that: $$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$

Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$. [i]Ray Li.[/i]

Let's call a set of numbers [i]lucky[/i] if it cannot be divided into two nonempty groups so that the product of the sum of the numbers in one group and the sum of the numbers in the other is positive. The teacher wrote several integers on the blackboard. Prove that the children can add another integer to the existing ones so that the resulting set is lucky. [i]A. Kuznetsov[/i]

A bucket filled with $25$ identical blocks weighs $35$ pounds. After three of the blocks are removed, the bucket of blocks weighs $31$ pounds. What is the weight in pounds of the empty bucket? $\text{(A) }\frac{2}{3}\text{ lbs}\qquad\text{(B) }1\frac{1}{3}\text{ lbs}\qquad\text{(C) }1\frac{2}{3}\text{ lbs}\qquad\text{(D) }2\frac{1}{3}\text{ lbs}\qquad\text{(E) }2\frac{2}{3}\text{ lbs}$

Regular hexagon $ABCDEF$ has area $1$. Starting with edge $AB$ and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side $AB$, the new point, $G$, is drawn so $AG = \tfrac12 AB$. This forms hexagon $GHIJKL$, as shown. What is the area of this new hexagon? [asy] size(120); pair A = (-1/2, sqrt(3)/2); pair B = (1/2, sqrt(3)/2); pair C = (1,0); pair D = (1/2, -sqrt(3)/2); pair EE = (-1/2, -sqrt(3)/2); pair F = (-1,0); pair G = (A+B)/2; pair H = (B+C)/2; pair I = (C+D)/2; pair J = (D+EE)/2; pair K = (EE+F)/2; pair L = (F+A)/2; draw(A--B--C--D--EE--F--cycle); draw(G--H--I--J--K--L--cycle); dot(A^^B^^C^^D^^EE^^F^^G^^H^^I^^J^^K^^L); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,SE); label("$E$",EE,SW); label("$F$",F,W); label("$G$",G,N); label("$H$",H,NE); label("$I$",I,SE); label("$J$",J,S); label("$K$",K,SW); label("$L$",L,NW); [/asy] $\textbf{(A) }\dfrac35\qquad\textbf{(B) }\dfrac57\qquad\textbf{(C) }\dfrac34\qquad\textbf{(D) }\dfrac79\qquad\textbf{(E) }\dfrac45$

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

a) A game master divides a group of $12$ players into two teams of six. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $3$ teammates and $1$ opposing player on each card (possibly in a mixed up order). Now he wants to write the names in such away that the players together cannot determine the two teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the two teams, if now each card contains the names of $4$ teammates and $1$ opposing player (possibly in a mixed up order)?

Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.

In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$ [i]CSJL[/i]

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.

Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$

Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that $$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover? (Bulgaria)

Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove: A) Prove that $M$ is the orthocenter of the triangle $ADE$. B) Prove that $TM$ cuts $DE$ in half.

Find all real polynomials $f$ and $g$, such that: \[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \] for all $x\in\mathbb{R}$.

A cloakroom in a cinema works with some breaks. The total time cloakroom worked today is 8 hours. The schedule of the cloakroom is such that it is possible to show any film of duration at most 12 hours such that the cloakroom will be open at least one hour before and after the film (the films are shown without breaks). Find the minimal possible amount of breaks in the schedule of cloakroom. [i]A. Voidelevich[/i]

Sally has five red cards numbered $ 1$ through $ 5$ and four blue cards numbered $ 3$ through $ 6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$? [i]2021 CCA Math Bonanza Team Round #9[/i]

Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

Integers $a$, $b$, $c$ are selected independently and at random from the set $ \{ 1, 2, \cdots, 10 \} $, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$. [i]Proposed by Evan Chen[/i]

Tickets for the football game are \$10 for students and \$15 for non-students. If 3000 fans attend and pay \$36250, how many students went?

For $i=0,1,\dots,5$ let $l_i$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i\frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_i$ is chosen uniformly at random from the intersection of $l_i$ with the unit disk. Consider the convex hull of the points $P_i$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$?

Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of $\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$ and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.