The five real numbers $v,w,x,y,s$ satisfy the system of equations \begin{align*} v&=wx+ys,\\ v^2&=w^2x+y^2s,\\ v^3&=w^3x+y^3s. \end{align*} Show that at least two of them are equal.

Let $n>2$ be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked. [asy]unitsize(.25cm); real s3=1.73205081; pair[] points={(-4,4*s3),(-2,4*s3),(0,4*s3),(2,4*s3),(4,4*s3),(-5,3*s3), (-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (5,3*s3), (-6,2*s3),(-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3),(6,2*s3),(-7,s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3),(7,s3),(-8,0), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (8,0),(-7,-s3),(-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (7,-s3), (-6,-2*s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (6,-2*s3), (-5,-3*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3), (5,-3*s3), (-4,-4*s3), (-2,-4*s3), (0,-4*s3), (2,-4*s3), (4,-4*s3)}; void draw_hexagon(pair p) { draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30))); } {for (int i=0;i<61;++i){draw_hexagon(points[i]);}} label((0,0), "\Large $*$"); [/asy] The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda and Magni take turns moving the stone: Anna begins, then Edda, then Magni, then Anna, and so on. The rules for each player's turn are: [list] [*] Anna has to move the stone to an adjacent tile, in any direction. [*] Edda has to move the stone straight by two tiles in any of the $6$ possible directions. [*] Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the $6$ possible directions. [/list] Find all $n$ for which Edda and Magni have a winning strategy.

An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves? [i]2020 CCA Math Bonanza Individual Round #1[/i]

On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses $\times$, the other $\circ$. The first who fills a $2\times 2$ square with his symbols wins. Can the player who starts always win?

Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is (a) a triangle with at least two sides longer than $1000$? (b) a triangle whose sides are all longer than $1000$? (c) a quadrangle?

(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.

Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$ [i]Proposed by Kyle Lee[/i]

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. [i]Proposed by Anthony Wang[/i]

Yoknapatawpha County has $500,000$ families. Each family is expected to continue to have children until it has a girl, at which point each family stops having children. If the probability of having a boy is $50\%$, and no families have either fertility problems or multiple children per birthing, how many families are expected to have at least $5$ children?

Find all the triples $(x,y,z)$ of real numbers such that \[2x\sqrt{y-1}+2y\sqrt{z-1}+2z\sqrt{x-1} \ge xy+yz+zx \]

At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.

Two real numbers between $0$ and $1$ are randomly chosen. Calculate the probability that any one of them is less than the square of the other.

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

On each side of a regular tetrahedron with edges of length $1$ one constructs exactly such a tetrahedron. This creates a dodecahedron with $8$ vertices and $18$ edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.

$x,y$ are real numbers such that $$x^2+y^2=1 , 20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk)

$$\int\frac{\ln^2(x)}{x}\,dx$$ [i]Proposed by Connor Gordon[/i]

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

Let $p=3\cdot 10^{10}+1$ be a prime and let $p_n$ denote the probability that $p\mid (k^k-1)$ for a random $k$ chosen uniformly from $\{1,2,\cdots,n\}$. Given that $p_n\cdot p$ converges to a value $L$ as $n$ goes to infinity, what is $L$? [i]Proposed by Vijay Srinivasan[/i]

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.

For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]