A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)\equal{}P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2\minus{}2xy\plus{}y^2$ satisfies this condition). Given that $ (x\minus{}y)$ is a factor of $ P(x,y)$, show that $ (x\minus{}y)^2$ is a factor of $ P(x,y)$.

Find all pairs $(x, y)$ of real numbers that satisfy the system of equations $$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\ y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$

Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{10}$ The chance of getting any given number is $\dfrac{1}{n}$ , so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

Two concentric circles have differing radii such that a chord of the outer circle which is tangent to the inner circle has length $18$. Compute the area inside the bigger circle which lies outside of the smaller circle.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.

If $a,b,c,d$ be any four positive real numbers, then prove that \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4. \]

* A bus route has $14$ stops (counting the first and the last). A bus cannot carry more than $25$ passengers. We assume that a passenger takes a bus from $A$ to $B$ if (s)he enters the bus at $A$ and gets off at $B$. Prove that for any bus route: a) there are $8$ distinct stops $A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4$ such that no passenger rides from $A_k$ to $B_k$ for all $k = 1, 2, 3, 4$ (#) b) there might not exist $10$ distinct stops $A_1, B_1, . . . , A_5, B_5$ with property (#).

Given a triangle $ABC$, the internal bisectors through $A$ and $B$ meet the opposite sides in $D$ and $E$, respectively. Prove that \[DE \le (3 - 2\sqrt2)(AB + BC + CA)\] and determine the cases of equality.

Given $n \ge 2$ points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position $P$ to one of the $n-1$ other points $Q$ and draws the segment $PQ$. This move is not allowed if the segment $PQ$ is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each $n$, which of the two players has a winning strategy.

Show that if $b = \frac{a+c}{2}$ in the triangle $ABC$, then $\cos (A-C) + 4 \cos B = 3$.

For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions: [list] [*] no two consecutive digits are equal, and [*] the last digit is a prime. [/list]

For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers.

For each real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer that does not exceed $x.$ For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_2 n\rfloor$ is a positive even integer.

In the figure, what is the ratio of the area of the gray squares to the area of the white squares? [asy] size((70)); draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0)); draw((-2.5,-7.5)--(7.5,2.5)); draw((-5,-5)--(5,5)); draw((-7.5,-2.5)--(2.5,7.5)); draw((-7.5,2.5)--(2.5,-7.5)); draw((-5,5)--(5,-5)); draw((-2.5,7.5)--(7.5,-2.5)); fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray); fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray); fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray); [/asy] $ \textbf{(A)}\ 3:10 \qquad\textbf{(B)}\ 3:8 \qquad\textbf{(C)}\ 3:7 \qquad\textbf{(D)}\ 3:5 \qquad\textbf{(E)}\ 1:1 $

Which of the following polygons has the largest area? [asy] size(330); int i,j,k; for(i=0;i<5; i=i+1) { for(j=0;j<5;j=j+1) { for(k=0;k<5;k=k+1) { dot((6i+j, k)); }}} draw((0,0)--(4,0)--(3,1)--(3,3)--(2,3)--(2,1)--(1,1)--cycle); draw(shift(6,0)*((0,0)--(4,0)--(4,1)--(3,1)--(3,2)--(2,1)--(1,1)--(0,2)--cycle)); draw(shift(12,0)*((0,1)--(1,0)--(3,2)--(3,3)--(1,1)--(1,3)--(0,4)--cycle)); draw(shift(18,0)*((0,1)--(2,1)--(3,0)--(3,3)--(2,2)--(1,3)--(1,2)--(0,2)--cycle)); draw(shift(24,0)*((1,0)--(2,1)--(2,3)--(3,2)--(3,4)--(0,4)--(1,3)--cycle)); label("$A$", (0*6+2, 0), S); label("$B$", (1*6+2, 0), S); label("$C$", (2*6+2, 0), S); label("$D$", (3*6+2, 0), S); label("$E$", (4*6+2, 0), S); [/asy] $ \textbf{(A)}\text{A}\qquad\textbf{(B)}\ \text{B}\qquad\textbf{(C)}\ \text{C}\qquad\textbf{(D)}\ \text{D}\qquad\textbf{(E)}\ \text{E} $

Two players take turns playing on a $3\times1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allows them to win, no matter how well his opponent plays.

$n$ is a positive integer, $K$ the set of polynoms of real variables $x_1,x_2,...,x_{n+1}$ and $y_1,y_2,...,y_{n+1}$, function $f:K\rightarrow K$ satisfies \[f(p+q)=f(p)+f(q),\quad f(pq)=f(p)q+pf(q),\quad (\forall)p,q\in K.\] If $f(x_i)=(n-1)x_i+y_i,\quad f(y_i)=2ny_i$ for all $i=1,2,...,n+1$ and \[\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}\] for any real $t$, prove, that for all $k=1,...,n+1$ \[f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}\]

Convex quadrilateral $ABCD$ with perpendicular diagonals satisfies $\angle B = \angle C = 90^\circ$, $BC=20$, and $AD=30$. Compute the square of the area of a triangle with side lengths equal to $CD$, $DA$, and $AB$. [i]2021 CCA Math Bonanza Tiebreaker Round #2[/i]

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those two children half of the combination. After the children's father performs four blends consecutively, what is the probability that the four children will all have the same amount of mashed potatoes?

Let $U$ be the set of all complex numbers $m$ such that the $4$ roots of $(x^2+2x+5)(x^2-2mx+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite union of some curves. Find the sum of the lengths of those curves (i.e. the perimeter of $U$).

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Take five good haikus Scramble their lines randomly What are the chances That you end up with Five completely good haikus (With five, seven, five)? Your answer will be m over n where m,n Are numbers such that m,n positive Integers where gcd Of m,n is 1. Take this answer and Add the numerator and Denominator. [i]Proposed by Jeff Lin[/i]