Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$.

In the governor elections, there were three candidates: $A$, $B$, and $C$. In the first round, $A$ received $44\%$ of the votes that were cast between $B$ and $C$. No candidate obtained the majority needed to win in the first round, and $C$ was the one who received the least votes of the three, so there was a runoff between $A$ and $B$. The voters for the runoff were the same as in the first round, except for $p\%$ of those who voted for $C$, who chose not to participate in the runoff; $p$ is an integer, $1 \leqslant p \leqslant 100$. It is also known that all those who voted for $B$ in the first round also voted for him again in the runoff, but it is unknown what those who voted for $A$ in the first round did. A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of $p$ the journalist is telling the truth. [b]Note:[/b] The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.

Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.

Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$

Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.

Let $ABC$ be an acute triangle inscribed on the circumference $\Gamma$. Let $D$ and $E$ be points on $\Gamma$ such that $AD$ is perpendicular to $BC$ and $AE$ is diameter. Let $F$ be the intersection between $AE$ and $BC$. Prove that, if $\angle DAC = 2 \angle DAB$, then $DE = CF$.

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$, \[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

Given a positive integer $k$. Prove that there are infinitely many positive integers $n$ satisfy the following conditions at the same time: a) $n$ has at least $k$ distinct prime divisors b) All prime divisors other than $3$ of $n$ have the form $4t+1$, with $t$ some positive integer. c) $n | 2^{\sigma(n)}-1$ Here $\sigma(n)$ demotes the sum of the positive integer divisors of $n$.

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*} $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8 $

Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

A random permutation $a = (a_1, a_2,...,a_{40})$ of $(1, 2,...,40)$ is chosen, with all permutations being equally likely. William writes down a $20 \times 20$ grid of numbers $b_{ij}$ such that $b_{ij} = \max (a_i, a_{j+20})$ for all $1 \le i, j \le 20$, but then forgets the original permutation $a$. Compute the probability that, given the values of $b_{ij}$ alone, there are exactly $2$ permutations $a$ consistent with the grid.

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

Cut a square into eight acute-angled triangles.

Let $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$. There is a unique value of $ y$ such that $ 0 < y < x$ and $ x^x \equal{} y^y$. What is the value of $ y$? Express your answer in the form $ \sqrt[c]{\frac{a}{b}}\,$, where $ a$ and $ b$ are relatively prime positive integers and $ c$ is a prime number.

Consider the polynomial $P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$ where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$ [b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$ [b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$

Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of $$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$ .

Prove that the number $a=2019^{2020}+4^{2019}$ is a composite number.

Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]