Find all pairs of positive integers $(a,b)$ such that there exists a finite set $S$ satisfying that any positive integer can be written in the form $$n = x^a + y^b + s$$where $x,y$ are nonnegative integers and $s \in S$ [i]CSJL[/i]

Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then \[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

In a triangle $ABC$, two angle trisectors of $A$ intersect with $BC$ at $D$ and $E$ respectively so that $B,D,E,C$ comes in order. If we have $BD = 3$, $DE = 1$ and $EC = 2$, find $\angle DAE$.

Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.

Find all pairs of real numbers $x, y$ that satisfy the following system of equations $$\begin{cases} x^2 + 3y = 10 \\ 3 + y = \frac{10}{ x} \end{cases}$$

If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

Players $1,2,\ldots,10$ are playing a game on Christmas. Santa visits each player's house according to a set of rules: -Santa first visits player $1$. After visiting player $i$, Santa visits player $i+1$, where player $11$ is the same as player $1$. -Every time Santa visits someone, he gives them either a present or a piece of coal (but not both). -The absolute difference between the number of presents and pieces of coal that Santa has given out is at most $3$ at every point in time. -If Santa has a choice between giving out a present and a piece of coal, he chooses with equal probability. Let $p$ be the probability that player $1$ gets a present before player $2$ does. If $p=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $100m+n$. [i]Proposed by Tristan Shin

What’s the greatest pyramid volume one can form using edges of length $2, 3, 3, 4, 5, 5$, respectively?

A pile of $2000$ coins is given on a table. In each step, we choose a pile with at least three coins, remove one coin from it, and divide the rest of this pile into two piles (not necessarily of the same size). Is it possible that after several steps each pile on the table has exactly three coins?

Let $x_1,x_2,x_3$ be three positive real roots of the equation $x^3+ax^2+bx+c=0$ $(a,b,c\in R)$ and $x_1+x_2+x_3\leq 1. $ Prove that $$a^3(1+a+b)-9c(3+3a+a^2)\leq 0$$

Let \( ABC \) be an acute-angled scalene triangle. Let \( B_1 \) and \( B_2 \) be points on the rays \( BC \) and \( BA \), respectively, such that \( BB_1 = BB_2 = AC \). Similarly, let \( C_1 \) and \( C_2 \) be points on the rays \( CB \) and \( CA \), respectively, such that \( CC_1 = CC_2 = AB \). Prove that if \( B_1B_2 \) and \( C_1C_2 \) intersect at \( K \), then \( AK \) is parallel to \( BC \).

A convex set $S$ on a plane, not lying on a line, is painted in $p$ colors. Prove that for every $n \ge 3$ there exist infinitely many congruent $n$-gons whose vertices are of the same color.

The sum of three numbers is $ 98$. The ratio of the first to the second is $ \frac {2}{3}$, and the ratio of the second to the third is $ \frac {5}{8}$. The second number is: $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33$

Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

The circle $C_1$ and the circle $C_2$ passing through the center of $C_1$ intersect each other at $A$ and $B$. The line tangent to $C_2$ at $B$ meets $C_1$ at $B$ and $D$. If the radius of $C_1$ is $\sqrt 3$ and the radius of $C_2$ is $2$, find $\dfrac{|AB|}{|BD|}$. $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {2\sqrt 3}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \dfrac {\sqrt 5}2 $

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes. $(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once? $(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts

Let $ z \in \mathbb{C}$ such that for all $ k \in \overline{1, 3}$, $ |z^k \plus{} 1| \le 1$. Prove that $ z \equal{} 0$.

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Let $f : R^+ \to R^+$ be a function such that for all $x, y \in R^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $R^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$. .

If 554 is the base $ b$ representation of the square of the number whose base $ b$ representation is 24, then $ b$, when written in base 10, equals $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

There are $27$ cardboard and $27$ plastic boxes. There are balls of certain colors inside the boxes. It is known that any two boxes of the same kind do not have a ball with the same color. Boxes of different kind have at least one ball of the same color. At each step we select two boxes that have a ball of same color and switch this common color into any other color we wish. Find the smallest number $n$ of moves required.