If you increase a number $X$ by $20\%,$ you get $Y.$ By what percent must you decrease $Y$ to get $X?$

Let $ a,b,c,d $ be four non-negative reals such that $ a+b+c+d = 4 $. Prove that $$ a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5} $$

For odd integers $n,$ two people play the game on $2\times n$ grid. Each people color one cell that is not colored before with white and black. When coloring is done, they count the number of ordered pairs of neighboring cells that have the same color and different color, respectively. If same color neighboring ordered pair of cells are more than different color neighboring ordered pair of cells, the person who first starts win and lose otherwise. (If the number is same, they are tied.) If both of them use the best strategy, find the result of the game.

Given an isosceles triangle $ABC$ with base $AB$, cirumcenter $O$, incenter $S$ and $\angle C<60^\circ$. The circumcircle of $AOS$ intersects $AC$ at $D$. Prove that $SD\parallel BC$ and $AS\perp OD$.

Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega$. Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A$, $\omega_B$, and $\omega_C$ meet in six points$-$two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC$, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC$. The side length of the smaller equilateral triangle can be written as $\sqrt{a}-\sqrt{b}$, where $a$ and $b$ are positive integers. Find $a+b$.

Prove that for all positive integers $n$, there exists a positive integer $m$ which is a multiple of $n$ and the sum of the digits of $m$ is equal to $n$.

Find the minimum value of the expression $f(a, b, c) = (a + b)^4 + (b + c)^4 + (c + a)^4 - \frac47 (a^4 + b^4 + c^4)$, as $a, b, c$ varies over the set of all real numbers

Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.

Consider a chessboard $6 \times 6$, made up of $36$ single squares. We want to place $6$ chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square. For each of those $30$ houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the $6$ rooks ($T$) as below, we have: (a) Is it possible to place the rooks so that there are $30$ green squares? (b) Is it possible to place the rooks so that there are $30$ yellow squares? (c) Is it possible to place the rooks so that there are $15$ green and $15$ yellow squares?

Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$

A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc connecting $ D$ and $ E$? [asy] defaultpen(linewidth(0.8)); pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); clip(B--Arc(O, 2, 30, 60)--cycle); draw(Circle(origin, 2)); draw((-2,0)--(2,0)^^(0,-2)--(0,2)); draw(A--D^^C--E); label("$A$", A, dir(point--A)); label("$C$", C, dir(point--C)); label("$O$", O, dir(point--O)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$B$", B, SW);[/asy] $ \textbf{(A) } \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad \textbf{(B) } \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad \textbf{(C) } \pi\left(2 \minus{} \sqrt {3}\right) \qquad \textbf{(D) } \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\ \qquad \indent \textbf{(E) } \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}$

Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)

Given $x \ge 0$, prove that $$\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x$$

Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a $\frac{1}{3}$ and $\frac{1}{4}$ chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a $\frac{9}{10}$ chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.

The lengths of the sides of a triangle are integers, whereas the radius of its circumscribed circle is a prime number. Prove that the triangle is right-angled.

[b]9.[/b] Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$. Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$. [b](F. 5)[/b]

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.

What is the fewest number of multiplications required to reach $x^{2000}$ from $x$, using only previously generated powers of $x$? For example $x\rightarrow x^{2}\rightarrow x^{4}\rightarrow x^{8}\rightarrow x^{16}\rightarrow x^{32}\rightarrow x^{64}\rightarrow x^{128}\rightarrow x^{256}\rightarrow x^{512}\rightarrow x^{1024}\rightarrow x^{1536}\rightarrow x^{1792}\rightarrow x^{1920}\rightarrow x^{1984}\rightarrow x^{2000}$ uses $15$ multiplications.

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons. (a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides. (b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.

How many factors of $(20^{12})^2$ less than $20^{12}$ are not factors of $20^{12}$ ?

In a $4\times 4$ grid of unit squares, five squares are chosen at random. The probability that no two chosen squares share a side is $\tfrac mn$ for positive relatively prime integers $m$ and $n$. Find $m+n$. [i]Proposed by David Altizio[/i]