Josie and Ross are playing a game on a $20 \times 20$ chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie’s turn, she selects any two different empty cells, and places one white stone in each of them. On Ross’ turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are $ 8$ consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?

Let $ABC$ be a triangle with orthocenter $H$. Let $P$ be any point of the plane of the triangle. Let $\Omega$ be the circle with the diameter $AP$ . The circle $\Omega$ cuts $CA$ and $AB$ again at $E$ and $F$ , respectively. The line $PH$ cuts $\Omega$ again at $G$. The tangent lines to $\Omega$ at $E, F$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $L$ be the point on $MG$ such that $AL$ and $MT$ are parallel. Prove that $LA$ and $LH$ are orthogonal. Lê Phúc Lữ

Construct the midpoint of a segment using an unmarked ruler and a [i]trisector[/i] that marks in a segment the two points that divide the segment in three equal parts.

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); draw((0,0)--(0,3)^^(1,0)--(1,3)^^(2,0)--(2,3)^^(3,0)--(3,3)^^(0,0)--(3,0)^^(0,1)--(3,1)^^(0,2)--(3,2)^^(0,3)--(3,3)); draw((0,12)--(0,15)^^(1,12)--(1,15)^^(2,12)--(2,15)^^(3,12)--(3,15)^^(0,12)--(3,12)^^(0,13)--(3,13)^^(0,14)--(3,14)^^(0,15)--(3,15)); draw((12,0)--(12,3)^^(13,0)--(13,3)^^(14,0)--(14,3)^^(15,0)--(15,3)^^(12,0)--(15,0)^^(12,1)--(15,1)^^(12,2)--(15,2)^^(12,3)--(15,3)); draw((12,12)--(12,15)^^(13,12)--(13,15)^^(14,12)--(14,15)^^(15,12)--(15,15)^^(12,12)--(15,12)^^(12,13)--(15,13)^^(12,14)--(15,14)^^(12,15)--(15,15)); draw((5,5)--(5,10)^^(6,5)--(6,10)^^(7,5)--(7,10)^^(8,5)--(8,10)^^(9,5)--(9,10)^^(10,5)--(10,10)^^(5,5)--(10,5)^^(5,6)--(10,6)^^(5,7)--(10,7)^^(5,8)--(10,8)^^(5,9)--(10,9)^^(5,10)--(10,10)); draw((3.5,.2)--(11.5,.2)^^(3.5,1.5)--(11.5,1.5)^^(3.5,13.5)--(11.5,13.5)^^(3.5,14.8)--(11.5,14.8), linetype("1 7")); draw((.2,3.5)--(.2,11.5)^^(1.5,3.5)--(1.5,11.5)^^(13.5,3.5)--(13.5,11.5)^^(14.8,3.5)--(14.8,11.5), linetype("1 7")); draw((3.5,3.5)--(4.5,4.5)^^(3.5,11.5)--(4.5,10.5)^^(11.5,3.5)--(10.5,4.5)^^(11.5,11.5)--(10.5,10.5), linetype("1 7")); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,black); fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black); fill((0,14)--(1,14)--(1,15)--(0,15)--cycle,black); fill((1,13)--(2,13)--(2,14)--(1,14)--cycle,black); fill((2,12)--(3,12)--(3,13)--(2,13)--cycle,black); fill((14,0)--(15,0)--(15,1)--(14,1)--cycle,black); fill((13,1)--(14,1)--(14,2)--(13,2)--cycle,black); fill((12,2)--(13,2)--(13,3)--(12,3)--cycle,black); fill((14,14)--(15,14)--(15,15)--(14,15)--cycle,black); fill((13,13)--(14,13)--(14,14)--(13,14)--cycle,black); fill((12,12)--(13,12)--(13,13)--(12,13)--cycle,black); fill((5,5)--(6,5)--(6,6)--(5,6)--cycle,black); fill((6,6)--(7,6)--(7,7)--(6,7)--cycle,black); fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black); fill((8,8)--(9,8)--(9,9)--(8,9)--cycle,black); fill((9,9)--(10,9)--(10,10)--(9,10)--cycle,black); fill((5,9)--(6,9)--(6,10)--(5,10)--cycle,black); fill((6,8)--(7,8)--(7,9)--(6,9)--cycle,black); fill((8,6)--(9,6)--(9,7)--(8,7)--cycle,black); fill((9,5)--(10,5)--(10,6)--(9,6)--cycle,black); [/asy] $ \textbf{(A)}\ 121\qquad\textbf{(B)}\ 625\qquad\textbf{(C)}\ 676\qquad\textbf{(D)}\ 2500\qquad\textbf{(E)}\ 2601 $

Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.

Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).

It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?

Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. [i]Proposed by Ilir Snopce[/i]

Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$ $\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D)}\ 70\sqrt2 \qquad\textbf{(E)}\ 100$

A pharmaceutical company produces a disease test that has a $95\%$ accuracy rate on individuals who actually have an infection, and a $90\%$ accuracy rate on individuals who do not have an infection. They use their test on a population of mathletes, of which $2\%$ actually have an infection. If a test concludes that a mathlete has an infection, then the probability that the mathlete actually does have an infection is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$

Can we arrange $5$ points in space to form a pentagon with equal sides such that the angle between each pair of adjacent edges is $90^o$?

$9$ chess players participate in a chess tournament. According to the regulation, each participant plays a single game with each of the others. At a certain moment of the competition it was found that exactly two participants played the same number of party. To prove that in this case, not a single chess player played any the game, or just one chess player played with everyone else.

How many subsets of the set $\{1,2,3,4,5,6,7,8,9,10\}$ are there that does not contain 4 consequtive integers? $ \textbf{(A)}\ 596 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 679 \qquad \textbf{(D)}\ 773 \qquad \textbf{(E)}\ 812$

All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list.

Prove that there are infinitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }10 \qquad $

Calculate the sum $1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?

There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true $$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$

How many different four-digit numbers can be formed by rearranging the four digits in $2004$? $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 81$