Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

The dates of three Sundays of a month were even numbers. What day of the week was the $20$th of the month?

Two angles of an isosceles triangle measure $ 70^\circ$ and $ x^\circ$. What is the sum of the three possible values of $ x$? $ \textbf{(A)}\ 95 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 180$

Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$. The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$. Show that the segments $AY$ and $BX$ have equal length.

In a country with $2015$ cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which $k$ flights belong to the same airline, what is the maximum value of $k$?

Let $ABC$ be an acute-angled triangle with $BC > AB$, such that the points $A$, $H$, $I$ and $C$ are concyclic (where $H$ is the orthocenter and $I$ is the incenter of triangle $ABC$). The line $AC$ intersects the circumcircle of triangle $BHC$ at point $T$, and the line $BC$ intersects the circumcircle of triangle $AHC$ at point $P$. If the lines $PT$ and $HI$ are parallel, determine the measures of the angles of triangle $ABC$.

Call a $2$-by-$2$ grid a [I]perfectly perfect square[/I] if it contains distinct positive integers such that the sum of each row is a perfect square and the sum of each column is a perfect square. Define $f(n)$ to be the number of perfectly perfect squares whose entries sum to $n.$ Let $m$ be the smallest integer such that $f(m) > m.$ Find $f(m).$

Find all positive integers $N$ having only prime divisors $2,5$ such that $N+25$ is a perfect square.

Prove that every positive integer has a multiple whose decimal representation involves all ten digits.

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$