Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$. On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen. The lines $AL$ and $BK$ intersect at point $R$, which isn't on $XY$. The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$. The lines $BM$ and $CL$ intersect at point $P$, which isn't on $YZ$. The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$. The lines $CK$ and $AM$ intersect at point $Q$, which isn't on $ZX$. The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$. Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$. Prove that $RR_1=RR_2$.

The diagonals of a trapezoid $ABCD$ with $AD \parallel BC$ intersect at point $P$. Point $Q$ lies between the parallel lines $AD$ and $BC$ such that the line $CD$ separates points $P$ and $Q$, and $\angle AQD=\angle CQB$. Prove that $\angle BQP = \angle DAQ$.

A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number of cubic feet of dirt needed to form the pile can be written as $m + n \pi$ where $m$ and $n$ are positive integers. Find $m + n.$

Triangle $\triangle PQR$, with $PQ=PR=5$ and $QR=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{QR}$ which is tangent to both $\omega$ and $\overline{PQ}$.

Let $n$ be a positive integer. Consider the sum $x_1y_1 + x_2y_2 +\cdots + x_ny_n$, where that values of the variables $x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n$ are either 0 or 1. Let $I(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum of the number is odd, and let $P(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum is an even number. Show that: \[ \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1} \]

Given a positive integer $n\ge 3$, Arándano and Banana play a game. Initially, numbers $1,2,3,\dots,n$ are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise. Determine, in terms of $n$, who has a winning strategy.

Let $A$ be a $n \times n$ matrix of real numbers such that $A^2+\mathbb{I}=A$, where $\mathbb{I}$ is the identity $n \times n$ matrix. Prove that the matrix $A^{3n}$ , where $\nu\in\mathbb{Z}$ takes only two values and find those values.

Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent. (N. Beluhov)

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

Compute the number of dates in the year $2023$ such that when put in $\text{MM / DD / YY}$ form, the three numbers are in strictly increasing order. For example, $06/18/23$ is such a date since $6 < 18 < 23,$ while today, $11/11/23,$ is not.

Do there exist positive integers $a_1<a_2<\ldots<a_{100}$ such that for $2\le k\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is greater than the least common multiple of $a_k$ and $a_{k+1}$?

Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$. Prove the inequality \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB\equal{}52$, $ BC\equal{}12$, $ CD\equal{}39$, and $ DA\equal{}5$. The area of $ ABCD$ is [asy] pair A,B,C,D; A=(0,0); B=(52,0); C=(38,20); D=(5,20); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--D--cycle); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,N); label("52",(A+B)/2,S); label("39",(C+D)/2,N); label("12",(B+C)/2,E); label("5",(D+A)/2,W);[/asy] $ \text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260$

Find all pairs of positive integers $(n, k)$ such that all sufficiently large odd positive integers $m$ are representable as $$m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}$$ for some non-negative integers $a_1, a_2, \ldots, a_{k+1}$.

Prove that there exist (a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones; (b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.

Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.

A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?

A analityc function $f:\mathbb{C}\to\mathbb{C}$ is call interesting if $f(z)$ is real along the parabola $Re (z) = (Im (z))^2$. a) Find an example of a non constant interesting function. b) Show that every interesting function $f$ satisfy that $f'(-3/4) = 0.$

A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight $ L$ is placed in the left pan and another of weight $ R$ in the right pan, the pointer stops at the number $ R \minus{} L$ on the graduated ruler. There are $ n, (n \geq 2)$ bags of coins, each containing $ \frac{n(n\minus{}1)}{2} \plus{} 1$ coins. All coins look the same (shape, color, and so on). $ n\minus{}1$ bags contain real coins, all with the same weight. The other bag (we don’t know which one it is) contains false coins. All false coins have the same weight, and this weight is different from the weight of the real coins. A legal weighing consists of placing a certain number of coins in one of the pans, putting a certain number of coins in the other pan, and reading the number given by the pointer in the graduated ruler. With just two legal weighings it is possible to identify the bag containing false coins. Find a way to do this and explain it.

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and \[2f(m)f(n)-f(n-m)-1\] is a perfect square for all integer $m,n.$

Let $n\ge5$ be an integer. A trapezoid with base lengths of $1$ and $r$ is tiled by $n$ (not necessarily congruent) equilateral triangles. In terms of $n$, find the maximum possible value of $r$. [i]Linus Tang[/i]

Charlotte the cat is placed at the origin of the coordinate plane, such that the positive $x$ direction is pointing east, and the positive $y$ direction is pointing north. Then, every single lattice square (a unit square with vertices all on lattice points) in the first quadrant whose southwest vertex is a lattice point with both odd coordinates is completely removed. Charlotte can traverse the coordinate plane by drawing a segment between two valid vertices, given that they do not intersect a lattice square that has not been removed. Let $P_1$ and $P_2$ denote the distances of the first and second shortest paths Charlotte can take to $(5,7),$ respectively. Find $P_1-P_2.$ [i]Tiebreaker #4[/i]

Prove that among any $18$ consecutive positive integers not exceeding $2005$ there is at least one divisible by the sum of its digits.