Let $ A_1,A_2,...$ be a sequence of infinite sets such that $ |A_i \cap A_j| \leq 2$ for $ i \not\equal{}j$. Show that the sequence of indices can be divided into two disjoint sequences $ i_1<i_2<...$ and $ j_1<j_2<...$ in such a way that, for some sets $ E$ and $ F$, $ |A_{i_n} \cap E|\equal{}1$ and $ |A_{j_n} \cap F|\equal{}1$ for $ n\equal{}1,2,... .$ [i]P. Erdos[/i]

Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many $5$-digit numbers does Kelvin like?

Given that \[\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3\] for certain real numbers $a,b,c,d$, determine the value of \[\frac{a-d}{b-c}.\]

Prove that a positive integer can be expressed in the form $3x^2+y^2$ iff it can also be expressed in form $u^2+uv+v^2$, where $x,y,u,v$ are all positive integers.

Chords $A_1A_2, A_3A_4, A_5A_6$ of a circle $\Omega$ concur at point $O$. Let $B_i$ be the second common point of $\Omega$ and the circle with diameter $OA_i$ . Prove that chords $B_1B_2, B_3B_4, B_5B_6$ concur.

Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if (i) $a_n < a_{n+1}$ for all $n\ge 1$, (ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.

Let $f(n)$ be a function from integers to integers. Suppose $f(11) = 1$, and $f(a)f(b) = f(a +b) + f(a - b)$, for all integers $a, b$. Find $f(2013)$.

A circus act consists of $2024$ bamboo sticks of pairwise different heights placed in some order, with a monkey standing atop one of them. The circus master can then give commands to the monkey as follows: [color=#FFFFFF]___[/color]$\bullet$ Left! : When given this command, the monkey locates the closest bamboo stick to the left taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. [color=#FFFFFF]___[/color]$\bullet$ Right! : When given this command, the monkey locates the closest bamboo stick to the right taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. The circus master claims that given any two bamboo sticks, if the monkey is originally atop the shorter stick, then after giving at most $c$ commands he can reposition the monkey atop the taller stick. What is the smallest possible value of $c$? [i]Proposed by Archit Manas[/i]

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Suppose that $c\in\left(\frac{1}{2},1\right)$. Find the least $M$ such that for every integer $n\ge 2$ and real numbers $0<a_1\le a_2\le\ldots \le a_n$, if $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$, then we always have that $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$ where $m=[cn]$

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be.

$n$ people live in a town, and they are members of some clubs (residents can be members of more than one club). No matter how we choose some (but at least one) clubs, there is a resident of the town who is the member of an odd number of the chosen clubs. Prove that the number of clubs is at most $n$. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$ [i]Proposed by Kyle Lee[/i]

Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.

Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation $$x^3 + ax^2 + bx + c = 0$$ has three real roots creating an arithmetic progression.

Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score. Let $p$ be the probability that his score is a multiple of 13. There exists relatively prime positive integers $a$ and $b,$ with $b$ as small as possible, such that $|p - a/b| < 10^{-10}.$ What is $a/b?$ [i]Proposed by Dilhan Salgado, Daniel Li[/i]

Let $n$ be a positive integer and $d(n)$ is the number of positive integer divisors of $n$. For every two positive integer divisor $x,y$ of $n$, the remainders when $x,y$ divided by $d(n)+1$ are pairwise distinct. Show that either $d(n)+1$ is equal to prime or $4$.

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied \[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\] Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$

Let $\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}$ be two infinite sequences of real numbers, such that $\varepsilon_i \in \{-1,1\}$ for all $i$, and the numbers $\theta_i$ obey$$\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1$$and $\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}$. Compute the sum of all possible values of $$\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)$$ [i]Proposed by Grant Yu[/i]