Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.

Which of the following are true? $\textbf{(A)}~\text{For every }n\in\mathbb N,n^3-n\text{ is divisible by }6$ $\textbf{(B)}~\text{For every }n\in\mathbb N,n^7-n\text{ is divisible by }42$ $\textbf{(C)}~\text{Every perfect square is of the form }3m\text{ or }3m+1\text{ for some }n\in\mathbb N$ $\textbf{(D)}~\text{None of the above}$

Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

Suppose that $a_1 = 1,$ $a_2 = 2$, and for any $n \ge 3$, $a_n = a_1 + a_2 + \cdots + a_{n-1}$. Find $\frac{a_{2021}}{a_{2020}}$. [i]Proposed by Andrew Wu[/i]

Let $a$ and $b$ be two positive integers satisfying the equation \[ 20\sqrt{12} = a\sqrt{b}. \] Compute the sum of all possible distinct products $ab$. [i]Proposed by Lewis Chen[/i]

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer)

Let $n$ be a positive integer. Determine the maximum number of positive integers less than or equal to $n^2$ that can be colored red in such a way that if $a$ and $b$ are red, with $a \neq b$, then $a \cdot b$ is [b]not[/b] red.

Let $[a,b]$ be a given interval of real numbers not containing integers. Prove that there exists $N>0$ such that $[Na,Nb]$ does not contain integer numbers and the length of the interval $[Na,Nb]$ exceedes $\dfrac 16$.

Let $H$ and $O$ be the orthocenter and circumcenter of an acute-angled triangle $ABC$, respectively. The perpendicular bisector of $BH$ meets $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. Prove that $OB$ bisects the angle $A_1OC_1$.

In the right triangle $ABC,$ $CF$ is the altitude based on the hypotenuse $AB.$ The circle centered at $B$ and passing through $F$ and the circle with centre $A$ and the same radius intersect at a point of $CB.$ Determine the ratio $FB : BC.$

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.

Let $p$ be a prime such that $3|p+1$. Show that $p|a-b$ if and only if $p|a^3-b^3$

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer. (a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$. (b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$

Let $ABC$ be a triangle with circumcircle $w_1$ and incenter $I$. Suppose $w_2$ is a circle tangent to $AB,AC$ at $X,Y$, and internally tangent to $w$ at $D$. Let the parallel to the exterior angle bisector of $A$ through $D$ meet $w_2$ at $P$. Show that $AP, DI$ intersect on $w_2$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]