The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$.

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

Find all integer solutions to $m^2=n^6+1$.

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]