This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2018 Taiwan TST Round 2, 1
Tags: inequalities , Taiwan
Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that \[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]

2016 Math Prize for Girls Problems, 9
Tags: Math Prize for Girls
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.)

1998 Miklós Schweitzer, 2
Tags: polynomial , real analysis
For any polynomial f, denote by $P_f$ the number of integers n for which f(n) is a (positive) prime number. Let $q_d = max P_f$ , where f runs over all polynomials with integer coefficients with degree d and reducible over $\mathbb{Q}$. Prove that $\forall d\geq 2$ , $q_d = d$.

2023 MMATHS, 11
Tags: Yale , MMATHS
Suppose we have sequences $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ and the function $f(x)=\tfrac{1}{x}$ such that for all $n$ we have: [list] [*]$a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n))$ [*]$a_{n+2} = f(1-a_n) - f(1+a_n)$ [*]$b_{n+2} = f(1-b_n) - f(1+b_n)$ [/list] Given that $a_0=\tfrac{1}{6}$ and $b_0=\tfrac{1}{7},$ then $b_5=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the sum of the prime factors of $mn.$

2004 Indonesia MO, 4
Tags:
There exists 4 circles, $ a,b,c,d$, such that $ a$ is tangent to both $ b$ and $ d$, $ b$ is tangent to both $ a$ and $ c$, $ c$ is both tangent to $ b$ and $ d$, and $ d$ is both tangent to $ a$ and $ c$. Show that all these tangent points are located on a circle.

the 11th XMO, 3
Tags: prime numbers , number theory
Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

2017 ELMO Shortlist, 1
Tags: combinatorics
Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$, and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$. For each $\{m,n\}$, determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$, or prove that no such $k$ exists. [i]Proposed by Michael Ren

2014 Dutch IMO TST, 1
Tags: number theory proposed , number theory
Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

1961 Putnam, B2
Tags: Putnam , probability , lines
Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$

2008 Serbia National Math Olympiad, 4
Tags: geometry , circumcircle , search , combinatorics proposed , combinatorics
Each point of a plane is painted in one of three colors. Show that there exists a triangle such that: $ (i)$ all three vertices of the triangle are of the same color; $ (ii)$ the radius of the circumcircle of the triangle is $ 2008$; $ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.

1985 IMO Longlists, 44
Tags: analytic geometry , trigonometry , geometry , coordinate geometry , IMO Shortlist
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?

1998 Brazil Team Selection Test, Problem 2
Tags: number theory , ratio
There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

2007 Junior Balkan Team Selection Tests - Romania, 3
Tags: graph theory , combinatorics proposed , combinatorics
At a party there are eight guests, and each participant can't talk with at most three persons. Prove that we can group the persons in four pairs such that in every pair a conversation can take place.

MOAA Team Rounds, 2021.18
Tags: MOAA 2021 , team
Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

2021 Lusophon Mathematical Olympiad, 3
Tags: geometry , circumcircle
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$. Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.

2009 Harvard-MIT Mathematics Tournament, 8
Tags:
There are $5$ students on a team for a math competition. The math competition has $5$ subject tests. Each student on the team must choose $2$ distinct tests, and each test must be taken by exactly two people. In how many ways can this be done?

1976 Euclid, 3
Tags: geometry , circumcircle
Source: 1976 Euclid Part B Problem 3 ----- $I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.

2016 Macedonia JBMO TST, 3
Tags: JMMO , Junior , 2016 , Macedonia , combinatorics
We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?

2001 Regional Competition For Advanced Students, 1
Tags: number theory , Digit , Sum of powers , Sum , powers
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

2024 VJIMC, 1
Tags: inequalities , function , exponential , calculus
Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

2013 Today's Calculation Of Integral, 881
Tags: calculus , integration , trigonometry , calculus computations , Definite integral
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2006 Pre-Preparation Course Examination, 3
Tags: function , real analysis , real analysis unsolved
Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$. Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.

2008 Indonesia MO, 4
Tags: function , algebra , functional equation , algebra proposed
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$

2014-2015 SDML (High School), 9
Tags:
What is the smallest number of queens that can be placed on an $8\times8$ chess board so that every square is either occupied or can be reached in one move? (A queen can be moved any number of unoccupied squares in a straight line vertically, horizontally, or diagonally.) $\text{(A) }4\qquad\text{(B) }5\qquad\text{(C) }6\qquad\text{(D) }7\qquad\text{(E) }8$

2010 Saudi Arabia IMO TST, 2
Tags: ratio , geometry , equal angles
Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$