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.

AND:
OR:
NO:

Found problems: 85335

2018 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: number theory , coprime , coprime numbers , sum of powers , divide
Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2007 Postal Coaching, 6
Tags: digit , number theory , divide , divisible
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

2019 BMT Spring, 6
Tags: probability , combinatorics
At a party, $2019$ people decide to form teams of three. To do so, each turn, every person not on a team points to one other person at random. If three people point to each other (that is, $A$ points to $B$, $B$ points to $C$, and $C$ points to $A$), then they form a team. What is the probability that after $65, 536$ turns, exactly one person is not on a team

1990 IMO Longlists, 89
Tags: function , algebra , domain , combinatorics
Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

2009 Indonesia TST, 2
Tags: combinatorics
Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.

2008 ITest, 74
Tags: analytic geometry
Points $C$ and $D$ lie on opposite sides of line $\overline{AB}$. Let $M$ and $N$ be the centroids of $\triangle ABC$ and $\triangle ABD$ respectively. If $AB=841$, $BC=840$, $AC=41$, $AD=609$, and $BD=580$, find the sum of the numerator and denominator of the value of $MN$ when expressed as a fraction in lowest terms.

2005 IMO Shortlist, 1
Tags: coefficient , algebra , polynomial , root , quadratic
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

2017 China Northern MO, 1
Tags: sequence , inequalities
A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1,\) \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}\). Prove that, for all \(n \geq 1\), \(a_1 + a_2 + ... + a_n < \frac{34}{21}\).

2020 European Mathematical Cup, 4
Tags: algebra , functional equation , function
Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

2006 Germany Team Selection Test, 3
Tags: modular arithmetic , number theory
Is the following statement true? For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.

2016 NIMO Summer Contest, 7
Tags:
Suppose that $a$ and $b$ are real numbers such that $\sin(a)+\sin(b)=1$ and $\cos(a)+\cos(b)=\frac{3}{2}$. If the value of $\cos(a-b)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, determine $100m+n$. [i]Proposed by Michael Ren[/i]

2022 District Olympiad, P1
Tags: algebra , logarithm
Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

2015 Geolympiad Spring, 2
Tags:
Let $ABC$ be a triangle and $w$ its incircle. $w$ touches $BC,CA$ at $A_1,B_1$ respectively. The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$. Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.

2019 Purple Comet Problems, 8
Tags: geometry
The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]

2009 Poland - Second Round, 2
Tags: combinatorics
Find all integer numbers $n\ge 4$ which satisfy the following condition: from every $n$ different $3$-element subsets of $n$-element set it is possible to choose $2$ subsets, which have exactly one element in common.

1982 All Soviet Union Mathematical Olympiad, 327
Tags: geometry , circles , area
Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .

2010 Purple Comet Problems, 15
Tags: number theory , relatively prime
Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions $\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime $\star$ all other pairs of the five integers [b]are[/b] relatively prime.

1955 Miklós Schweitzer, 8
Tags: 3d geometry , tetrahedron , geometry
[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]

2012 Albania National Olympiad, 3
Tags: invariant , arithmetic sequence , algebra
Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression \[\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)\] does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.

2009 ITAMO, 2
Tags: circumcircle , projective geometry , geometry
Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.

MOAA Gunga Bowls, 2021.19
Tags:
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\] can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2024 Nigerian MO Round 2, Problem 6
Tags: number theory
Create $2024$ by $2024$ grid of integers numbered from $1$ to $2024^2$ such that the integer in row $n$ and column $k$ is $2024(n-1)+k$. Let the sum of the numbers in the grid be $S$. Arthur picks 2024 numbers such that their sum is $\dfrac{S}{2024}$. Prove that in every case, using the remaining numbers, Bob can create 2023 sets of 2024 numbers with equal numbers. For clarification, All numbers can only be used once

2019 Kosovo National Mathematical Olympiad, 4
Tags: number theory
Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.

2011 Iran Team Selection Test, 11
Tags: vector , geometry , geometric transformation , homothety
Let $ABC$ be a triangle and $A',B',C'$ be the midpoints of $BC,CA,AB$ respectively. Let $P$ and $P'$ be points in plane such that $PA=P'A',PB=P'B',PC=P'C'$. Prove that all $PP'$ pass through a fixed point.

2018 VJIMC, 1
Tags: geometry , rectangle , coloring , combinatorics
Every point of the rectangle $R=[0,4] \times [0,40]$ is coloured using one of four colours. Show that there exist four points in $R$ with the same colour that form a rectangle having integer side lengths.