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

2012 Greece JBMO TST, 2
Tags: number theory , least common multiple , coprime , diophantine equation
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

KoMaL A Problems 2021/2022, A. 810
Tags: algebra , combinatorics , series
For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$

2020 Putnam, A1
Tags:
How many positive integers $N$ satisfy all of the following three conditions?\\ (i) $N$ is divisible by $2020$.\\ (ii) $N$ has at most $2020$ decimal digits.\\ (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.

2017 Iran MO (3rd round), 1
Tags: algebra , functional equation
Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

MOAA Gunga Bowls, 2023.4
Tags:
An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

2018 BMT Spring, 4
Tags: number theory
What is the remainder when $201820182018... $ [$2018$ times] is divided by $15$?

2009 Balkan MO Shortlist, A8
Tags:
For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} [list=a] [*] Prove that $K_m \geq 0$ for every odd positive integer $m$ [*] Let $M$ $= \prod_{cyc} (x-y)^2$. Prove, $K_7+M^2 K_1 \geq M K_4$

2014 USAJMO, 4
Tags: function , inequalities , pigeonhole principle
Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

2002 German National Olympiad, 1
Tags: algebra , system of equations
Find all real numbers $a,b$ satisfying the following system of equations \begin{align*} 2a^2 -2ab+b^2 &=a\\ 4a^2 -5ab +2b^2 & =b. \end{align*}

2018 Hanoi Open Mathematics Competitions, 7
Tags: pentagon , geometry
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

2018 Malaysia National Olympiad, A2
Tags: prime number , number theory
Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

1970 Putnam, B6
Tags: geometry , area , quadrilateral
Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

1990 AMC 12/AHSME, 7
Tags: calculus , integration , geometry , perimeter
A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

2012 Online Math Open Problems, 19
Tags: geometry , trapezoid , relatively prime , number theory
In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

2011 Princeton University Math Competition, A1 / B5
Tags: algebra
A polynomial $p$ can be written as \begin{align*} p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d. \end{align*} Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.

2011 Greece Team Selection Test, 3
Tags: function , functional equation , algebra
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold: $$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$ $$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$ for all $x,y \in \mathbb{Q}$.

2017 Dutch BxMO TST, 5
Tags: number theory
Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

2011 Brazil Team Selection Test, 1
Tags: geometry , circumcircle , geometric transformation , reflection
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

Kvant 2023, M2741
Tags: combinatorics
Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

2012 Junior Balkan Team Selection Tests - Moldova, 3
Tags: angle bisector , perpendicular bisector , geometry
Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.

2014 Purple Comet Problems, 15
Tags: function
Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.

1962 AMC 12/AHSME, 27
Tags:
Let $ a @ b$ represent the operation on two numbers, $ a$ and $ b$, which selects the larger of the two numbers, with $ a@a \equal{} a.$ Let $ a ! b$ represent the operator which selects the smaller of the two numbers, with $ a ! a \equal{} a.$ Which of the following three rules is (are) correct? $ \textbf{(1)}\ a@b \equal{} b@a \qquad \textbf{(2)}\ a@(b@c) \equal{} (a@b)@c \qquad \textbf{(3)}\ a ! (b@c) \equal{} (a ! b) @ (a ! c)$ $ \textbf{(A)}\ (1)\text{ only} \qquad \textbf{(B)}\ (2) \text{ only} \qquad \textbf{(C)}\ \text{(1) and (2) only} \qquad \textbf{(D)}\ \text{(1) and (3) only} \qquad \textbf{(E)}\ \text{all three}$

2007 Peru Iberoamerican Team Selection Test, P2
Tags: number theory
Find all positive integer solutions of the equation $n^5+n^4=7^{m}-1$

2006 Victor Vâlcovici, 1
Tags: monotone , function , real analysis , ftc , definite integral , calculus , integration
Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]

2007 Swedish Mathematical Competition, 6
Tags: equal area , geometry , area
In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.