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: 451

1966 IMO Longlists, 34
Tags: number theory , equation
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

1989 IMO Shortlist, 3
Tags: rectangle , calculus , algebra , equation , area
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

1991 IMO Shortlist, 17
Tags: algebra , modular arithmetic , number theory , equation
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

1984 IMO Longlists, 43
Tags: equation , divisibility , power of 2 , number theory
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1978 Romania Team Selection Test, 4
Tags: trigonometry , algebra , trigonometric equation , equation
Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $

2017 IMO Shortlist, N7
Tags: equation , john berman more like jawnorz , polynomial , number theory
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

1997 IMO Shortlist, 17
Tags: equation , number theory , prime factorization
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

1963 IMO, 1
Tags: algebra , equation , parametric equation
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

2024 Bangladesh Mathematical Olympiad, P3
Tags: algebra , equation
Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?

1999 Bosnia and Herzegovina Team Selection Test, 1
Tags: algebra , root , equation , real root
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions

2009 Moldova National Olympiad, 12.3
Tags: trigonometry , trigonometric equation , equation , algebra
Find all pairs $(a,b)$ of real numbers, so that $\sin(2009x)+\sin(ax)+\sin(bx)=0$ holds for any $x\in \mathbf {R}$.

1957 Czech and Slovak Olympiad III A, 1
Tags: equation , parameterization , algebra , parametric equation
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

2013 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: trigonometry , algebra , equation
Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$

1960 Putnam, B1
Tags: integer , equation , number theory
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

1974 All Soviet Union Mathematical Olympiad, 194
Tags: algebra , equation
Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.

2004 Nicolae Coculescu, 2
Tags: algebra , floor function , equation
Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

1994 Swedish Mathematical Competition, 1
Tags: algebra , equation , digit
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?

2017 Junior Regional Olympiad - FBH, 3
Tags: equation , square root
Find all real numbers $x$ such that: $$ \sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}$$

2017 Vietnamese Southern Summer School contest, Problem 2
Tags: root , equation , polynomial , algebra
Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

2023 China Northern MO, 3
Tags: algebra , floor function , equation
Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

2002 All-Russian Olympiad, 1
Tags: functional equation , algebra , equation , polynomial , quadratic , ring theory
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

2009 Belarus Team Selection Test, 3
Tags: algebra , equation , number theory
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

1979 Vietnam National Olympiad, 5
Tags: algebra , integer part , equation
Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.

2012 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , equation , rational
Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.

2002 All-Russian Olympiad, 1
Tags: functional equation , algebra , polynomial , quadratic , ring theory , equation
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.