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

1984 IMO, 3
Tags: number theory , equation , divisibility , power of 2
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 $a=1$.

1989 IMO Shortlist, 25
Tags: number theory , diophantine equation , quadratic , equation
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2004 Gheorghe Vranceanu, 3
Tags: function , equation , inversability , algebra
Consider the function $ f:(-\infty,1]\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,& \quad x<-1 \\ 3^{\sqrt{1-x^2}} ,& \quad x\in [-1,1] \end{matrix} \right. . $$ [b]a)[/b] For a fixed parameter, find the roots of $ f-m. $ [b]b)[/b] Study the inversability of the restrictions of $ f $ to $ (-\infty,-1] $ and $ [-1,1] $ and find the inverses of these that admit them. [i]D. Zaharia[/i]

2014 IMO Shortlist, N2
Tags: number theory , equation
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2019 Irish Math Olympiad, 4
Tags: algebra , equation
Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy $$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$

2021 Malaysia IMONST 1, 18
Tags: algebra , equation
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?

2015 Hanoi Open Mathematics Competitions, 3
Tags: algebra , equation
Suppose that $a > b > c > 1$. One of solutions of the equation $\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$ is (A): $-1$, (B): $-2$, (C): $0$, (D): $1$, (E): None of the above.

2016 India PRMO, 2
Tags: algebra , factorial , equation , diophantine , diophantine equation , wilson
Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

1967 Czech and Slovak Olympiad III A, 1
Tags: algebra , equation , complex number
Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

2009 Germany Team Selection Test, 1
Tags: algebra , number theory , equation
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]

1978 Putnam, B4
Tags: integer , equation
Prove that for every real number $N$ the equation $$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$ has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$

2021 Hong Kong TST, 1
Tags: am-gm , equation , algebra
Find all real triples $(a,b,c)$ satisfying \[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]

1996 Akdeniz University MO, 1
Tags: equation , number theory
Solve the equation for real numbers $x,y,z$ $$(x-y+z)^2=x^2-y^2+z^2$$

2012 District Olympiad, 1
Tags: equation , algebra , fractional part , integer part , floor
Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

2003 Estonia National Olympiad, 2
Tags: equation , algebra , logarithm
Solve the equation $\sqrt{x} = \log_2 x$.

1995 Denmark MO - Mohr Contest, 4
Tags: algebra , equation
Solve the equation $$(2^x-4)^3 +(4^x-2)^3=(4^x+2^x-6)^3$$ where $x$ is a real number.

2004 Estonia National Olympiad, 1
Tags: equation , algebra
Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

1988 Bundeswettbewerb Mathematik, 4
Tags: number theory , diophantine equation , equation , algebra
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

1997 Akdeniz University MO, 1
Tags: equation , number theory
Prove that, $$15x^2-7y^2=9$$ equation has any solutions in integers.

2011 India IMO Training Camp, 2
Tags: modular arithmetic , number theory , equation , lte lemma
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2004 Nicolae Coculescu, 2
Tags: floor function , equation , algebra
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]

1966 IMO Longlists, 29
Tags: number theory , summation , equation
A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

1997 IMO, 5
Tags: equation , prime factorization , number theory
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2016 Kazakhstan National Olympiad, 2
Tags: algebra , equation
Find all rational numbers $a$,for which there exist infinitely many positive rational numbers $q$ such that the equation $[x^a].{x^a}=q$ has no solution in rational numbers.(A.Vasiliev)

2006 Petru Moroșan-Trident, 2
Tags: equation , diophantine equation , algebra
Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]