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

2012 USA Team Selection Test, 3
Tags: function , linear algebra , matrix , number theory , equation
Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

2011 Croatia Team Selection Test, 4
Tags: number theory , relatively prime , diophantine equation , equation
Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

1991 Austrian-Polish Competition, 1
Tags: algebra , equation , austrian polish , combination
Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.

1998 Junior Balkan MO, 3
Tags: induction , algebra , number theory , equation
Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

2015 German National Olympiad, 1
Tags: system of equations , algebra , equation
Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10,\\ y^3+xy^2 &=2. \end{align*}

1992 IMO Longlists, 68
Tags: trigonometry , number theory , relatively prime , algebra , equation
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

1977 Chisinau City MO, 135
Tags: algebra , equation
Solve the equation: $$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$

2016 Macedonia National Olympiad, Problem 1
Tags: equation , number theory
Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$

2010 Laurențiu Panaitopol, Tulcea, 1
Tags: quadratic , algebra , equation
Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

2000 Tournament Of Towns, 4
Tags: even , algebra , equation
Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation $$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$ for all values of $x$ for which the lefthand side of the equation makes sense. (a) Prove that $n$ is even. (b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist? (M Skopenko)

2021 EGMO, 6
Tags: number theory , asymptotics , algebra , equation , counting
Does there exist a nonnegative integer $a$ for which the equation \[\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a\] has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers? [i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]

2016 District Olympiad, 1
Tags: equation , algebra , logarithm
Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$

2008 Hanoi Open Mathematics Competitions, 4
Tags: number theory , equation , relatively prime , algebra
Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation \[x^3-nx+mn=0\] has three distint integer roots.

1978 Putnam, B4
Tags: equation , integer
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 $N.$

1966 IMO Longlists, 10
Tags: trigonometry , algebra , equation , trigonometric equation
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

1986 Traian Lălescu, 1.1
Tags: system of equations , algebra , equation
Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$

2015 District Olympiad, 3
Tags: equation , complex number , algebra
Solve in $ \mathbb{C} $ the following equation: $ |z|+|z-5i|=|z-2i|+|z-3i|. $

2016 Middle European Mathematical Olympiad, 8
Tags: equation , modular arithmetic , number theory
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

2015 Belarus Team Selection Test, 2
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]

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.

2011 Bosnia and Herzegovina Junior BMO TST, 1
Tags: positive integer , number theory , equation
Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

2004 IMO Shortlist, 1
Tags: equation , divisor , number theory
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

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

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)

2016 District Olympiad, 2
Tags: perfect square , number theory , fractional part , equation , algebra
If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $ 1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.