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

2018 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , algebra , equation
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions

2018 Junior Regional Olympiad - FBH, 1
Tags: time , equation
When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?

2017 Balkan MO, 1
Tags: equation , algebra
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

2004 Nicolae Coculescu, 1
Tags: equation , algebra
Find all pairs of integers $ (a,b) $ such that the equation $$ |x-1|+|x-a|+|x-b|=1 $$ has exactly one real solution. [i]Florian Dumitrel[/i]

1983 IMO Shortlist, 22
Tags: trigonometry , number theory , equation , trigonometric identities , trigonometric equation
Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

1985 Traian Lălescu, 1.1
Tags: equation , square root
Solve the equation $ \frac{\sqrt{2+x} +\sqrt{2-x}}{\sqrt{2+x} -\sqrt{2-x}} =\sqrt 3. $

2012 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , rational , equation
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 $x = y$.

2017 China Team Selection Test, 1
Tags: combinatorics , equation , generating functions
Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

1966 IMO Longlists, 25
Tags: trigonometry , algebra , trigonometric identities , equation
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

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

2006 Petru Moroșan-Trident, 2
Tags: diophantine equation , algebra , equation
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]

2002 India Regional Mathematical Olympiad, 2
Tags: equation
Solve for real $x$ : \[ (x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3. \]

2018 Junior Regional Olympiad - FBH, 1
Tags: costing , equation , percent
Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost?

1966 IMO Shortlist, 31
Tags: function , parameterization , algebra , equation
Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?

2002 China Girls Math Olympiad, 6
Tags: induction , number theory , equation , algebra
Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

1980 IMO Shortlist, 3
Tags: number theory , equation , algebra , diophantine equation
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2001 Estonia National Olympiad, 4
Tags: algebra , equation , absolute value
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.

1989 ITAMO, 1
Tags: algebra , equation , rational
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

2011 Belarus Team Selection Test, 2
Tags: number theory , equation , multiplication
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]

1970 IMO Shortlist, 7
Tags: modular arithmetic , number theory , digit , equation
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

2013 India IMO Training Camp, 2
Tags: modular arithmetic , number theory , equation
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

2011 District Olympiad, 1
Tags: increasing function , algebra , easy , district olympiad , equation
Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

2005 India IMO Training Camp, 2
Tags: number theory , equation , divisor
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 $n$.

2015 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , equation , algebra , diophantine equation
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers

1966 IMO Shortlist, 40
Tags: parametric equation , algebra , equation
For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]