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

1998 Croatia National Olympiad, Problem 1
Tags: complex number , algebra , equation
Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.

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

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

2007 Mathematics for Its Sake, 2
Tags: equation , system of equations , algebra
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones. [i]Mugur Acu[/i]

2017 Junior Regional Olympiad - FBH, 4
Tags: equation , remainder
If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

2017 Junior Regional Olympiad - FBH, 2
Tags: milk , equation
In three cisterns of milk lies $780$ litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern $\frac{3}{7}$ of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?

2006 Petru Moroșan-Trident, 2
Tags: equation , system of equations , algebra , monotone
Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

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]

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]

2005 Germany Team Selection Test, 1
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 $n$.

2012 India IMO Training Camp, 1
Tags: algebra , equation , sequence
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

2016 District Olympiad, 2
Tags: equation , algebra , perfect square , number theory , fractional part
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.

2019 Lusophon Mathematical Olympiad, 4
Tags: algebra , equation
Find all the real numbers $a$ and $b$ that satisfy the relation $2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$

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

1991 Denmark MO - Mohr Contest, 4
Tags: equation , algebra
Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.

2022 Thailand Online MO, 1
Tags: algebra , equation
Determine, with proof, all triples of real numbers $(x,y,z)$ satisfying the equations $$x^3+y+z=x+y^3+z=x+y+z^3=-xyz.$$

1966 IMO Shortlist, 48
Tags: algebra , equation , number theory , root , parametric equation
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2018 VJIMC, 1
Tags: equation , power , real number
Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

1989 IMO Longlists, 22
Tags: algebra , irrational number , number theory , equation
$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

2002 IMO Shortlist, 4
Tags: algebra , number theory , equation , infinitely many solutions , vieta jumping , pell equation
Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

2016 District Olympiad, 1
Tags: equation , diophantine equation , algebra
Solve in $ \mathbb{N}^2: $ $$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$

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]

2018 Ramnicean Hope, 2
Tags: equation , monotone , trigonometry , algebra
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

2004 Thailand Mathematical Olympiad, 5
Tags: algebra , sum , equation , radical
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =| 2nx - n - 2n^2|$

1966 IMO Shortlist, 9
Tags: trigonometric equation , trigonometry , equation , algebra
Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.