This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 451

2023 Mexican Girls' Contest, 3
Tags: equation , algebra
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies: \begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}

1939 Moscow Mathematical Olympiad, 046
Tags: equation , algebra , radical , parameter
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.

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?

2000 District Olympiad (Hunedoara), 1
Tags: system of equations , equation , number theory , algebra
[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

2017 IMO, 6
Tags: number theory , equation , john berman more like jawnorz , polynomial
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]

2012 IMO, 6
Tags: number theory , equation
Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1. \] [i]Proposed by Dusan Djukic, Serbia[/i]

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

2001 Bosnia and Herzegovina Team Selection Test, 2
Tags: equation , positive integer , number theory
For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

2019 Teodor Topan, 1
Tags: number theory , equation , diophantine equation
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $ [i]Dragoș Crișan[/i]

1968 IMO Shortlist, 11
Tags: equation , algebra , root
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

2019 Purple Comet Problems, 15
Tags: equation , number theory
Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

2014 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: positive integer , equation , number theory
Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

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]

2016 District Olympiad, 3
Tags: equation , depressed cubic , cubic equation , cubic function , polynomial , algebra
[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution. [b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.

1989 IMO Shortlist, 9
Tags: number theory , algebra , irrational number , 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.$

2023 District Olympiad, P1
Tags: equation , algebra
Determine all real numbers $x{}$ satisfying $2^{x-1}+2^{1/\sqrt{x}}=3$.

1962 IMO, 4
Tags: trigonometry , equation , trigonometric equation , algebra
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

2015 Hanoi Open Mathematics Competitions, 8
Tags: equation , algebra
Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$

1980 IMO Longlists, 14
Tags: sequence , equation , algebra
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

1984 IMO Longlists, 13
Tags: polynomial , diophantine equation , equation , quadratic , number theory
Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

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

2014 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: equation , algebra , logarithm
Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

2016 Dutch BxMO TST, 1
Tags: equation , divisor , odd , number theory
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

2004 Nicolae Coculescu, 1
Tags: system of equations , monotone , equation , algebra
Solve in the real numbers the system: $$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$ [i]Eduard Buzdugan[/i]

2014 Regional Competition For Advanced Students, 1
Tags: equation , algebra
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .