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

1949-56 Chisinau City MO, 53
Tags: radical , algebra , equation
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$

2000 Romania National Olympiad, 4
Tags: equation , fermat , ring theory , superior algebra , finite ring , abstract algebra , fermat s last theorem
Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $

1968 IMO, 2
Tags: number theory , decimal representation , digit , equation
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2017 Baltic Way, 4
Tags: algebra , equation , polynomial
A linear form in $k$ variables is an expression of the form $P(x_1,...,x_k)=a_1x_1+...+a_kx_k$ with real constants $a_1,...,a_k$. Prove that there exist a positive integer $n$ and linear forms $P_1,...,P_n$ in $2017$ variables such that the equation $$x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}$$ holds for all real numbers $x_1,...,x_{2017}$.

1987 All Soviet Union Mathematical Olympiad, 460
Tags: rotation , function , plot , equation
The plot of the $y=f(x)$ function, being rotated by the (right) angle around the $(0,0)$ point is not changed. a) Prove that the equation $f(x)=x$ has the unique solution. b) Give an example of such a function.

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

2016 Kosovo National Mathematical Olympiad, 4
Tags: equation
Solve equation in real numbers $\log_{2}(4^x+4)=x+\log_{2}(2^{x+1}-3)$

2010 JBMO Shortlist, 2
Tags: algebra , number theory , equation
[b]Determine all four digit numbers [/b]$\bar{a}\bar{b}\bar{c}\bar{d}$[b] such that[/b] $$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}$$

1971 IMO Longlists, 31
Tags: equation , root , polynomial , algebra
Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

1986 Traian Lălescu, 2.1
Tags: equation , algebra , parameterization , parametric equation
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.

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]

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. $

2016 Azerbaijan IMO TST First Round, 3
Tags: equation
Find the solution of the equation $8x(2x^2-1)(8x^4-8x^2+1)=1$ in the interval $[0,1]$?

1984 IMO Shortlist, 16
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 $a=1$.

1984 IMO Longlists, 25
Tags: number theory , equation , product , perfect square
Prove that the product of five consecutive positive integers cannot be the square of an integer.

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

1967 IMO Shortlist, 1
Tags: algebra , equation , root , exponential equation
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

1984 IMO Shortlist, 10
Tags: number theory , equation , product , perfect square
Prove that the product of five consecutive positive integers cannot be the square of an integer.

1965 Czech and Slovak Olympiad III A, 3
Tags: algebra , parameter , radical , equation , real root
Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.

2017 Germany, Landesrunde - Grade 11/12, 1
Tags: algebra , polynomial , equation
Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.

1978 Germany Team Selection Test, 3
Tags: algebra , recurrence relation , sequence , equation , floor function
Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2023 Romania National Olympiad, 2
Tags: algebra , equation , number theory
Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations: \begin{align*} a^2 + a = b + c, \\ b^2 + b = a + c, \\ c^2 + c = a + b. \end{align*}

2017 Junior Regional Olympiad - FBH, 5
Tags: equation
Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

2006 Petru Moroșan-Trident, 1
Tags: floor function , algebra , equation
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $ [i]Constantin Nicolau[/i]

2012 IMO Shortlist, N4
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.