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

1976 Dutch Mathematical Olympiad, 4
Tags: algebra , parameter , parameterization , equation , trinomial
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?

2017 Romania National Olympiad, 1
Tags: algebra , fractional part , integer part , floor , logarithm , equation
Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

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

2009 ISI B.Math Entrance Exam, 2
Tags: algebra , equation , calculus
Let $c$ be a fixed real number. Show that a root of the equation \[x(x+1)(x+2)\cdots(x+2009)=c\] can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.

2007 Mathematics for Its Sake, 2
Tags: system of equations , algebra , equation
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]

1980 IMO Longlists, 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.

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 $\mathbb{R}$.

2013 IFYM, Sozopol, 8
Tags: number theory , equation , set
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ : $[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$. Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?

2004 Nicolae Coculescu, 3
Tags: matrices , linear algebra , cayley-hamilton , equation
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

2020 Iran MO (3rd Round), 4
Tags: polynomial , equation , algebra
We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that $$P(x)=\sum_{}{x^{n_i}}+1$$ Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.

1980 All Soviet Union Mathematical Olympiad, 301
Tags: floor function , algebra , equation
Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)

1982 IMO Shortlist, 16
Tags: number theory , equation , diophantine equation , divisibility
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

1980 IMO Shortlist, 14
Tags: algebra , sequence , equation
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 $x_r - x_s = k.$

1994 IMO, 3
Tags: function , number theory , equation , binary representation
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

2021 Malaysia IMONST 1, 18
Tags: algebra , equation
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?

2010 IMO Shortlist, 1
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]

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*}

2000 Denmark MO - Mohr Contest, 5
Tags: algebra , polynomial , equation
Determine all possible values of $x+\frac{1}{x}$ , where the real number $x$ satisfies the equation $$x^4+5x^3-4x^2+5x+1=0$$ and solve this equation.

2015 Hanoi Open Mathematics Competitions, 3
Tags: algebra , equation
Suppose that $a > b > c > 1$. One of solutions of the equation $\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$ is (A): $-1$, (B): $-2$, (C): $0$, (D): $1$, (E): None of the above.

2021-IMOC, A1
Tags: algebra , equation
Find all real numbers x that satisfies$$\sqrt{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}+\sqrt{1-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}=x.$$ [url=https://artofproblemsolving.com/community/c6h2645263p22889979]2021 IMOC Problems[/url]

2002 All-Russian Olympiad, 1
Tags: algebra , polynomial , quadratic , ring theory , equation , functional equation
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

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

2009 Moldova National Olympiad, 12.3
Tags: trigonometry , algebra , equation , trigonometric equation
Find all pairs $(a,b)$ of real numbers, so that $\sin(2009x)+\sin(ax)+\sin(bx)=0$ holds for any $x\in \mathbf {R}$.

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]

1949-56 Chisinau City MO, 54
Tags: algebra , equation
Solve the equation: $$\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)$$