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

1989 IMO Shortlist, 4
Tags: algebra , equation , polynomial , diophantine equation , rational roots
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

1971 Bulgaria National Olympiad, Problem 2
Tags: algebra , equation
Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

2019 EGMO, 1
Tags: algebra , equation
Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and $$a^2b + c = b^2c + a = c^2a + b.$$

2005 Estonia National Olympiad, 4
Tags: equation , algebra , trinomial
Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

2023 AMC 12/AHSME, 23
Tags: equation
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\] $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{an infinite number}$

1997 Akdeniz University MO, 3
Tags: equation , number theory , sequence
$(x_n)$ be a sequence with $x_1=0$, $$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$. Prove that for $k \geq 2$ $x_k$ is a natural number.

2012 Bogdan Stan, 3
Tags: algebra , equation
Find the real numbers $ x,y,z $ that satisfy the following: $ \text{(i)} -2\le x\le y\le z $ $ \text{(ii)} x+y+z=2/3 $ $ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $ [i]Cristinel Mortici[/i]

1997 Romania National Olympiad, 4
Tags: floor function , algebra , equation
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$ If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.

2011 Dutch BxMO TST, 3
Tags: absolute value , equation , algebra
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

2017 Mathematical Talent Reward Programme, MCQ: P 1
Tags: equation , algebra
The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]

2023 Bulgaria EGMO TST, 5
Tags: bound , algebra , equation , calculate
The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

2016 District Olympiad, 3
Tags: algebra , polynomial , equation , depressed cubic , cubic equation , cubic function
[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.

2013 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , equation
Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

1984 IMO Shortlist, 11
Tags: number theory , equation , polynomial , diophantine equation
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

1954 Moscow Mathematical Olympiad, 272
Tags: trigonometry , equation , algebra
Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.

2023 China Northern MO, 3
Tags: floor function , algebra , equation
Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

2009 Belarus Team Selection Test, 3
Tags: algebra , number theory , equation
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

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

2012 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: equation , parameterization , algebra
Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter

2004 Nicolae Coculescu, 1
Tags: equation , system of equations , monotone , 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]

2006 Belarusian National Olympiad, 2
Tags: algebra , equation
Find all triples $(x, y,z)$ such that $x, y, z \in (0,1)$ and $$\left(x+\frac{1}{2x}-1\right) \left(y+\frac{1}{2y}-1\right) \left(z+\frac{1}{2z}-1\right) = \left(1-\frac{xy}{z}\right)\left(1-\frac{yz}{x}\right)\left(1-\frac{zx}{y}\right)$$ (D. Bazylev)

1980 IMO Longlists, 12
Tags: number theory , diophantine equation , algebra , equation , polynomial
Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

1996 Tuymaada Olympiad, 5
Tags: algebra , radical , equation
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

2009 Romania National Olympiad, 1
Tags: unique solution , algebra , equation , function
[b]a)[/b] Show that two real numbers $ x,y>1 $ chosen so that $ x^y=y^x, $ are equal or there exists a positive real number $ m\neq 1 $ such that $ x=m^{\frac{1}{m-1}} $ and $ y=m^{\frac{m}{m-1}} . $ [b]b)[/b] Solve in $ \left( 1,\infty \right)^2 $ the equation: $ x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} . $

1960 Putnam, A1
Tags: integer , equation
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$\frac{xy}{x+y}=n?$$