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

1961 IMO Shortlist, 3
Tags: trigonometry , algebra , equation , trigonometric equation
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

2012 Greece JBMO TST, 1
Tags: algebra , equation , system of equations
Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$

2011 Greece JBMO TST, 1
Tags: inequality , equation , algebra
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$. b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$

1954 Czech and Slovak Olympiad III A, 1
Tags: quadratic polynomial , equation , parameter , parameterization
Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.

1999 Bosnia and Herzegovina Team Selection Test, 1
Tags: algebra , root , equation , real root
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions

2017 EGMO, 5
Tags: number theory , equation
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

1957 Moscow Mathematical Olympiad, 353
Tags: algebra , floor function , equation
Solve the equation $x^3 - [x] = 3$.

1972 Bulgaria National Olympiad, Problem 2
Tags: algebra , system of equations , equation
Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$. [i]H. Lesov[/i]

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 IMO Shortlist, 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 $n$.

2019 Ramnicean Hope, 1
Tags: equation , monotone , algebra
Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/i]

2018 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: algebra , equation , floor function , frac function
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

2017 Balkan MO Shortlist, N1
Tags: algebra , equation
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

1983 Spain Mathematical Olympiad, 4
Tags: algebra , equation , parameter
Determine the number of real roots of the equation $$16x^5 - 20x^3 + 5x + m = 0.$$

2004 Unirea, 1
Tags: equation , trigonometry , algebra
Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $

1985 Traian Lălescu, 1.4
Tags: equation , algebra , floor , logarithm
Let $ a $ be a non-negative real number distinct from $ 1. $ [b]a)[/b] For which positive values $ x $ the equation $$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$ is true? [b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $

2004 Gheorghe Vranceanu, 2
Tags: equation , monotone , inequalities
Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

1969 IMO Longlists, 37
Tags: trigonometric equation , algebra , trigonometry , equation
$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

2013 Dutch Mathematical Olympiad, 4
Tags: number theory , equation , product , positive , divisor
For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$. (a) Find all positive integers $n$ satisfying $P(n) = 15n$. (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.

2016 Middle European Mathematical Olympiad, 8
Tags: equation , modular arithmetic , number theory
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

1997 Romania National Olympiad, 4
Tags: equation , floor function , algebra
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.

2023 Turkey Team Selection Test, 3
Tags: number theory , equation
For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?

2014 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: algebra , equation
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$

1995 IMO Shortlist, 2
Tags: algebra , system of equations , equation
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

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