Olympiad problems

1967 IMO Shortlist, 3

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

2017 Romania National Olympiad, 1

Tags: equation , algebra , fractional part , integer part , floor , logarithm

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]

1983 IMO Shortlist, 22

Tags: trigonometric equation , trigonometry , number theory , equation , trigonometric identities

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

2023 China Second Round, 5

Tags: trigonometry , algebra , inequalities , equation

Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$

2018 Junior Regional Olympiad - FBH, 1

Tags: costing , equation , percent

Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost?

2016 Kosovo National Mathematical Olympiad, 2

Tags: equation

Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .

1983 IMO Longlists, 63

Tags: trigonometric equation , trigonometry , number theory , equation , trigonometric identities

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

1990 India National Olympiad, 1

Tags: algebra , equation

Given the equation \[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\] has four real, positive roots, prove that (a) $ pr \minus{} 16s \geq 0$ (b) $ q^2 \minus{} 36s \geq 0$ with equality in each case holding if and only if the four roots are equal.

2004 Nicolae Coculescu, 1

Tags: equation , system of equations , algebra , monotone

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]

2022 Moldova EGMO TST, 1

Tags: equation

Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

2025 Bangladesh Mathematical Olympiad, P2

Tags: equation , algebra , exponential equation

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

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}} . $

1999 Moldova Team Selection Test, 5

Tags: equation

Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.

2018 Dutch BxMO TST, 5

Tags: algebra , equation

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

1997 IMO Shortlist, 17

Tags: number theory , equation , prime factorization

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

1996 Spain Mathematical Olympiad, 4

Tags: algebra , equation , radical

For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.

1987 USAMO, 1

Tags: algebra , diophantine equation , equation

Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

2023 Moldova EGMO TST, 5

Tags: equation , system of equations

Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$

2000 Moldova National Olympiad, Problem 5

Tags: algebra , equation

Solve in real numbers the equation $$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$

2018 Polish Junior MO First Round, 6

Tags: number theory , equation , algebra

Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.

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]

2011 IMO Shortlist, 2

Tags: algebra , equation , sequence

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

1966 IMO Longlists, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

1970 IMO Shortlist, 5

Tags: geometry , 3d geometry , tetrahedron , volume , equation

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

1982 IMO, 1

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 $n=2891$.