Olympiad problems

2002 India Regional Mathematical Olympiad, 2

Tags: equation

Solve for real $x$ : \[ (x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3. \]

2010 IMO Shortlist, 1

Tags: equation , number theory , 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]

1989 IMO Longlists, 83

Tags: equation , diophantine equation , quadratic , number theory

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

1997 Croatia National Olympiad, Problem 1

Tags: equation , algebra

Let $n$ be a natural number. Solve the equation $$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$

1966 IMO Shortlist, 9

Tags: equation , trigonometry , trigonometric equation , algebra

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

2011 Dutch BxMO TST, 3

Tags: algebra , equation , absolute value

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

2012 District Olympiad, 1

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

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

2017 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: equation , root , algebra

If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: equation , radical , rational , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

1995 Denmark MO - Mohr Contest, 4

Tags: equation , algebra

Solve the equation $$(2^x-4)^3 +(4^x-2)^3=(4^x+2^x-6)^3$$ where $x$ is a real number.

2020 Abels Math Contest (Norwegian MO) Final, 3

Tags: equation , algebra , solutions , real solutions

Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .

2017 District Olympiad, 2

Tags: equation , logarithm , system

Solve in $ \mathbb{Z} $ the system: $$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$

1977 IMO Longlists, 28

Tags: equation , floor function , recurrence relation , sequence , algebra

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 \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2006 Mathematics for Its Sake, 1

Tags: algebra , equation , fractional part , floor

Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.

2018 Bundeswettbewerb Mathematik, 2

Tags: algebra , equation , real number , function , floor function

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

1969 IMO Shortlist, 17

Tags: arithmetic sequence , equation , number theory , algebra

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

2023 AMC 12/AHSME, 23

Tags: equation

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

2015 IFYM, Sozopol, 8

Tags: equation , algebra

The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations: $a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$ are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.

VII Soros Olympiad 2000 - 01, 9.5

Tags: equation , algebra , parameterization , cubic

For all valid values of $a$ and $b$, solve the equation $$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$

1978 Putnam, B4

Tags: equation , integer

Prove that for every real number $N$ the equation $$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$ has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$

1977 IMO Shortlist, 11

Tags: algebra , equation , floor function , sequence , recurrence relation

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 \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2004 IMO Shortlist, 1

Tags: equation , number theory , 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 $n$.

2016 Kosovo National Mathematical Olympiad, 2

Tags: equation

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

1997 Israel Grosman Mathematical Olympiad, 3

Tags: algebra , equation , radical

Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.

2021 EGMO, 6

Tags: equation , asymptotics , number theory , counting , algebra

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]