Olympiad problems

2011 Dutch BxMO TST, 3

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

1994 IMO Shortlist, 5

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-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]

2016 Junior Regional Olympiad - FBH, 4

Tags: number theory , equation

In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$

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.

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 $x\in \mathbf {R}$.

1966 IMO Longlists, 40

Tags: algebra , equation , parametric equation

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

2003 Estonia National Olympiad, 2

Tags: floor function , equation , algebra

Find all positive integers $n$ such that $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$

2015 Germany Team Selection Test, 1

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]

1997 Israel Grosman Mathematical Olympiad, 3

Tags: radical , equation , algebra

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

2014 Contests, 1

Tags: algebra , equation

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

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 $$\frac{xy}{x+y}=n?$$

1980 IMO Longlists, 12

Tags: number theory , diophantine equation , algebra , equation , polynomial

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2018 Junior Regional Olympiad - FBH, 1

Tags: time , equation

When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?

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$.

1980 IMO, 19

Tags: diophantine equation , algebra , equation , polynomial , number theory

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: polynomial , root , equation , algebra

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

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 $2$.

2018 Junior Regional Olympiad - FBH, 1

Tags: buying , splitting money , equation

Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?

2005 India IMO Training Camp, 2

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 $n$.

1993 Swedish Mathematical Competition, 4

Tags: algebra , equation

To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$. Solve the equation $x*36 = 216$.

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]

1994 All-Russian Olympiad, 1

Tags: algebra , radical , equation

Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.

1996 IMO Shortlist, 4

Tags: floor function , number theory , equation , algebra

Find all positive integers $ a$ and $ b$ for which \[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]

1988 IMO Shortlist, 22

Tags: number theory , relatively prime , quadratic , equation

Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \] [b]OR[/b] Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]