Olympiad problems

2013 IFYM, Sozopol, 6

For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?

1966 IMO Shortlist, 10

Tags: trigonometry , algebra , equation , trigonometric equation

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

1978 Vietnam National Olympiad, 1

Tags: number theory , digit , equation

Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.

2004 Gheorghe Vranceanu, 4

Tags: complex number , algebra , equation

Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $

1980 IMO, 19

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

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

2013 Bogdan Stan, 4

Tags: equation , algebra , logarithm

Solve in the real numbers the equation $ 3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1. $ [i]Ion Gușatu[/i]

2016 India National Olympiad, P5

Tags: geometry , inradius , trigonometry , equation

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

2003 Estonia National Olympiad, 2

Tags: equation , algebra , logarithm

Solve the equation $\sqrt{x} = \log_2 x$.

1984 IMO Shortlist, 2

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

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

1997 Croatia National Olympiad, Problem 1

Tags: algebra , equation

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

1970 IMO Longlists, 20

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 $PQRS$).

1957 Moscow Mathematical Olympiad, 353

Tags: algebra , floor function , equation

Solve the equation $x^3 - [x] = 3$.

1958 Czech and Slovak Olympiad III A, 1

Tags: equation

Find all real solutions of equation $x + \sqrt{2p - x^2} = 8$ with real parameter $p$.

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]

2013 IFYM, Sozopol, 7

Tags: algebra , equation

Let $a,b,c,$ and $d$ be real numbers and $k\geq l\geq m$ and $p\geq q\geq r$. Prove that $f(x)=a(x+1)^k (x+2)^p+b(x+1)^l (x+2)^q+c(x+1)^m (x+2)^r-d=0$ has no more than 14 positive roots.

2006 Petru Moroșan-Trident, 1

Tags: floor , algebra , equation , monotone

Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $ [i]Nedelcu Ion[/i]

1968 IMO Shortlist, 15

Tags: floor function , number theory , equation , summation

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

1989 IMO Shortlist, 9

Tags: algebra , irrational number , number theory , equation

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

2005 Germany Team Selection Test, 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 $n$.

2006 Mathematics for Its Sake, 1

Tags: fractional part , algebra , floor , equation

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

1965 German National Olympiad, 1

Tags: parameter , equation , algebra

For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.

2013 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , trigonometry

Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$

2017 Junior Regional Olympiad - FBH, 5

Tags: equation

Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

1968 IMO Shortlist, 22

Tags: number theory , decimal representation , digit , equation

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , positive integer , equation

Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers