Olympiad problems

1995 IMO Shortlist, 4

Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.

2002 Baltic Way, 1

Tags: symmetric , equation , algebra

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

2023 China Second Round, 5

Tags: algebra , trigonometry , equation , inequalities

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

1989 IMO Longlists, 3

Tags: area , equation , geometry , rectangle , algebra

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

1985 Traian Lălescu, 1.4

Tags: equation , logarithm , floor , algebra

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. $

2000 District Olympiad (Hunedoara), 2

Tags: equation , floor , iteration , algebra , number theory

[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation $$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$ has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $ [b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.

2016 Latvia National Olympiad, 3

Tags: equation , egyptian fractions , algebra

Prove that for every integer $n$ ($n > 1$) there exist two positive integers $x$ and $y$ ($x \leq y$) such that $$\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$$

2018 Hanoi Open Mathematics Competitions, 4

Tags: equation , algebra

Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$. A. $0$ B. $1$ C. $2$ D. $3$ E. $4$

2023 China Second Round, 6

Tags: algebra , equation , inequalities

Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$

1978 Germany Team Selection Test, 3

Tags: equation , recurrence relation , floor function , 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 \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

1968 IMO Shortlist, 15

Tags: equation , number theory , summation , floor function

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]

2012 USA Team Selection Test, 3

Tags: equation , linear algebra , matrix , function , number theory

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: equation , algebra , floor function , frac function

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

2006 Estonia National Olympiad, 4

Tags: floor function , equation , algebra

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

2004 Thailand Mathematical Olympiad, 4

Tags: equation , radical , algebra

Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$

1967 IMO Shortlist, 3

Tags: equation , algebra , summation , diophantine equation , polynomial

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 $S(x,n) = 0.$

1966 IMO Shortlist, 12

Tags: equation , decimal representation , number theory

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

1995 IMO Shortlist, 4

2002 Baltic Way, 1

2023 China Second Round, 5

1989 IMO Longlists, 3

1985 Traian Lălescu, 1.4

2000 District Olympiad (Hunedoara), 2

2016 Latvia National Olympiad, 3

2018 Hanoi Open Mathematics Competitions, 4

2023 China Second Round, 6

1978 Germany Team Selection Test, 3

1968 IMO Shortlist, 15

2012 USA Team Selection Test, 3

2018 Bosnia And Herzegovina - Regional Olympiad, 3

2006 Estonia National Olympiad, 4

2004 Thailand Mathematical Olympiad, 4

1967 IMO Shortlist, 3

1966 IMO Shortlist, 12

2021 Malaysia IMONST 1, 18

2010 IMO Shortlist, 2

2020 Middle European Mathematical Olympiad, 4#

1957 Moscow Mathematical Olympiad, 353

2019 Lusophon Mathematical Olympiad, 4

2017 Balkan MO, 1

1966 IMO Shortlist, 48

1970 IMO Longlists, 59