Olympiad problems

2006 Cezar Ivănescu, 1

Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

2017 Balkan MO Shortlist, N1

Tags: algebra , equation

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

2017 Junior Regional Olympiad - FBH, 4

Tags: equation , remainder

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

2011 Brazil Team Selection Test, 2

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]

2012 IMO Shortlist, N4

Tags: modular arithmetic , number theory , equation

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

2003 Gheorghe Vranceanu, 4

Tags: set equation , equation , set theory , algebra

Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $

1985 Traian Lălescu, 1.4

Tags: algebra , floor , logarithm , equation

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. $

2004 German National Olympiad, 1

Tags: system of equations , equation , real , algebra

Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}

1989 IMO Longlists, 3

Tags: algebra , geometry , area , rectangle , equation

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?

2017 IMO, 6

Tags: number theory , polynomial , equation , john berman more like jawnorz

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

1966 IMO Longlists, 29

Tags: number theory , summation , equation

A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

2001 Bosnia and Herzegovina Team Selection Test, 2

Tags: positive integer , number theory , equation

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

2011 Greece JBMO TST, 1

Tags: inequality , equation , algebra

a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$. b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$

1995 IMO Shortlist, 4

Tags: algebra , number theory , equation , greatest common divisor

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

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: floor function , floor , algebra , monotone functions , equation , trigonometric equation

Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $ [i]Petre Guțescu[/i]

2011 Dutch BxMO TST, 3

Tags: absolute value , equation , algebra

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

1978 Germany Team Selection Test, 3

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

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 Gheorghe Vranceanu, 2

Tags: inequalities , equation , monotone

Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

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

2019 LIMIT Category A, Problem 10

Tags: equation , algebra

Number of solutions of the equation $3^x+4^x=8^x$ in reals is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~\infty$

1957 Czech and Slovak Olympiad III A, 1

Tags: equation , parametric equation , algebra , parameterization

Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.

1967 IMO Shortlist, 1

Tags: algebra , equation , root , exponential equation

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

2019 Ramnicean Hope, 1

Tags: algebra , equation , monotone

Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/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 $x+y = 0$.

1989 IMO Shortlist, 2

Tags: algebra , geometry , area , rectangle , equation

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?