Olympiad problems

1983 IMO Shortlist, 22

Tags: equation , trigonometric identities , trigonometric equation , trigonometry , number theory

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

2023 Romania National Olympiad, 2

Tags: equation , number theory , algebra

Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations: \begin{align*} a^2 + a = b + c, \\ b^2 + b = a + c, \\ c^2 + c = a + b. \end{align*}

2013 Poland - Second Round, 4

Tags: equation , algebra

Solve equation $(x^4 + 3y^2)\sqrt{|x + 2| + |y|}=4|xy^2|$ in real numbers $x$, $y$.

2011 Morocco TST, 1

Tags: equation , lte lemma , modular arithmetic , number theory

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2003 Estonia National Olympiad, 2

Tags: logarithm , equation , algebra

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

2002 IMO Shortlist, 4

Tags: infinitely many solutions , vieta jumping , pell equation , equation , algebra , number theory

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

1980 IMO, 2

Tags: sequence , equation , algebra

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

2004 Unirea, 1

Tags: equation , trigonometry , algebra

Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , parameterization , equation

Solve equation $$x^2-\sqrt{a-x}=a$$ where $x$ is real number and $a$ is real parameter

1992 IMO Longlists, 68

Tags: trigonometry , equation , algebra , relatively prime , number theory

Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

1965 German National Olympiad, 1

Tags: parameter , equation , algebra

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

2006 Petru Moroșan-Trident, 2

Tags: algebra , diophantine equation , equation

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

1980 IMO Shortlist, 14

Tags: algebra , equation , sequence

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

1992 IMO Longlists, 16

Tags: equation , number theory

Find all triples $(x, y, z)$ of integers such that \[\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23\]

1969 IMO Longlists, 17

Tags: equation , arithmetic sequence , 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 $p.$

2018 Polish Junior MO First Round, 6

Tags: equation , number theory , algebra

Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.

1990 India National Olympiad, 1

Tags: algebra , equation

Given the equation \[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\] has four real, positive roots, prove that (a) $ pr \minus{} 16s \geq 0$ (b) $ q^2 \minus{} 36s \geq 0$ with equality in each case holding if and only if the four roots are equal.

2010 Laurențiu Panaitopol, Tulcea, 1

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

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

1953 Moscow Mathematical Olympiad, 256

Tags: factorial , equation , algebra

Find roots of the equation $$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$

2011 Belarus Team Selection Test, 2

Tags: equation , lte lemma , number theory , modular arithmetic

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2011 Bosnia and Herzegovina Junior BMO TST, 1

Tags: equation , positive integer , number theory

Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

1984 IMO Shortlist, 10

Tags: equation , number theory , product , perfect square

Prove that the product of five consecutive positive integers cannot be the square of an integer.

2021 Turkey Team Selection Test, 6

Tags: equation , algebra , construction , n-variable equation

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

1966 IMO Longlists, 29

Tags: equation , summation , number theory

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.

1941 Moscow Mathematical Olympiad, 079

Tags: absolute value , equation , algebra

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.