Olympiad problems

1970 IMO Shortlist, 7

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

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]

1966 IMO Shortlist, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

1950 Moscow Mathematical Olympiad, 180

Tags: equation , radical , algebra

Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.

1963 IMO, 1

Tags: algebra , equation , parametric equation

Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

2000 District Olympiad (Hunedoara), 1

Tags: equation , system of equations , number theory , algebra

[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

2004 Nicolae Coculescu, 3

Tags: equation , matrices , linear algebra , cayley-hamilton

Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

1984 IMO Longlists, 43

Tags: number theory , equation , divisibility , power of 2

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1998 Tuymaada Olympiad, 2

Tags: algebra , radical , equation

Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$

2011 Belarus Team Selection Test, 2

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

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]

1979 Romania Team Selection Tests, 6.

Tags: modular arithmetic , algebra , number theory , equation

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

2008 IMO Shortlist, 1

Tags: algebra , number theory , equation

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

1993 Moldova Team Selection Test, 4

Tags: equation

Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$.

1980 IMO Longlists, 14

Tags: algebra , sequence , equation

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 $x_r - x_s = k.$

2007 Nicolae Coculescu, 1

Tags: equation , floor , fractional part , algebra

Let be two real numbers $ x,y, $ and a natural number $ n_0 $ such that $ \{ n_0x \} = \{ n_0y \} $ and $ \{ (n_0+1)x \} = \{ (n_0+1)y \} ,$ where $ \{\} $ denotes the fractional part. Show that $ \{ nx \} =\{ ny \} , $ for any natural number $ n. $ [i]Ovidiu Pop[/i]

2018 Bundeswettbewerb Mathematik, 2

Tags: real number , algebra , equation , floor function , function

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

1992 IMO Longlists, 16

Tags: number theory , equation

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

1966 IMO Longlists, 25

Tags: trigonometry , algebra , trigonometric identities , equation

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

2009 Swedish Mathematical Competition, 2

Tags: algebra , equation

Find all real solutions of the equation \[ \left(1+x^2\right)\left(1+x^3\right)\left(1+x^5\right)=8x^5 \]

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 $p$.

1980 IMO, 3

Tags: number theory , equation , algebra , diophantine equation

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2002 All-Russian Olympiad, 1

Tags: polynomial , quadratic , ring theory , equation , functional equation , algebra

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

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 $PQRS$).

1994 Swedish Mathematical Competition, 1

Tags: digit , equation , algebra

$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?

2019 Purple Comet Problems, 15

Tags: equation , number theory

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.