Olympiad problems

2009 Brazil Team Selection Test, 1

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]

1949 Moscow Mathematical Olympiad, 161

Tags: radical , equation , algebra , parameter

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

1991 Denmark MO - Mohr Contest, 4

Tags: equation , algebra

Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.

1977 IMO Longlists, 28

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 \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

1954 Czech and Slovak Olympiad III A, 1

Tags: quadratic polynomial , equation , parameter , parameterization

Solve the equation $$ax^2+2(a-1)x+a-5=0$$ in real numbers with respect to (real) parametr $a$.

1998 Tuymaada Olympiad, 2

Tags: algebra , radical , equation

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

1994 IMO Shortlist, 5

Tags: function , number theory , equation , binary representation

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

1999 Moldova Team Selection Test, 5

Tags: equation

Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.

1963 IMO Shortlist, 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.

1969 IMO Longlists, 17

Tags: arithmetic sequence , algebra , number theory , equation

$(CZS 6)$ 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 $p.$

2019 Danube Mathematical Competition, 1

Tags: diophantine equation , algebra , equation

Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ [i]Lucian Petrescu[/i]

1989 IMO Shortlist, 5

Tags: algebra , polynomial , root , equation , diophantine equation

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2015 Ukraine Team Selection Test, 10

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

1998 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , diophantine equation , equation

Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ [i]Adrian Zanoschi[/i]

2025 Bangladesh Mathematical Olympiad, P2

Tags: equation , algebra , exponential equation

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

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 $a=1$.

1974 All Soviet Union Mathematical Olympiad, 194

Tags: algebra , equation

Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.

VII Soros Olympiad 2000 - 01, 9.5

Tags: parameterization , algebra , cubic , equation

For all valid values of $a$ and $b$, solve the equation $$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$

1986 IMO Shortlist, 4

Tags: number theory , diophantine equation , equation , algebra

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.