Olympiad problems

1951 Moscow Mathematical Olympiad, 197

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

2023 Bulgaria National Olympiad, 4

Tags: area , convex quadrilateral , geometry

Prove that there exists a unique point $M$ on the side $AD$ of a convex quadrilateral $ABCD$ such that \[\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}\] if and only if $AB\parallel CD$.

2013 Stanford Mathematics Tournament, 4

Tags:

What is the smallest number over 9000 that is divisible by the first four primes?

2016 Israel Team Selection Test, 4

Tags: combinatorics

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?

2018 AMC 12/AHSME, 12

Tags: geometry , angle bisector

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? $ \textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20 \qquad $

2009 Stanford Mathematics Tournament, 9

Tags: parabola , calculus , algebra , polynomial , derivative , quadratic , conic

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

2015 Hanoi Open Mathematics Competitions, 13

Tags: algebra , rational , equation

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

2012 District Olympiad, 1

Tags: limit , real analysis , sequence

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

1998 Romania Team Selection Test, 3

Tags: inequalities , induction , divisibility theory

Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

2014 Online Math Open Problems, 27

Tags: ratio , probability , expected value , number theory , relatively prime , algebra

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]