This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 106

2020 India National Olympiad, 5
Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon. (a) Show that $3, 4, 6$ are frameable. (b) Show that any integer $n \geqslant 7$ is not frameable. (c) Determine whether $5$ is frameable. [i]Proposed by Muralidharan[/i]
trigonometry rational galois theory tiling niven theorem descent number theory bijection complex numbers construction coordinate method geometry mathematical induction modular arithmetic

2002 Junior Balkan Team Selection Tests - Romania, 1
Let $a$ be an integer. Prove that for any real number $x, x^3 < 3$, both the numbers $\sqrt{3 -x^2}$ and $\sqrt{a - x^3}$ cannot be rational.
radical rational irrational algebra inequalities

2006 MOP Homework, 1
Let $S$ be a set of rational numbers with the following properties: (a) $\frac12$ is an element in $S$, (b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$. Prove that $S$ contains all rational numbers in the interval $(0, 1)$.
rational rational number strong induction set number theory Euclidean algorithm mathematical induction

2018 Danube Mathematical Competition, 3
Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.
rational positive integer number theory sum product

2012 Junior Balkan Team Selection Tests - Romania, 2
Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.
rational equation algebra casework number theory

2015 Poland - Second Round, 1
Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.
rational irrational number polynomial sum algebra polynomials