Olympiad problems

1956 Putnam, A6

i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances. ii) Show that the corresponding statement for the line is false.

Let $x, y, z$ non-zero real numbers such that $xy$, $yz$, $zx$ are rational. [list=a] [*] Show that the number $x^{2}+y^{2}+z^{2}$ is rational. [*] If the number $x^{3}+y^{3}+z^{3}$ is also rational, show that $x$, $y$, $z$ are rational. [/list]

PEN F Problems, 15

Tags: rational number , trigonometry , algebra , polynomial

Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.

2022 Korea Junior Math Olympiad, 8

Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

2006 MOP Homework, 1

Tags: set , rational , number theory , strong induction , rational number

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 $(0, 1)$.

PEN F Problems, 11

Tags: rational number

Let $S=\{x_0, x_1, \cdots, x_n\} \subset [0,1]$ be a finite set of real numbers with $x_{0}=0$ and $x_{1}=1$, such that every distance between pairs of elements occurs at least twice, except for the distance $1$. Prove that all of the $x_i$ are rational.

2019 India IMO Training Camp, P1

Tags: rational number , set , algebra

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

PEN F Problems, 13

Tags: rational number , factorial

Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

PEN F Problems, 8

Tags: rational number , algebra , polynomial , analytic geometry , function , induction

Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.

PEN F Problems, 16