Olympiad problems

2022 Math Prize for Girls Olympiad, 1

Tags: MP4G 2022 Olympiad

Let $a$, $b$, $c$ be positive integers with $a \le 10$. Suppose the parabola $y = ax^2 + bx + c$ meets the $x$-axis at two distinct points $A$ and $B$. Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$.

2022 Math Prize for Girls Olympiad, 3

Tags: MP4G 2022 Olympiad

Serena has written 20 copies of the number 1 on a board. In a move, she is allowed to $\quad *$ erase two of the numbers and replace them with their sum, or $\quad *$ erase one number and replace it with its reciprocal. Whenever a fraction appears on the board, Serena writes it in simplest form. Prove that Serena can never write a fraction less than 1 whose numerator is over 9000, regardless of the number of moves she makes.

2022 Math Prize for Girls Olympiad, 2

Tags: MP4G 2022 Olympiad , mew

Determine, with proof, whether or not there exists a [i]non-isosceles[/i] trapezoid $ABCD$ such that the lengths $AC$ and $BD$ both lie in the set $\{ DA+AB, AB+BC, BC+CD, CD+DA, AB+CD, BC+DA \}$.

2022 Math Prize for Girls Olympiad, 4

Tags: MP4G 2022 Olympiad

Let $n > 1$ be an integer. Let $A$ denote the set of divisors of $n$ that are less than $\sqrt n$. Let $B$ denote the set of divisors of $n$ that are greater than $\sqrt n$. Prove that there exists a bijective function $f \colon A \to B$ such that $a$ divides $f(a)$ for all $a \in A$. (We say $f$ is [i]bijective[/i] if for every $b \in B$ there exists a unique $a \in A$ with $f(a) = b$.)