Found problems: 27
2023 Poland - Second Round, 3
Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
2006 Flanders Math Olympiad, 2
Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on $\left[AB\right]$.
$Q$ is the point on $BC$ such that $PQ$ is perpendicular to $AB$. $R$ is the point on $AC$ such that $QR$ is perpendicular to $BC$. And $S$ is the point on $AB$ such that $RS$ is perpendicular to $AC$.
$Q'$ is the point on $BC$ such that $PQ'$ is perpendicular to $BC$. $R'$ is the point on $AC$ such that $Q'R'$ is perpendicular to $AC$. And $S'$ is the point on $AB$ such that $R'S'$ is perpendicular to $AB$.
Determine $\frac{|PB|}{|AB|}$ if $S=S'$.