How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$

Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.

Let $\Gamma$ be a circumference and $A$ a point outside it. Let $B$ and $C$ be points in $\Gamma$ such that $AB$ and $AC$ are tangent to $\Gamma$. Let $P$ be a point in $\Gamma$. Let $D$, $E$ and $F$ be points in $BC$, $AC$ and $AB$ respectively, such that $PD \perp BC$, $PE \perp AC$, and $PF \perp AB$. Show that $PD^2 = PE \cdot PF$

Let $m,n,p$ be rational numbers such that $\sqrt{m} + \sqrt{n} + \sqrt{p}$ is a rational number. Prove that $\sqrt{m}, \sqrt{n}, \sqrt{p}$ are also rational numbers