SomeElementaryProblems

Consider a quadrangle with two perpendicular diagonals of the same (unknown) length. Assume that the lengths of the sides are known. Find the area of the quadrangle. (Proof)
Let $S$ be the set ${1, 2, \dots, n}$ , $n \geq 2$ , and $p$ a positive integer with $2 p \leq n + 1$ . Find the number of $p$ element subsets $T$ ( $p$ -subsets) of $S$ with the property that $T$ contains no consecutive numbers. (Proof)
For $n \geq 2$ , $\sin (\frac{π}{n}) \cdot \sin (2 \cdot \frac{π}{n}) \cdot \dots \cdot \sin ((n - 1) \cdot \frac{π}{n}) = \frac{n}{2^{n - 1}}$ . (Proof)
Find an uncountable collection of subsets of integers such that the intersection of any two distinct members of the collection is either empty or finite. (Solution)
A nontrivial Hermitian singular circulant matrix. (Solution)
Let $a$ , $b$ and $c$ be positive integers with $gcd (a, b, c) = 1$ . Find $a^{'}$ and $b^{'}$ with $a^{'} \equiv a (\mod c)$ and $b^{'} \equiv b (\mod c)$ with $gcd (a^{'}, b^{'}) = 1$ . (Solution)