# 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 $\{1,2,\dots,n\}$, $n\geq2$, and $p$ a positive integer with $2p\leq n+1$. Find the number of $p$ element subsets $T$ ($p$-subsets) of $S$ with the property that $T$ contains no consecutive numbers. (Proof)
• For $n \geq 2$, $\sin(\frac{\pi}n)\cdot \sin(2\cdot\frac{\pi}n)\cdot\dots\cdot \sin((n-1)\cdot\frac{\pi}n) = \frac n{2^{n-1}}$. (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 $a'\equiv a\pmod c$ and $b'\equiv b\pmod c$ with $\gcd(a',b')=1$. (Solution)