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SomeElementaryProblems

Filed in: MathWiki.SomeElementaryProblems · Modified on : Wed, 19 Jun 13

  • 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)

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