MathDB

2

Part of 2013 IMC

Problems(2)

Imc 2013/1/2

Source: Imc 2013 Problem 2

8/8/2013
Let f:R→R\displaystyle{f:{\cal R} \to {\cal R}} be a twice differentiable function. Suppose f(0)=0\displaystyle{f\left( 0 \right) = 0}. Prove there exists ξ∈(−π2,π2)\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)} such that f′′(ξ)=f(ξ)(1+2tan⁡2ξ).\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.
Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.
trigonometryIMCcollege contestsBad Latex
IMC 2013/2/2

Source: Imc 2013 Problem 7

8/9/2013
Let p,q\displaystyle{p,q} be relatively prime positive integers. Prove that ∑k=0pq−1(−1)⌊kp⌋+⌊kq⌋={0 if pq is even1if pq odd\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}
Proposed by Alexander Bolbot, State University, Novosibirsk.
floor functionPutnamnumber theoryrelatively primeIMCcollege contests