Olympiad problems

2013 IMC, 2

Let $\displaystyle{f:{\cal R} \to {\cal R}}$ be a twice differentiable function. Suppose $\displaystyle{f\left( 0 \right) = 0}$. Prove there exists $\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)}$ such that \[\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.\] [i]Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.[/i]

1993 AMC 12/AHSME, 3

Tags: AMC , Bad Latex

$\frac{15^{30}}{45^{15}}=$ $ \textbf{(A)}\ \left(\frac{1}{3}\right)^{15} \qquad\textbf{(B)}\ \left(\frac{1}{3}\right)^2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 3^{15} \qquad\textbf{(E)}\ 5^{15}$

2004 Harvard-MIT Mathematics Tournament, 1

Tags: inequalities , Bad Latex

How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities? \begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}

2013 F = Ma, 19

Tags: trigonometry , Bad Latex

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. At what angle $\theta_g$ during the swing is the tension in the rod the greatest? $\textbf{(A) } \text{The tension is greatest at } \theta_g = \theta_0.\\ \textbf{(B) } \text{The tension is greatest at }\theta_g = 0.\\ \textbf{(C) } \text{The tension is greatest at an angle } 0 < \theta_g < \theta_0.\\ \textbf{(D) } \text{The tension is constant.}\\ \textbf{(E) } \text{None of the above is true for all values of } \theta_0 \text{ with } 0 < \theta_{0} < \frac{\pi}{2}$