2017-09-11 Advanced Practice Problems Online

In an $8 \times 8$ grid of points, what is the maximum number of points that we can select such that no four selected points are the corners of a rectangle whose sides are parallel to the edges of the grid?

If the three blue points are selected, then the red point cannot be selected.

Consider a uniform rod of mass $M$ and length $L,$ free to rotate around a frictionless axis passing through its center and going into the page. Initially, the rod is stationary in the horizontal position, as shown in the diagram below.

Now, a small bullet of mass $m$ moving with velocity $v$ hits the rod at its extreme end and sticks to it. The system rotates vertically through some angle $\theta$ before it momentarily comes to rest. If this angle can be expressed (in degrees) as $\theta = \alpha + \arcsin \left(\frac {\beta mv^2}{( M+\gamma m)gL}\right),$ where $g$ denotes the gravitational acceleration and $\alpha$ , $\beta$ , and $\gamma$ are positive integer constants with $\alpha$ in degrees, then find the value of $\alpha + \beta + \gamma$ .

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ which satisfies

$f(x)$ is bijective;
$f(x)$ is neither non-increasing nor non-decreasing;
$f\big(f(x)\big)$ is non-decreasing?

Let $\{a_{n}\}$ be a sequence of real numbers satisfying $\begin{cases} a_{0}=1 \\ a_{n+1}=\sqrt{4+3a_{n}+a_{n}^{2}}-2 & \text{for } n \ge 0. \end{cases}$ Let $\displaystyle S=\sum_{n=0}^{\infty} a_{n}$ .

If $S$ converges, submit your answer as $\big\lfloor 100S \big\rfloor$ .
If $S$ diverges, submit your answer as $-1$ .

This problem is based on a recent Putnam contest problem.

A ball with mass $m$ is thrown from the origin at speed $V_{0}$ toward the right on an exotic planet where the strength of gravity is $g^\prime = \frac{g}{10} = \SI[per-mode=symbol]{1}{\meter\per\second\squared}.$

Let $\alpha$ be the largest possible angle such that, for all $\theta<\alpha$ , the distance between the ball and its launch point will be strictly increasing for $t>0$ .

What is $\tan^{2} \alpha,$ to two decimal places?

Details and Assumptions:

$V_{0}=100 \text{ m/s}.$
$m=1 \text{ kg}.$

Bonus: Generalize this angle for arbitrary values of $V_{0}$ , $m$ , and $g^\prime$ .

Problems of the Week

2017-09-11 Advanced