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# Problems of the Week

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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

1. $$f(x)$$ is bijective;
2. $$f(x)$$ is neither non-increasing nor non-decreasing;
3. $$f\big(f(x)\big)$$ 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 $$g^\prime$$.

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