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2017-09-11 Advanced


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. 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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