2018-01-08 Advanced

         

Consider the equation \[x^2+y^2=3z^2.\] Are there any other integer solutions besides the solution where \(x=y=z=0?\)

A liquid is kept in a cylindrical vessel, which is being rotated about its axis. The liquid rises at the side, as shown in the diagram.

If the radius of the vessel is \(0.05\text{ m}\) and the speed of rotation is \(2\) revolutions per second, find the difference in the heights of the liquid at the center of the vessel and its sides (in centimeters).

Take \(g = \pi^2 \text{ m/s}^2.\)

Find the number of pairs of positive integers \((a,b)\) with \(1\leq a < b \leq 100\) such that

there is at least one positive integer \(m\) with \(a<m<b\) such that \(m\) is divisible by every common divisor of \(a\) and \(b.\)

\[ \large { \sqrt[30]{20x +\sqrt[30]{20x + \sqrt[30]{20x + 17}}} = 17 } \]

Let \(N\) be the sum of all the real solutions to the above equation. If \(N = \dfrac{a^b-a}{c},\) where \(a,\) \(b,\) and \(c\) are positive integers and \(a\) and \(c\) are coprime, then what is \(a+b+c?\)

A guitar string of length \(l\) stretched along the \(x\)-axis is plucked in the middle. The initial deflection of the string has the shape \[ u(x, t = 0) = u_0(x) = \begin{cases} \frac{2 A_0 x}l & x < \frac l2 \\\\ \frac{2 A_0 (l - x)}l & x \geq \frac l2, \end{cases} \] where \(A_0\) is in the middle (see diagram). The string is then released and swings freely for time \(t > 0,\) and the resulting string vibration can be written as a superposition of standing waves \[u(x, t) = \sum_{n = 1}^\infty A_n \cos(2 \pi f_n t) \sin\left( \frac{n\pi}{l} x \right),\] with eigenfrequencies \(f_n = n \cdot \nu\) and fundamental frequency \(\nu\).

What is the relative amount of vibrational energy stored in the fundamental mode \(f = \nu?\) To do this, you'll have to determine the amplitudes \(A_n\).

Hint:

  • The set of sine functions has the property \[ \int_0^l \sin\left( \frac{n\pi}{l} x \right) \sin\left( \frac{m\pi}{l} x \right) dx = \begin{cases}\frac l2 & n = m \\ 0 & n \not= m, \quad n, m \in \mathbb{N}. \end{cases} \]
  • The kinetic energy of a small section of the string at position \(x\) of length \(dx\) is given by \(\frac12 \overbrace{\left(\frac{m}{l}dx\right)}^\textrm{mass} \left(\frac{du(x,t)}{dt}\right)^2,\) where \(m\) is the mass of the string. Assume that this holds along the entire string.
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