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2018-01-08 Advanced

         

Consider the equation x2+y2=3z2.x^2+y^2=3z^2. Are there any other integer solutions besides the solution where x=y=z=0?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 m0.05\text{ m} and the speed of rotation is 22 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=π2 m/s2.g = \pi^2 \text{ m/s}^2.

Find the number of pairs of positive integers (a,b)(a,b) with 1a<b1001\leq a < b \leq 100 such that

there is at least one positive integer mm with a<m<ba<m<b such that mm is divisible by every common divisor of aa and b.b.

20x+20x+20x+17303030=17 \large { \sqrt[30]{20x +\sqrt[30]{20x + \sqrt[30]{20x + 17}}} = 17 }

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

A guitar string of length ll stretched along the xx-axis is plucked in the middle. The initial deflection of the string has the shape u(x,t=0)=u0(x)={2A0xlx<l22A0(lx)lxl2, 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 A0A_0 is in the middle (see diagram). The string is then released and swings freely for time t>0,t > 0, and the resulting string vibration can be written as a superposition of standing waves u(x,t)=n=1Ancos(2πfnt)sin(nπlx),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 fn=nνf_n = n \cdot \nu and fundamental frequency ν\nu.

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

Hint:

  • The set of sine functions has the property 0lsin(nπlx)sin(mπlx)dx={l2n=m0nm,n,mN. \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 xx of length dxdx is given by 12(mldx)mass(du(x,t)dt)2,\frac12 \overbrace{\left(\frac{m}{l}dx\right)}^\textrm{mass} \left(\frac{du(x,t)}{dt}\right)^2, where mm is the mass of the string. Assume that this holds along the entire string.
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