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Consider the equation x2+y2=3z2. Are there any other integer solutions besides the solution where x=y=z=0?
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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 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=π2 m/s2.
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Find the number of pairs of positive integers (a,b) with 1≤a<b≤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.
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3020x+3020x+3020x+17=17
Let N be the sum of all the real solutions to the above equation. If N=cab−a, where a, b, and c are positive integers and a and c are coprime, then what is a+b+c?
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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)=u0(x)=⎩⎪⎨⎪⎧l2A0xl2A0(l−x)x<2lx≥2l, where A0 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)=n=1∑∞Ancos(2πfnt)sin(lnπx), with eigenfrequencies fn=n⋅ν and fundamental frequency ν.
What is the relative amount of vibrational energy stored in the fundamental mode f=ν? To do this, you'll have to determine the amplitudes An.
Hint:
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