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2017-05-22 Advanced

         

For Mother's Day, Little Albert wants to show just how much he loves his mom by making a card for her with a big, colorful heart on the front. He gets out his paper, uncaps a marker, and gets to work.

The curve that Little Albert drew is called a limaçon. (The extra loop inside the heart will be invisible to his mom after coloring.) The polar equation for the particular limaçon that he drew is r=36sinθ.r = 3 - 6 \sin \theta. The area of the region that Little Albert colored in is equal to aπ+b3c,a \pi + \frac{b \sqrt{3}}{c}, where a,a, b,b, and cc are positive integers, and bb and cc are coprime.

Find a+b+c.a + b + c.

In the diagram above, suppose that BA1=A1A2=A2C BA_{1} = A_1A_2 = A_2C and AC1=C1BAC_1 = C_1B . If C1Y=2017C_1Y = 2017, then XY=pqXY = \frac{p}{q} , where pp and qq are coprime positive integers. Find p+qp + q.

2017 identical metallic plates (initially uncharged), each having area of cross section AA, are each separated by a distance of d,d, as shown in the figure above.

Plates 2 and 2016 are given charges 2Q2\mathbf Q and Q,\mathbf Q, respectively. Plates 1 and 2017 are both earthed via a very thin conducting wire.

If the potential difference between plates 1728 and 1729 is of the form aQdbAϵ0\dfrac{a Qd}{bA\epsilon_0}, where aa and bb are coprime positive integers, then submit your answer as a+ba+b.


Details and Assumptions:

  • ϵ0\epsilon_0 denotes the permittivity of free space.
  • The plates are very large, and the phenomena of fringing of electric field lines are neglected.
  • Because of the large size, the electric field due to any particular plate is considered as that of an infinitely large, thin conducting plate.

Let f(x)=x+ex1f(x) = x + e^x - 1, and let f1(x)f^{-1} (x) denote the inverse function of f(x)f(x) .

Then the integral e1+e2f1(x)dx \displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx evaluates to k0+k1e1+k2e2++knen, k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n , where k0,k1,,knk_0, k_1, \ldots, k_n are rational numbers and e2.718e\approx 2.718 is Euler's number.

If the sum k0+k1+k2++knk_0 + k_1 + k_2 + \cdots + k_n can be expressed as AB,\frac{A}{B}, where AA and BB are coprime positive integers, what is A+B+n?A+B+n?

Two uniform circular disks of masses MM and 4M4M and radii RR and 2R,2R, respectively, are connected by a massless spring of force constant kk and placed at different heights so that the spring is parallel to the horizontal, as shown above.

Initially, the spring is compressed, and then the system is released from rest. Assuming friction is large enough to prevent slipping, find the angular frequency of oscillation of the system.

If you answer comes in the form of akbM,\sqrt{ \dfrac{ak}{bM}}, where aa and bb are coprime positive integers, then enter a+ba+b as the final answer.

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