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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=3−6sinθ. The area of the region that Little Albert colored in is equal to aπ+cb3, where a, b, and c are positive integers, and b and c are coprime.
Find a+b+c.
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In the diagram above, suppose that BA1=A1A2=A2C and AC1=C1B. If C1Y=2017, then XY=qp, where p and q are coprime positive integers. Find p+q.
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2017 identical metallic plates (initially uncharged), each having area of cross section A, are each separated by a distance of d, as shown in the figure above.
Plates 2 and 2016 are given charges 2Q and 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 bAϵ0aQd, where a and b are coprime positive integers, then submit your answer as a+b.
Details and Assumptions:
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Let f(x)=x+ex−1, and let f−1(x) denote the inverse function of f(x).
Then the integral ∫e1+e2f−1(x)dx evaluates to k0+k1e1+k2e2+⋯+knen, where k0,k1,…,kn are rational numbers and e≈2.718 is Euler's number.
If the sum k0+k1+k2+⋯+kn can be expressed as BA, where A and B are coprime positive integers, what is A+B+n?
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Two uniform circular disks of masses M and 4M and radii R and 2R, respectively, are connected by a massless spring of force constant k 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 bMak, where a and b are coprime positive integers, then enter a+b as the final answer.
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