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2017-09-18 Advanced


Given a sphere, a circle is called a great circle if it is the intersection of the sphere with a plane passing through its center.

Now, 5 distinct great circles dissect a sphere into nn pieces. If mm and MM are the minimum and maximum values of nn, respectively, then find m+Mm+M.

A cylindrical vessel with radius 90 cm90 \text{ cm} filled with 3-meter deep water is sitting on a tower, as shown in the diagram. The vessel has a hole on the bottom with a cross-sectional area of 6.3 cm26.3 \text{ cm}^2.

How long (in minutes) will it take before the vessel is completely drained through the hole?

Assume that the gravitational acceleration is g=9.8 m/s2.g = 9.8 \text{ m/s}^2.

Consider the Euler's totient function ϕ(n)\phi(n) which is the number of integers from 1 to nn coprime to nn.

If nn is divisible by ϕ(n),\phi(n), what is the product of all possible values of nϕ(n)?\frac{n}{\phi(n)}?

Let p(x)p(x) be a non-constant polynomial function such that there exists a real number aa for which p(a)0,p(a)=0,p(a)=0.p(a) \ne 0,\quad p'(a)=0,\quad p''(a)=0.

True or False?

At least one root of p(x)p(x) is non-real.

A ball thrown in the horizontal direction bounces off the ground, as illustrated in the diagram. Eventually, the ball stops bouncing and rolls on the ground. What is the total bounced distance x,x_\infty, rounded to the nearest integer?

Details and Assumptions:

  • The initial height of the ball is h0=0.8 m.h_0 = 0.8 \text{ m}.
  • The horizontal velocity of the ball is vx=2.6 m/s.v_x = 2.6 \text{ m/s}.
  • The gravitational acceleration is g=10 m/s2.g = 10 \text{ m/s}^2.
  • After each bounce, the ball loses 19% of its vertical kinetic energy 12mvy2\frac{1}{2} m v_y^2, whereas the horizontal motion is unaffected.


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