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2018-06-04 Advanced


AA and BB are distinct numbers such that AnBn A^n - B^n is an integer for all positive integers n.n.

Do both AA and BB have to be integers?

Steph Curry regularly wows crowds with his incredible 3-point shooting, as he did in Game 1 of the 2018 NBA championship last Thursday when he made a shot from half-court right at the buzzer.

When he shoots, he aims the ball at the center of the rim, horizontal distance dd ahead of him and vertical distance hh above. There is some freedom in making the shot — there are an infinite number of pairs of shot velocity and shot angle that will result in the ball passing through the target point (d,h)(d,h). But the key to a consistent shot is minimizing the effect of small errors in execution. In fact, there's a magic angle at which the variation in both dd and hh with respect to θ\theta is minimized.

At what angle (in degrees) should he take his shots for the most reproducible three-pointer?

Assumptions and Details:

  • d=14.5 md = \SI{14.5}{\meter} and h=1.15 m.h = \SI{1.15}{\meter}.
  • Neglect air resistance.
  • In general, a player needs to be consistent in their shot angle and shot velocity but, for simplicity, we focus on shot angle and assume that Curry can set his shot velocity with no error.

f(x)=x+sin(x).f(x) = x+\sin(x).

What is the area of one of the infinitely many closed regions between the curves y=f(x)y = f(x) and its inverse function y=f1(x)?y=f^{-1}(x)?

Triangle ABCABC has vertices A=(0,1),A = (0,1), B=(1,0),B=(1,0), and C=(0,0)C=(0,0) in the coordinate plane.

The insides of the sides are lined with mirrors, and then a laser beam is fired from the origin with a slope of 314379.\frac{314}{379}.

Which corner of the triangle will the laser beam hit first?

Two unit circles are joined by many equally spaced strings of length 22 to form a cylinder.

The upper circle is rotated by 9090 degrees while the lower is fixed. If there were infinitely many strings, they would serve as the surface of a 3D figure, together with the two circles.

What is the volume inside this 3D figure?


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