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2017-06-26 Advanced


True or False?

The infinite sequence 1×2017, 2×2017, 3×2017, 4×2017, \big\lfloor 1\times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 2 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 3 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 4 \times \sqrt{ 2017 } \big\rfloor,\ \ldots contains infinitely many perfect squares.

Notation: \lfloor \cdot \rfloor denotes the floor function.

Pi Han has written down a perfect square between 1 to 300 inclusive. Calvin wants to know the integer, and is allowed to ask "yes/no" questions to determine it. Pi Han would immediately answer the question, and then Calvin can use that information to ask his next question. As a point of pride, Calvin would not ask questions to which he already knows the answer. Calvin asks the following questions:

  1. Is it less than 100?
  2. Is it even?
  3. Is it 2-digit?
  4. Is the second digit from the left a 6?

At this point, Calvin declares that he knows the answer. However, he turns out to be incorrect because Pi Han gave the wrong answer to each of the 4 questions!

What is the number in the envelope? If you think it cannot be uniquely determined, enter 0 as your answer.

Clarification: Before the questions are asked, Calvin knows that Pi Han wrote down a perfect square between 1 and 300 inclusive.

Parallelogram ABCD ABCD with AB=10,BC=16,ABC=60AB = 10, BC =16, \angle ABC = 60^{\circ} has an inscribed ellipse such that the tangent point E E satisfies CE=34CD CE = \frac{3}{4} CD .

Find the length of the semi-major axis of this ellipse.

Three points are chosen at random on the circumference of a circle with radius 1. Find the expected value, vv, of the area of the triangle formed by the three points.

If v=abπv = \dfrac{a}{b \pi}, where aa and bb are coprime positive integers, submit a+ba+b.


In the equation above, rr and ss are both rational numbers and their difference is an integer.

What is the number of solutions to the equation?


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