2015 IMC Day 1 Problem 3

Posted on 29 May 2016 by kjytay

2015 IMC 1.3. Let F(0) = 0, F(1) = \frac{3}{2}, and F(n) = \frac{5}{2}F(n-1) - F(n-2) for n \leq 2.

Determine whether or not \displaystyle\sum_{n=0}^\infty \frac{1}{F(2^n)} is a rational number.

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[Soln] 2015 IMC Day 2 Problem 6

Posted on 28 May 2016 by kjytay

2015 IMC 2.6. Prove that

\displaystyle\sum_{n = 1}^\infty \frac{1}{\sqrt{n}(n+1)} < 2.

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2015 IMC Day 2 Problem 6

Posted on 27 May 2016 by kjytay

2015 IMC 2.6. Prove that

\displaystyle\sum_{n = 1}^\infty \frac{1}{\sqrt{n}(n+1)} < 2.

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[Soln] 2015 IMC Day 1 Problem 1

Posted on 20 May 2016 by kjytay

2015 IMC 1.1. For any integer n \geq 2 and two n \times n matrices with real entries A, B that satisfy the equation

A^{-1} + B^{-1} = (A+B)^{-1}.

Prove that \det (A) = \det (B).

Does the same conclusion follow for matrices with complex entries?

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2015 IMC Day 1 Problem 1

Posted on 19 May 2016 by kjytay

2015 IMC 1.1. For any integer n \geq 2 and two n \times n matrices with real entries A, B that satisfy the equation

A^{-1} + B^{-1} = (A+B)^{-1}.

Prove that \det (A) = \det (B).

Does the same conclusion follow for matrices with complex entries?

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[Soln] Problem-Solving Strategies Ch 9 Problem 25

Posted on 18 May 2016 by kjytay

Engel 9.25. The sequence x_n is defined by x_1 = 1/2, x_{k+1} = x_k^2 + x_k. Find the integer part of the sum

\displaystyle\frac{1}{x_1 + 1} + \frac{1}{x_2+1} + \dots + \frac{1}{x_{100}+1}.

Note: This problem was taken from Arthur Engel’s Problem-Solving Strategies.

Click for solution

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Problem-Solving Strategies Ch 9 Problem 25

Posted on 17 May 2016 by kjytay

Engel 9.25. The sequence x_n is defined by x_1 = 1/2, x_{k+1} = x_k^2 + x_k. Find the integer part of the sum

\displaystyle\frac{1}{x_1 + 1} + \frac{1}{x_2+1} + \dots + \frac{1}{x_{100}+1}.

Note: This problem was taken from Arthur Engel’s Problem-Solving Strategies.

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[Soln] Zero Determinant

Posted on 16 May 2016 by kjytay

Let A and B be two 3 \times 3 matrices with complex entries, such that A^2 = AB + BA. Prove that \det (AB-BA) = 0.

(Credits: I learnt of this problem from this link.)

Click for solution

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Zero Determinant

Posted on 15 May 2016 by kjytay

Let A and B be two 3 \times 3 matrices with complex entries, such that A^2 = AB + BA. Prove that \det (AB-BA) = 0.

(Credits: I learnt of this problem from this link.)

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[Soln] 2016 Japan Math Olympiad Preliminary Problem 10

Posted on 5 May 2016 by kjytay

2016 JMO Prelim 10. Boy A and 2016 flags are on the circumference of a circle, with circumference length 1. He wants to get all the flags by moving on the circumference. Find the smallest possible l such that for any position of Boy A and the flags, he can get all flags by moving distance \leq l.

(Note that Boy A does not have to return to the starting point to return collected flags.)

Click for solution

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