Pre RMO practice sheet

Question 1:

What is the smallest positive integer k such that k(3^{3}+4^{3}+5^{3})=a^{n} for some positive integers a and n with n>1?

Solution 1:

We have k \times 216= a^{n} so that k \times 6^{3} = a^{n} giving k=1.

Question 2:

Let S_{n} = \sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}.

What is the value of \sum_{n=1}^{99} \frac{1}{S_{n}+S_{n-1}}

Answer 2:

Given

S_{n} = \sum_{k=0}^{n} \frac{1}{\sqrt{k+1}+\sqrt{k}} \times \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k+1}-\sqrt{k}}

S_{n} = \sum_{k=0}^{n} (\sqrt{k+1}-\sqrt{k}) = \sqrt{n+1}

Similarly,

S_{n-1} = \sum_{k=0}^{n-1}(\sqrt{k+1}-\sqrt{k}) = \sqrt{n}

Now, \sum_{n=1}^{99} \frac{1}{S_{n}+S_{n-1}} = \sum_{n=1}^{99}\frac{1}{\sqrt{n+1}-\sqrt{n}} \times \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}

which in turn is equal to \sum_{n=1}^{99} (\sqrt{n+1}-\sqrt{n}) = (\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+ (\sqrt{4}-\sqrt{3}+ \ldots + (\sqrt{100}-\sqrt{99})) = \sqrt{100} -\sqrt{1}=10-1=9

Question 3: Homework:

It is given that the equation x^{2}+ax+20=0 has integer roots. What is the sum of all possible values of a?

Cheers,

Nalin Pithwa

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