## Challenges so far

## October challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 31 October. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Show that the sequence @d \left(\sqrt{13}, \sqrt{13+\sqrt{5}}, \sqrt{13+\sqrt{5+\sqrt{13}}}, \sqrt{13+\sqrt{5+\sqrt{13+\sqrt{5}}}}, \sqrt{13+\sqrt{5+\sqrt{13+\sqrt{5+\sqrt{13}}}}},\ldots \right)@d converges, and calculate its limit.

## November challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 30 November. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Find algebraically the solutions of the equation

@d \sqrt{3}\left(\frac1{\sin x}+\frac1{\cos
x}\right)+\left(\frac1{\sin x}-\frac1{\cos x}\right)=
4\sqrt{2}.@d

## February challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 28 February. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Let @ia\in\mathbb{N}@i. Find all solutions @ix\in\mathbb{Z}@i of
the equation

@d(x+a)^3 = x^4-a^4.@d

## December challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 31 December. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Let @i x,\,y,\,z>0 @i. Show:

@d x+y+z \; \le \; \frac{x^2+y^2}{x+y} + \frac{y^2+z^2}{y+z}
+\frac{x^2+z^2}{x+z} \; \le \; 3\cdot
\frac{x^2+y^2+z^2}{x+y+z}.@d

## March challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 31 March. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Let @i n @i be a natural number and @i p>5 @i a prime with @i
p\mathbin| n^2+5 @i.

Show that the tens digit of @i p @i is an even number.

## April challenge

To win a BB$50-voucher to be used in the UWI Bookshop, answer the following question and submit your answer to Dr Bernd Sing by 30 April. Only complete (and well-structured!) solutions will be considered, and it must be your own work (no plagiarism!). Tie-break rule: If more than one complete and correct solution is received, the most elegant solution will receive the prize money (if still no decision can be reached, the solution that was received earlier will win.)

Show @d \sum\limits_{n=1}^{\infty} \arctan\left(\frac{4\,n^2}{n^4-4}\right) \ =\ \frac{\pi}2.@d