MYTS 2017 - Grade 10 - Round 2

Real numbers x,y satisfy x+y=1. Find the maximum of $A=\frac{x}{1+y^2}+\frac{y}{1+x^2}$.

Find all pairs (p,n), where p is prime and n is a positive integer, such that $p^n+144$ is a perfect square.

f(x) is a polynomial of degree 2016 satisfying $f(k)=1/k$ for k=1,2,…,2017. Find f(2018).

100 athletes enter a knockout table-tennis tournament: a loser is immediately eliminated, and there are no draws. Two athletes may play only if their numbers of matches already played differ by at most 1. Eventually exactly one champion remains and everyone else is eliminated. Can the champion win more than 9 matches? Explain.

The diagram is a pyramid of 21 squares. Mr Thuận fills each square with a natural number, with every entry above the bottom row equal to the sum of the two entries immediately below it. The filled diagram is an example. What is the smallest possible number of even entries in the pyramid? Explain why.

Triangle ABC is inscribed in circle γ. D lies on the arc BC not containing A. E and F lie on sides AB and AC respectively, with BE=BD and CF=CD. G is the midpoint of EF. Prove that BG is perpendicular to GC.