MYTS 2017 - Grade 8 - Round 2

Find all ordered quadruples of positive integers (a,b,c,d) satisfying both $ab=c+d$ and $a+b=cd$.

Four real numbers, not necessarily distinct, have sum 172, and every pair has nonnegative sum. Let ℓ be the smallest of them. Find the minimum possible value of ℓ.

Suppose a polynomial q(x) satisfies $x^{17}=(x^3-x^2-x+1)q(x)+ax^2+bx+c$, where a,b,c are real numbers. Find b.

ABC is a right triangle with hypotenuse BC and altitude AH. M lies on HC with MC=2MH. The line ℓ through H perpendicular to AM meets the extended line AB at N. On the ray opposite to NH, choose K so that NH=NK. Let I be the intersection of BC and AK. Find IK/AK.

Seven students play table tennis. On the first day, each may have played no matches, one match or several matches. Each student wins at most one match that day. Prove that (a) among any four students, at least two have not played each other; (b) after the first day, the seven students can always be divided into three groups such that no two students in a group have played each other.

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.