MYTS 2018 - Grade 8 - Round 2

Find all three-digit numbers equal to the cube of their digit sum.

Nonzero real numbers a,b satisfy a+b=3 and (1+a)²/b+(1+b)²/a=35. Find a⁷+b⁷.

Write 1,2,…,2018 on a board. Repeatedly erase some numbers and replace them by the remainder when their sum is divided by 31. Eventually two numbers remain, one of which is 96. Find the other.

ABCD is a square with centre O. On the ray opposite to CD take M, and on the ray opposite to DC take N, such that CM+DN=CD. Let I=AM∩BN and P=OI∩AB. Prove that angle MPN=90°.

In the plane there is a quadrilateral ABCD and 19 distinct pairwise nonparallel lines. Each line is equidistant from two vertices of the quadrilateral. Prove that at least three of the lines are concurrent.

For each integer n≥2, let F_n be the set of fractions a/b with 0≤a$<$b≤n and gcd(a,b)=1. For example F_2={0/1,1/2}, F_3={0/1,1/3,1/2,2/3}, F_4={0/1,1/4,1/3,1/2,2/3,3/4}. Prove that F_n has an even number of elements.