MYTS 2018 - Grade 7 - Round 2

Diệp thinks of a three-digit number abc. The sum of the five numbers acb,bac,bca,cab,cba is 3496. What number did Diệp choose?

Tuấn walks to a friend’s house at constant speed. Returning along the same route, a speed increase of 200 m/h would make the return time 20/21 of the outward time; a decrease of 400 m/h would take 2 minutes longer than the outward journey. Find the one-way distance in kilometres.

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.

ABC is right-angled at A. Points F,D lie on BC such that angle ACB=2 angle BAF and angle BAD=3 angle BAF. On the bisector of angle ACB choose E inside triangle ABC with AE=AD. Prove that AEF is equilateral.

Ten people sit around a circular table. For every person, that person’s age plus the ages of the two immediate neighbours is odd. Prove that all ten ages are odd.

Find all pairwise coprime positive integers a$<$b$<$c such that a+b+c+ab+bc+ca is divisible by abc.