MYTS 2019 - Grade 9 - Round 2

Numbers $x,y$ have sum 2. Prove that $(x^2+x+1)(y^2-y+1)\ge3$. When does equality hold?

Integers $a,b,c$ satisfy that $ab+bc+ca+1$ is divisible by 4. Prove that $a+b+c+abc$ is also divisible by 4.

An acute triangle $ABC$ is inscribed in circle $(O)$. Choose $D$ on the minor arc $BC$, distinct from $B,C$. Draw $DX\perp BC$ and $DY\perp AC$. Let $M,N$ be the midpoints of $AB,XY$, respectively. Prove that $\angle DNM=90^\circ$.

Positive integers $m,n$ and a prime $p$ satisfy $p=\frac{m+n}{2}+3\sqrt{mn}$. Prove that $p+4m$ and $p+4n$ are perfect squares.

There are 2019 cards bearing distinct positive integers at most 2019. Does there exist a positive integer $n$ such that all cards can be placed in $n$ boxes, with the cards in each box divisible into two groups so that one group’s label sum equals $\frac27$ of the other group’s label sum?

For a group containing one card, its sum is that card’s label.

Consider all tickets bearing nine-digit numbers formed from the digits 1, 2, 3. Two tickets are called “separated” if their numbers $\overline{a_1a_2\cdots a_9}$ and $\overline{b_1b_2\cdots b_9}$ satisfy $a_i\ne b_i$ for every $i=1,2,\ldots,9$.

(a) Prove that all tickets can be coloured blue, red or yellow, one colour per ticket, so that separated tickets always have different colours.

(b) Consider any colouring satisfying (a). Suppose ticket 122222222 is red and ticket 222222222 is yellow. What colour is ticket 123123123?