MYTS 2019 - Grade 7 - Round 2

Positive numbers $x,y$ satisfy $\frac{x-5y}{x+y}=\frac12$. Find $S=\frac{x-4y}{x+3y}$.

Integers $a,b,c$ satisfy $a+b+c=bc$. Prove that $(a+b)(a+c)$ is divisible by 4.

Let $x,y$ be rational numbers. Prove that if $x+y$ and $xy$ are integers, then $x,y$ are integers.

There are 50 cards, each labelled with a positive integer at most 50, with distinct labels on different cards. Does there exist a positive integer $n$ such that all 50 cards can be placed in $n$ boxes, and in every box the cards can be split into two groups whose label sums are equal?

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

In the figure, line $AB$ is parallel to line $CD$. Point $E$ is the intersection of the angle-bisector rays of $\angle DAB$ and $\angle CBA$, and $\angle ADC$ and $\angle BCD$ are acute. Given $AD=5$ cm and $EC=ED$, find $BC$. (Enter the number of centimetres.)

Positive real numbers $a_1\le a_2\le\cdots\le a_{2018}\le a_{2019}$ satisfy the following condition: for every $1\le i<j<k\le2019$, the product $a_i a_j a_k$ is one of the 2019 listed numbers. Prove that $a_2=a_3=\cdots=a_{2018}=1$.