MYTS 2019 — Grade 8 — Round 1

Real numbers $a,b,c$ satisfy $a-b=1,b-c=2$. Find $P=a^2+b^2+c^2-ab-bc-ca$.

A rectangle has perimeter 10 cm and diagonal $\sqrt{13}$ cm. Find its area in cm².

A box contains blue, red and yellow pencils. The total number of red and yellow pencils exceeds the number of blue pencils by 5. The total number of blue and yellow pencils exceeds the number of red pencils by 13. How many yellow pencils are there?

A real number $a$ satisfies $a^2+2a-1=0$. Find $Q=a^4-4a^2+4a+3$.

In the figure, $ABCD$ is a square of side length 2 and triangle $BEC$ is right isosceles at $E$. Find $AE$.

Let $A$ be the sum of all positive divisors of $7^{2019}$. Find the remainder when $A$ is divided by 3.

A mathematics club has 15 members from grades 7, 8 and 9. Together they donate 40 books to the school library. Each grade-7 pupil gives 1 book, each grade-8 pupil 4 books, and each grade-9 pupil 6 books. Find the number of pupils from each grade.

Find the coefficient of $x$ when $(x+1)(2x+1)(3x+1)\cdots(10x+1)$ is expanded.

A pan holds at most four pieces of tofu. Cooking one side of a piece takes 1 minute. What is the least time needed to cook both sides of ten pieces?

Huy Hoàng swims the length of a pool at constant speed. At 120% of that speed, he would save 2 seconds. If he instead keeps the original speed for the first half and increases it by 25% for the second half, how many seconds does he save?

For a positive integer $n$, let $s(n)$ be its digit sum; for example $s(2019)=2+0+1+9=12$. How many three-digit numbers satisfy $s(n)>s(n+1)$?

In trapezoid $ABCD$, let $O$ be the intersection of the diagonals. Triangle $ABO$ has area 3 cm² and triangle $CDO$ area 4 cm². Find the area of triangle $ADO$ in cm².

Real numbers $x,y,z$ satisfy $x^2+2y^2+9z^2=12$. Find the maximum of $T=xy+3yz$.

Let $M=11\times21\times31\times\cdots\times91\times101$. Find its remainder upon division by 100.

For each positive integer $k$, let $f(k)=\frac1{(k+2)\times(k!)}$. Find the largest positive integer $n$ such that

$$f(1)+f(2)+\cdots+f(n)<\frac12-\frac1{2019!}.$$

($k!$ is the product of the first $k$ positive integers.)

In the figure, $ABCD$ and $AEFG$ are squares. Find $\frac{CF}{DE}$.

A box contains 100 marbles, coloured yellow, blue, white and red. Any selection of 90 marbles is guaranteed to contain all four colours. What is the least number that must be drawn to guarantee at least three colours?

In the figure, $ABCD$ is a square of side length 2, $O$ is the midpoint of $AD$, $EF$ is perpendicular to $CO$ at $M$, and $OM=1$. Find $EF$.

How many ways can the squares $A,B,C,D$ be coloured blue, red, purple or yellow so that squares sharing an edge have different colours?

Let $N$ be the product of ten distinct integers greater than 1. What is the smallest possible number of positive divisors of $N$?