MYTS 2019 — Grade 9 — Round 1

Solve $\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}=2019x+1$.

Find the natural number $n$ such that both $n$ and $n+19$ are perfect squares.

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

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

Let $A$ be the sum of all positive divisors of $2^{2019}$. Find its remainder upon division by 3.

Find all values of $a$ for which $x^2-9x+a=0$ has real roots $x_1,x_2$ satisfying $(x_1-2)^2+(x_2-3)^2=10$.

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?

Two circles $(O_1)$ and $(O_2)$ touch each other, and $AB$ is a common external tangent. Given $AB=18$ and the radius of $(O_1)$ equal to 3, find the radius of $(O_2)$.

Nonzero pairwise distinct numbers $a,b,c$ satisfy $a+\frac4b=b+\frac4c=c+\frac4a$. Find $|abc|$.

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.)

A nonzero real number is written at each vertex of the octagon shown. Each black vertex equals the sum of its two adjacent white vertices, and each white vertex equals the product of its two adjacent black vertices. Find the sum of all eight numbers.

In the figure, $ABCD$ is a square of side length 2, $O$ is the midpoint of $AD$, and $EF$ is tangent to the semicircle with diameter $AD$ at $M$. Find $EF$.

Real numbers $x,y$ satisfy $x+y=12$ and $[x]+2[y]=13$. Find all possible values of $[x]$.

($[x]$ denotes the floor of $x$.)

In the figure, $H$ is the orthocentre of triangle $ABC$. The areas of triangles $ABE$ and $ACD$ are 4 and 6, respectively, and $AH=4$. Find $BC$.

Find all real numbers $a$ such that both $a+\sqrt5$ and $a^2+\sqrt5$ are rational.

What is the largest number of positive divisors of $20^{2019}$ that can be chosen so that no chosen number is a multiple of another chosen number?

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?

What is the largest number of integers that can be selected from 1 through 30 so that no selected integer divides the product of all the other selected integers?

Eight squares are numbered as shown. A jump from square $m$ to square $n$ has length $|m-n|$.

Starting at square 0, a frog jumps to other squares, each jump having length 1, 2 or 4, without visiting any square more than once. What is the largest possible total jump length?