MYTS 2019 — Grade 7 — Round 1

From 9 a.m. to 9 p.m. in one day, how many times do the hour and minute hands lie on the same straight line?

Find $S=\frac4{1\times3}+\frac{16}{3\times5}+\frac{36}{5\times7}+\cdots+\frac{2500}{49\times51}$.

A clock has hour, minute and second hands. At 8:10, what is the smallest angle, in degrees, formed by a pair of its hands?

Two workers, An and Bình, share a payment. An observes: “If my share increased by 6%, Bình’s share would decrease by 5%.” If Bình’s share increased by 30%, by what percentage would An’s share decrease?

Ten numbers $a_1,a_2,\ldots,a_{10}$ satisfy

$$\frac{a_1-1}{10}=\frac{a_2-2}{9}=\frac{a_3-3}{8}=\cdots=\frac{a_{10}-10}{1},\qquad a_1+a_2=41.$$

Find $S=a_1-a_2+a_3-a_4+\cdots+a_9-a_{10}$.

In the figure, triangle $ABC$ has area 24 cm². Find the area of triangle $ADE$ in cm².

Numbers $x,y,z$ satisfy $\frac{x-2y}{z-y}=-5$. Find $\frac{x-2z}{y-z}$.

Find all natural-number triples $(x,y,z)$ satisfying $x\le y\le z$ and $x^2+y^2+z^2=34$.

Initially each square of a $4\times4$ board contains one grasshopper. They all simultaneously jump to an adjacent square, with several permitted to land together. What is the greatest possible number of empty squares afterward?

(Two squares are adjacent if they share an edge.)

Triangle $ABC$ is isosceles at $A$, with $\angle A=45^\circ$ and $AB=4$ cm. Square $ACDE$ is constructed externally as shown. Find $BD$ in centimetres.

Hà takes 20 minutes to walk from home to the market; her mother takes 30 minutes on the same route. One day her mother leaves at 7 a.m., and Hà leaves 5 minutes later. At what time does Hà catch up?

Four numbers are written in order, with mean 20. The first three have mean 23 and the last three have mean 19. Find the mean of the first and last numbers.

Among the natural numbers from 1 to 2019, how many perfect squares are not divisible by 3?

A two-digit natural number is written on the board.

- An says: “It is even but not divisible by 4.”
- Bình says: “It is a perfect square.”
- Cường says: “It is a power of 3.”

Exactly one statement is false and the other two are true. What is the number?

Find the largest six-digit number of the form $\overline{a2019b}$ that is a multiple of 33.

Five natural numbers, not necessarily distinct, have the following ten pairwise sums:

$4,5,7,7,9,9,10,11,12,14$.

Find the product of the five numbers.

Seven cities and the roads joining them are shown. Starting from city 1, how many routes visit every other city exactly once and then return to city 1?

Find the smallest natural number that is a sum of five consecutive positive integers, a sum of seven consecutive positive integers, and a sum of nine consecutive positive integers.

Square $ABCD$ has side length 2 cm. A moving point $E$ lies outside the square, with triangle $ABE$ right-angled at $E$. Find the greatest possible length of $DE$ in centimetres.

A container holds 20 blue balls, 20 red balls and 20 yellow balls. Within each colour, the balls are numbered $1,2,\ldots,20$. What is the least number a blindfolded person must draw to guarantee three balls of one colour with consecutive labels?