MYTS 2019 — Grade 6 — Round 1

An arithmetic sequence is $A,B,C,D,E,F,G,H,I,K$. Given that $E+F=28$, find the sum of all terms.

How many rectangles have positive integer length and width in metres, and area 48 square metres?

Let $N=\underbrace{4\times4\times\cdots\times4}_{15\text{ factors}}\times\underbrace{125\times125\times\cdots\times125}_{7\text{ factors}}$. How many consecutive zeros occur at the end of its decimal representation?

How many matchsticks are there in the figure?

A restaurant has rectangular tables seating 8 people and round tables seating 5. How many tables are needed to seat exactly 36 guests with no empty seats?

In the figure, $D,E,F$ divide $AB,BC,CA$, respectively, in the ratio $1:2$. Find the area ratio $\frac{S_{DEF}}{S_{ABC}}$.

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?

A class has 20 pupils good at Mathematics and 22 good at Literature. Eight are good at both, and five are good at neither. How many pupils are in the class?

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

A triangular garden is shown in the figure. An and Bình start together at $A$ and walk around its boundary in opposite directions: An counterclockwise and Bình clockwise. An walks twice as fast as Bình. Let $D$ be their second meeting point after departure (not counting the initial meeting). Find $BD$ in metres.

Find the smallest natural number $n$ that is both the sum of five consecutive positive integers and the sum of seven consecutive positive integers.

Initially each square of a $3\times3$ board contains one grasshopper. All grasshoppers simultaneously jump to an adjacent square. Several may land in the same square. What is the greatest possible number of empty squares after the jumps?

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

Two machines start together. The first repeatedly runs for 3 hours and rests for 1 hour; the second runs for 4 hours and rests for 1 hour. During the first 100 hours, for how many hours do both machines rest simultaneously?

How many four-digit numbers with distinct digits can be formed from 1,2,3,4,5,6 and be divisible by 25?

How many ways can the circles in the figure be coloured blue, red or yellow so that any two circles joined by a line segment have different colours?

A container holds 20 balls numbered $1,2,\ldots,20$. What is the least number a blindfolded person must draw to guarantee three balls whose labels are three consecutive integers?

How many natural numbers $N=\overline{3a4b}$ satisfy that $N-17$ is a multiple of 33, where $a,b$ are digits?

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

- An says: “It is even but not divisible by 4.”
- Bình says: “It is the product of two equal numbers.”
- Cường says: “It is a power of 3.”

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

In a plane there are 17 distinct points and a line $\ell$ through none of them. What is the greatest number of segments, each joining two of the 17 points, that $\ell$ can intersect?

How many ordered pairs of natural numbers $(m,n)$ satisfy $m\mid210$ and $n\mid m$?