MYTS 2018 — Grade 5 — Round 1

Eight equal circles lie in a rectangle as shown. Find the ratio of the solid-shaded area to the hatched area.

Khôi initially has $7/8$ as many sweets as Nguyên. They give sweets to children, and Khôi gives away $7/8$ as many as Nguyên gives away. Who has more sweets left?

An pays with a 200-thousand-đồng note for a fountain pen costing 17 thousand đồng. The seller has notes of 50,20,10,5,2 and 1 thousand đồng. What is the minimum number of change notes?

How many triangles are in the figure?

Lan writes the largest four-digit multiple of 9 with pairwise distinct digits. Hương writes the smallest number with the same properties. Find their sum.

Thảo is one of three triplet sisters and has twin older brothers. The five siblings’ ages total 36. Each brother is 1.5 times Thảo’s age. How old is Thảo?

Tuấn has two electric digital clocks. Both stop during a power cut. When power returns, the first restarts at 00:00 and the second resumes from its stopped reading. Power fails at 2:30 a.m. and returns at 4:30 a.m. When Tuấn wakes at 7:00 a.m., how many minutes apart are the readings? There is only one power cut that night.

How many products in $1\times2\times3,\ 2\times3\times4,\ 3\times4\times5,\ldots,97\times98\times99$ end in digit 0?

A rectangle is divided into four smaller rectangles as shown. Three have areas 6,10,15; the fourth has greater area than any of them. Find the entire area.

Points $A,B,C,D$ divide a circular track into equal arcs as shown. Hải starts at $A$ and Phượng at $B$, simultaneously and clockwise. Hải’s speed is $6/5$ of Phượng’s. How many laps has Hải run when first catching Phượng?

Thu, Thảo and Tuấn each have a card showing a natural number and cannot see the other two numbers. They learn that the three numbers are distinct, sum to 13 and have odd product. Thu and Thảo can immediately determine the product, but Tuấn cannot. What number is on Tuấn’s card?

Thảo draws lines dividing each face of a cube of side 4 cm into sixteen unit squares, including the cube’s edges. What is the total drawn length in cm?

My uses blue, red and yellow for the segments of T, and purple and orange for the segments of S, as shown. Segments meeting at a circle must have different colours. How many colourings are possible?

A box contains 30 red,20 blue,10 yellow,8 white and4 black marbles. What is the minimum draw guaranteeing at least 5 marbles of each of two colours?

Gifts are prepared for Lan, Hương, Thúy and Quỳnh. Exactly one of them opens the boxes. They say:
- Lan: Hương did not open them.
- Hương: Lan or Quỳnh did.
- Thúy: Hương did.
- Quỳnh: Lan or Thúy did.
Exactly one person lies. Who opened the gifts?

Hà’s odometer reads 15951 km, a palindrome. After exactly 2 hours of continuous driving, it again shows a palindrome. The speed limit is 70 km/h. Find the average speed over those 2 hours in km/h.

A program generates a sequence of figures following a pattern; the first three are shown. How many triangles are in the 30th figure?

An ant follows the broken line from $A$ to $B$ in the figure. Adjacent parallel lines are 1 cm apart. How long is the path in cm?

Two groups donate 102 books altogether. In the first group Dũng gives 5, and An, Đăng and Tường give 10 each. In the second group everyone gives at least one book, with pairwise different amounts. What is the largest possible total number of people?

Six boxes are numbered 1 through 6. Hồng puts cards numbered 1 through 2018 into them: card 1 into box 1, card 2 into box 2, …, card 6 into box 6; card 7 into box 5, card 8 into box 4, …, continuing back and forth. Which box receives card 2018?