MYTS 2018 — Grade 6 — Round 1

How many digits does $10^{2018}-2018$ have?

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

A photo shop charges 6000 đồng per print; for every 20 prints, 3 are free. How many đồng must An pay for 61 prints?

Each side of the rectangle is divided into 2,3 or 4 equal parts as shown. Find the ratio of the area of the solid-line quadrilateral to that of the dashed-line quadrilateral.

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?

A program generates figures following the pattern shown in the first four figures below. The first square has side length 1 unit. What is the hatched area in the sixth figure?

Let $A=10\times11\times\cdots\times99$. Reverse the tens and units digits of every factor to get $B=01\times11\times21\times\cdots\times99$. Find $A/B$.

A rectangular garden is surrounded by a path 1 m wide as shown. The garden’s side lengths in metres are integers, and its area equals the path’s area. What is the garden’s minimum perimeter in metres?

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?

Twenty-five products have an average price of 120 thousand đồng. None costs less than 40 thousand đồng, and exactly 10 cost less than 100 thousand đồng. What is the highest possible price of one product, in đồng?

A rectangle is tiled by squares as shown. Every square side length is a positive integer. What is the smallest possible perimeter of the rectangle?

Four children sing and accompany one another. For each song, one accompanies and the other three sing. An sings 9 songs, Bình 5, Lan 6 and Hoa 7. How many songs does Bình accompany?

The odd numbers are arranged in rows:

$1$

$3\quad5$

$7\quad9\quad11$

$13\quad15\quad17\quad19$

$21\quad23\quad25\quad27\quad29$

$\ldots$

Find the sum of row 20.

Tuyết writes the largest four-digit number with distinct digits that is a multiple of 9, whose last three digits form a three-digit multiple of 3 and whose last two digits form a two-digit multiple of 2. Hương writes the smallest such number. Find their sum.

Two cubes of side 4 are built from unit cubes. Fourteen small cubes are removed from one as shown. Find the difference between the surface areas of the resulting two solids.

An has thirteen cards numbered 1 through 13. What is the largest number of cards that can be selected so that the product of their numbers is a perfect square?

Lan has 37 sweets, Thảo 49 and Mai 67. Each day one of them gives one sweet to every other classmate, excluding these three. When no further such distribution is possible, all three have equal remainders. What is the largest possible class size?

Rabbit tells the truth on Monday and Tuesday and lies on other days. Tiger tells the truth on Wednesday, Thursday and Friday and lies on other days. Lion tells the truth on Tuesday and Friday and lies on other days. One day all three say, “The day after tomorrow I will lie.” What day is it?

Points $E,F$ lie on side $CD$ of rectangle $ABCD$ with $DE=EF=FC$. Segments $AE,AF$ meet $BD$ at $X,Y$, respectively. Find $XY/BD$.

The table below follows a pattern. How many ways can numbers be placed in cells $A,B$?