AMO 2027 Grades 6-7 SET 14 - Self-study

The number 64 has the property that it is divisible by its units (ones) digit. How many whole numbers between 10 and 50 have the same property?


A. 15

B. 16

C. 17

D. 18

E. None of the above

**Editorial note for this self-study exercise:** Enter the number of qualifying integers, not an option letter. A zero units digit cannot be used as a divisor.

A rectangular piece of paper is folded as shown below. If the area of the shaded part is 50cm², find the perimeter of the rectangular piece of paper when it is unfolded.

**Editorial note for this self-study exercise:** For this self-study exercise, assume the unlabelled rectangular flap hanging below the shaded triangle is a square. This extra condition is needed because its height is not marked in the source. Enter the perimeter in centimeters.

Jane had $\frac13$ as many stickers as Mary. Mary had 120% as many stickers as Paul. After Mary gave 16 stickers to Paul, Mary had the same number of stickers as Paul. How many stickers did the 3 children have altogether?

The figure ABCD is a rectangle. The area of triangles AEB, AED and CDE are 81cm², 36cm² and 27cm² respectively. Find the area of triangle BCE.

Consider the following pattern:

$1=1^2$

$1+3=2^2$

$1+3+5=3^2$

$1+3+5+7=4^2$

$\ldots$

$1+3+5+\cdots+(2k-1)=144$

$\ldots$

a) Write down the 8th line in the pattern.

b) Find the value of k.

**Editorial note for this self-study exercise:** Write the full equation for part (a), then give k for part (b). This response is marked manually so equivalent mathematical notation can be credited.

Henry, Carol and James have an average of $\$260$. Altogether, Henry and Carol have $\$80$ more than James. Carol has $\$130$ more than Henry. How much does James have?

Town A and Town B are 378km apart. At 9 : 45 a.m, Roy set off from Town B to Town A at an average speed of 88km/h. 15 minutes later, Sam set off from Town A to Town B at an average speed of 90km/h.

a) How far from Town A was Roy at 10a.m?

b) At what time did Sam and Roy pass at each other?

**Editorial note for this self-study exercise:** Assume both travel toward one another along the same route at the stated constant speeds, with no stops before meeting. Enter the distance in kilometers, then the meeting time as HH:MM on a 24-hour clock, separated by a semicolon with no spaces.

The teacher writes all integers from 1 to 100 in the board. In how many ways can Lisa erase two numbers such that the sum of the remaining 98 numbers is an even number?

**Editorial note for this self-study exercise:** Count each unordered pair of distinct erased numbers once; changing the order of erasing the same two numbers does not make a new way.

In the following, x, y, z are whole numbers greater than 0. What is the product of x, y and z?

$\displaystyle\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z+\dfrac1z}}}=1-\dfrac{1}{4+\dfrac{1}{4+\dfrac14}}$


A. 6

B. 8

C. 12

D. 24

E. None of the above

**Editorial note for this self-study exercise:** Enter the numeric product, not an option letter.

For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the third digit was any integer from 1 to 9.

a) How many area codes were possible?

b) How many area codes starting with a 4 were possible?

**Editorial note for this self-study exercise:** Use exactly the digit rules stated here, with both endpoints included; this is a mathematical counting model, not a claim about present-day telephone numbering. Enter the answers to (a) and (b), in that order, separated by a semicolon with no spaces.

Take an example of evaluating LCM(12, 18, 30).

**Editorial note for this self-study exercise:** Compute the least common multiple of the three stated positive integers.

If the four-digit number $\overline{4N79}$ is divisible by 21, find N.

A square is labelled anticlockwise ABCD. The point E is outside the square so that CDE is an equilateral triangle. Find angle $\angle BED$.

In a basketball game, Marc, Jackson, and Cole scored 106 points altogether. Marc scored 15 more points than Jackson. Jackson scored 7 fewer than Cole. How many points did Cole score?

Point M is placed inside the square ABCD in such a way that

$\angle MAD:\angle MBA:\angle MCD=2:4:7$.

What is measure of $\angle MCA$ in degrees?