AMO 2027 Grades 6-7 SET 16 - Self-study

Positive integer p has 6 positive divisors. Positive integers q has 4 positive divisors. What is the least number of positive divisors of p × q may have?

Mr. Wong and Mrs. Wong left their house in the same car for Town P. Mr. Wong drove at a speed of 60km/h. Realizing that he left his lap-top at home, he let Mrs. Wong alight at a bus-stop and drove back to his house. Mrs. Wong walked from the bus. stop at a speed of 4km/h to Town P. It took her 45 minutes to reach Town P. Both Mr Wong and Mrs. Wong arrived Town P at the same time. Find the distance between their house and Town P.

**Editorial note for this self-study exercise:** Assume the car keeps the stated 60 km/h on both the return home and the final journey, the routes are along the same road, and turning or collecting the laptop takes no extra time.

ABCDEFGHI is a regular nine-sided polygon (called a ’nonagon’ or ’enneagon’). Find the angle $\angle FAE$.

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan’s distance from his home to his distance from the stadium?

**Editorial note for this self-study exercise:** Assume constant speeds on the same route and no extra time to get the bicycle. Enter the ratio as the numerical quotient (distance to home)/(distance to stadium), using a number or a fraction a/b, without units. Use a decimal point (.) if you enter a decimal.

Find the largest prime factor of 2499.

The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day was shown in the table below.
**Number of Contestants by Score and Day**

| Day | 5 out 5 | 4 out 5 | 3 out 5 | 2 out 5 | 1 out 5 | 0 out 5 | Total |
|---|---|---|---|---|---|---|---|
| Day 1 | 2 | 3 | 4 | 6 | 2 | 3 | 20 |
| Day 2 | 2 | 3 | 5 | 5 | 4 | 1 | 20 |
| Day 3 | 3 | 3 | 4 | 5 | 3 | 2 | 20 |
| Total | 7 | 9 | 13 | 6 | 9 | 6 | 60 |

What was the mean score of the contestants on Day 1?

**Editorial note for this self-study exercise:** Use the Day 1 row for the requested mean. The source table is preserved, including an erroneous bottom-row total in the two-point column; that total is not needed for this question. Use a decimal point (.) if you enter a decimal.

0.7 of the balls in a bag are black, 0.2 of the remainder are pink and the remainder are red. If there are 184 more black balls than red balls, how many balls are there in the bag together?

If the height of a cube is increased by 3cm, it will become a cuboid (rectangular box) and the total surface area will be increased by 72cm². What is the volume of the cube?


A. 27cm³

B. 64cm³

C. 125cm³

D. 216cm³

E. 343cm³

**Editorial note for this self-study exercise:** Enter the volume in cubic centimeters as a number, rather than an option letter. Only the height changes; the base remains the original square.

Simplify the following algebraic expressions.

a) $1-r+\frac{2s}{5}-3s+2r$

b) $\frac{2y^2}{3}+4-y^2+2y$

c) $4pq+2p-3pq-2q$

d) $d+8d-4d:3$

e) $10k^2:2k+5k-12k:4$

**Editorial note for this self-study exercise:** Read : as division, carried out before addition and subtraction. In (e), the divisor of the first quotient is 2k, and k is nonzero. Give all five simplified expressions; equivalent algebraic forms will be marked manually.

75% of the marbles in a bag were green. 60% of the remainder were red and the rest were blue. There were 234 more green marbles than blue marbles.

a) How many red marbles were there?

b) After some red marbles were removed from the bag. 10% of the remaining marbles in the bag were red. How many red marbles were removed from the bag?

**Editorial note for this self-study exercise:** Enter the answer to (a), then the answer to (b), as two integers separated by a semicolon with no spaces. Only red marbles are removed.

Find how many three-digit numbers satisfy all the following conditions:
• if it is divided by 2, the remainder is 1,
• if it is divided by 3, the remainder is 2,
• if it is divided by 4, the remainder is 3,
• if it is divided by 5, the remainder is 4,
• if it is divided by 8, the remainder is 7.

Rocky bought an equal number of apples and mangoes for $\$57.60$. Each mango cost $\$1.80$. Each apple cost $\$1.20$ less than a mango. How much money did he spend on the mangoes?

**Editorial note for this self-study exercise:** Enter the amount in dollars without a currency symbol or unit. Use a decimal point (.) if entering a decimal.

A sealed container with the measurements as shown in Figure 1 is filled with 10.2 liters of water. The sealed container is inverted as shown in Figure 2. Find the height of the water in the container from the base of the container.

**Editorial note for this self-study exercise:** Use the internal dimensions and the two connected rectangular-prism sections shown; ignore wall thickness. Measure from the lowest base in Figure 2, including the lower narrow section, not only from the shoulder of the wider section.

Rowan and Emma each had an equal number of popsicles. After Rowan ate 25 popsicles and Emma ate 31 popsicles, Rowan had twice as many as Emma. How many popsicles did they each have at first?

a) In the following figure, there are seven points on the circumference of a circle. How many triangles can be drawn with any three points as vertices?

b) In the following figure, there are eight points on the sides of a square. How many triangles can be drawn with any three points as vertices?

**Editorial note for this self-study exercise:** Use the marked points only, counting each choice of three distinct vertices once. Three collinear points do not form a triangle. Enter (a) then (b) as integers separated by a semicolon with no spaces.