AMO 2027 Grades 8-9 SET 8 - Self-study

What is the smallest positive integer which cannot occur as the difference between two positive prime numbers?

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?

Phoebe has some 32-cent stamps, some 29-cent stamps, and some 3-cent stamps. The number of 29-cent stamps is 10 less than the number of 32-cent stamps, while the number of 3-cent stamps is 5 less than the number of 29-cent stamps. The total value of the stamps is \(\$9.45\). How many 32-cent stamp does she have?

Find the largest prime factor of 2499.

The average cost of 2 mangoes and 5 pineapples is \(\$2.30\). The average cost of each pineapple is \(\$1.95\). The difference in cost between the 2 mangoes \(\$1.25\). Find the cost of each mango.

**Editorial input format:** Enter both prices in dollars, using decimal points and a comma between prices; for example, 1.20, 2.30.

**Editorial input note:** Enter the smaller price first, using two digits after each decimal point, a comma between prices, no spaces, and no currency symbols. Apply this no-spaces rule to the format example above.

In the diagram below, $ABCD$ is a square, $\angle CDE=72^\circ$ and $\angle CED=54^\circ$. Find the angle $\angle EAD$.

In a peak integer, the digits increase to a peak digit and then decrease. This peak is not the first or last digit. For example, 256850 is a peak integer and its peak digit is 8; 3456219 and 278864 are not peak integers. How many different 4-digit peak integers have a peak digit of 7?

Two pipes A and B are opened together to fill a tank. Both pipes fill the tank in a certain time. If A alone takes 36 minutes more than the time tank by A and B together to fill the tank. Find the time taken by A and B to fill the tank when both the pipes are opened together.

Given a set of 12 points, seven of which lie on the same line. Moreover, the rest five points don’t lie on that line and no three of them are collinear. How many triangles can be formed by choosing the vertices from this set?

Stacy has 63 marbles. She places them in three piles, with an odd number of marbles in each pile. In how many different ways can she accomplish this?

A “stair-step” figure is made of alternating black and white squares in each row. Rows 1 through 4 are shown. All rows begin and end with a white square. How many black squares are there in the 37th row?

Albert, Bernard, Cindy and David look at a number. Then, they give following statements.

- Albert: This is a square number.
- Bernard: The sum of its digit is 48.
- Cindy: This number is divisible by 9.
- David: 10201 times of this number is also a square number.

It is known that three of the four statements above are correct. Who gives incorrect statement?

Rectangle $ABCD$ is cut into four smaller rectangles, as shown in the figure. The four smaller rectangles have the properties:

a) The perimeters of three of them are 11, 16 and 19.

b) The perimeter of the fourth is neither the biggest nor the smallest of the four.

What is the perimeter of the original rectangle $ABCD$?

A tank is filled in 5 hours by three pipes A, B and C. C is twice as fast as B. B is twice as fast as A. How much time will pipe A alone take to fill the tank?

There are 5 red balls and 7 green balls in a box. We pick 3 balls from the box randomly at the same time. What is the probability that we pick 3 green balls?