AMO 2027 Grades 8-9 SET 12 - Self-study

Last year, three fifths of the Outing Club were girls, but this year the number of boys doubled and six new girls joined. There are now as many boys in the club as girls. How many members did the club have last year?

In the below figure, we have $AB=13$, $BC=15$ and $AC=4$. Find the area of the rectangle $BECD$.

What will be the probability for a leap year to have 52 Mondays and 53 Sundays?

$X$, $Y$ and $Z$ are distinct non-zero digits used to form the two 3-digit whole numbers $\overline{XYZ}$ and $\overline{ZYX}$ so that $\overline{XYZ}-\overline{ZYX}=297$. How many integer values of $\overline{XYZ}$ are possible?

If $p$ is a prime number, then which of the following is NOT ALWAYS true?

If the four-digit number $\overline{66N2}$ is divisible by 28, find $N$.

Three pipes A, B and C can fill the cistern in 12 hours, 20 hours and 15 hours respectively. If A is operated for all the time and pipe B and C are opened alternately which B being first. Find the time taken to fill the cistern.

**Editorial input format:** Explain the missing switching condition and give a correctly justified result under an explicitly stated schedule. This source omission requires manual grading.

Find the missing number.

**Editorial input format:** Use the editorial rules described in the solution: outside-number sum for a, main-diagonal symmetry for b, and equal row differences for c. Enter the three answers in a–c order, separated by commas.



**Editorial input note:** Use no spaces before or after commas.

In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both tea and coffee.

a) How many students like only tea?

b) How many students like only coffee?

c) How many students like neither tea nor coffee?

d) How many students like only one of tea or coffee?

e) How many students like at least one of the beverages?

**Editorial input format:** Enter all five counts in a–e order, separated by commas.

**Editorial input note:** Use no spaces before or after commas.

Find the next term of the following sequence:
$$2,12,1112,3112,132112,\ldots$$

The chart below provides information on how bonuses of certain amounts were distributed among the employees of a business:



a) What was the average bonus per employee?

b) If the median bonus is $m$ dollars, the average bonus $a$ dollars, and the mode bonus is $p$ dollars, which of the must be true?

A $m\lt a\lt p$.

B $m\lt a=p$.

C $m=p\lt a$.

D $m\lt p\lt a$.

**Editorial input format:** Enter the average for part a and the option letter for part b, separated by a comma.

**Editorial input note:** Use no spaces before or after commas.

Find the missing numbers.

What is the largest number of zeros that can occur at the end of $1^n+2^n+3^n+4^n$ for any positive integer $n$?

A rectangle has length $x$ and width $y$. A triangle has base 16 and height $x$. If the area of the rectangle is equal to the area of the triangle, then find the value of $y$.

A tank 64cm long, 46cm wide and 36cm high was $\frac18$ full of water. Water was then pumped out of it at a rate of $16\mathrm{cm}^3$ per second. How long would it take to empty all the water in the tank?

**Editorial input format:** Enter the time in seconds.