XXX_PMO2025_MOCK TEST

Let a, b, c be positive integers such that 3.14 = a + 1/(b + 1/c). What is a + b + c?

• 11

• 13

• 17

• 19

What is the area of a rhombus whose diagonals have lengths 14 and 48, respectively?

• 100

• 336

• 625

• 672

The arithmetic mean of 11 integers is 10. After adding 20 to each of the first four and subtracting 24 from each of the last seven, what is the new mean?

• 2

• 3

• 4

• 6

• 576

• 567

• 512

• 441

An urn contains two white and two black balls. John draws two balls simultaneously from the urn. If the balls are of different colors, he stops. Otherwise, he returns both balls to the urn and then repeats the process. What is the probability that he stops after exactly three draws?

• 1/16

• 2/27

• 1/8

• 4/27

• 1

• 2

• 3

• 4

There are 2024 coins laid out in a row. For some positive integer x, at least 2/3 of the first x coins are heads, and at least 4/5 of the last x coins are tails. What is the maximum possible value of x?

• 1012

• 1079

• 1288

• 1380

• 4

• 7

• 10

• 13

The integers from 1 to 2023 are written on a long blackboard in a straight line. Millie colors any number divisible by 23 in blue, any number divisible by 88 in yellow, any number divisible by both 23 and 88 in green, and any number not divisible by either in red. How many adjacent pairs of numbers have different colors?

• 222

• 220

• 218

• 216

• -1/5

• -1/3

• 1/5

• 1