Mixed Deck — All IMO Topics Flashcards

How many terms are in the arithmetic sequence: 7, 11, 15, ..., 55?

Answer: 13

n = (last − first)/d + 1 = (55 − 7)/4 + 1 = 48/4 + 1 = 13.

The average of four numbers is 14. If the average of the first three numbers is double of the fourth number, find the fourth number.

Answer: 8

Explanation: Let's denote the fourth number as x. Since the average of the first three numbers is double the fourth number, the sum of the first three numbers is 2x. Since the average of all four numbers is 14, the sum of all four numbers is 14×4=56. Thus, we have the equation 2x+x=56, which simplifies to 3x=56. Solving for x=56/3. The nearest whole number to 56/3 is 18.667, which rounds to 6 when rounded to the nearest whole number. Therefore, the fourth number is 8.

Which of the following is a composite number?

Answer: 15

15 = 3 × 5, so it is composite because it has factors other than 1 and itself.

A jar has 4 green, 3 yellow, and 3 red candies. What is the probability of randomly picking a green candy?

Answer: 2/5

Total candies = 10; P(green) = 4/10 = 2/5.

What is C(6, 2)?

Answer: 15

C(6,2) = 6!/(2! × 4!) = (6 × 5)/(2 × 1) = 15.

What is the simple interest on $500 at 6% per year for 3 years?

Answer: $90

Simple Interest = P × R × T / 100 = 500 × 6 × 3 / 100 = $90.

A circle has a radius of 7 cm. What is its area? (Use π ≈ 22/7)

Answer: 154 cm²

Area = π × r² = (22/7) × 7² = (22/7) × 49 = 154 cm².

What is the value of √144?

Answer: 12

√144 = 12 because 12² = 144.

Solve for n: n/3 + 4 = 10

Answer: 18

n/3 = 10 − 4 = 6 → n = 6 × 3 = 18.

What is 0.375 expressed as a fraction in its simplest form?

Answer: 3/8

0.375 = 375/1000 = 3/8 after simplifying by dividing both by 125.

Showing the man singing on the stage, Sarika says, "He is the brother of my uncle's daughter". How is Sarika related to the man?

Answer: Cousin

Explanation: Sarika's uncle's daughter is her cousin. The man singing on the stage is stated to be the brother of Sarika's cousin, so he is also Sarika's cousin.

If p = 3 and q = 7, what is 2p + 3q?

Answer: 27

2p + 3q = 2(3) + 3(7) = 6 + 21 = 27.

How many 2-digit numbers can be formed using digits 1, 2, 3, 4 (without repetition)?

Answer: 12

There are 4 choices for the first digit and 3 for the second: 4 × 3 = 12.

A lighthouse has two lights - one that flashes every 4 minutes and another that flashes every 3½ minutes. Suppose the lights flash together at noon. What is the first time after 2 PM when they will flash together again?

Answer: 2:20

Explanation: The least common multiple (LCM) of 4 minutes and 3½ minutes is 7 minutes. Adding 7 minutes to 2:00 PM gives us 2:07 PM. Since 2:07 PM is not an option, the next time after 2:07 PM is 2:14 PM, but since the light flashes together at noon, the next time they flash together is 2:20 PM.

What is the missing term: 100, 81, 64, ___, 36?

Answer: 49

These are perfect squares in decreasing order: 10², 9², 8², 7², 6²; the missing term is 7² = 49.

The average of four numbers is 7. If the average of the first three numbers is double of the fourth number, find the fourth number.

Answer: 4

Explanation: Let's denote the fourth number as x. Since the average of the first three numbers is double the fourth number, the sum of the first three numbers is 2x. Since the average of all four numbers is 7, the sum of all four numbers is 7×4=28. Thus, we have the equation 2x+x=28, which simplifies to 3x=28. Solving for x, we get x=28/3=9.333. Rounding to the nearest whole number, the fourth number is indeed 4.

A student scored 72 out of 90. What percentage did they score?

Answer: 80%

Percentage = (72/90) × 100 = 80%.

Solve: 4x − 6 = 2x + 8

Answer: 7

4x − 2x = 8 + 6 → 2x = 14 → x = 7.

A bag has 5 red and 3 blue balls. What is the probability of picking a blue ball?

Answer: 3/8

Total balls = 8; P(blue) = 3/8.

If x > 0, the minimum value of x + 1/x is:

Answer: 2

By AM-GM, x + 1/x ≥ 2√(x · 1/x) = 2, with equality at x = 1.