Math Olympiad: 60 MCQs with answers

A · 15

A multiple of both 3 and 5 must be divisible by 15. Among the options, 15 is divisible by both 3 and 5.

A · 26 cm

Perimeter of a rectangle is $2 \times (\text{length} + \text{width}) = 2 \times (8 + 5) = 2 \times 13 = 26$ cm.

C · $\frac{6}{8}$

Multiply numerator and denominator of $\frac{3}{4}$ by 2 to get $\frac{6}{8}$, which is equivalent.

C · 36 cm²

Area of a square is side × side = $6 \times 6 = 36$ cm².

D · 7

Divide 56 by 8: $56 \div 8 = 7$. So the missing number is 7.

A · 32

An even number is divisible by 2. Among the options, 32 is divisible by 2, so it is even.

B · False

A prime number has exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor, so it is not prime.

A · True

An odd number can be written as $2k+1$. The sum of two odd numbers is $(2k+1)+(2m+1)=2(k+m+1)$, which is even.

A · 16

The correct answer is $16$ because $9 + 7 = 16$.

D · Square

A square has four equal sides, while a rectangle has opposite sides equal.

B · 7

The correct answer is $7$ because $12 - 5 = 7$.

B · 15

The smallest number among the options is $15$.

A · 12

The correct answer is $12$ because $3 imes 4 = 12$.

D · 15

The correct answer is $15$, which is a two-digit number.

D · 4

The next number in the sequence is $4$, as it follows the pattern of counting.

C · 16 cm

The perimeter of a square is calculated as $4 imes ext{side length} = 4 imes 4 = 16$ cm.

D · 20

The correct answer is $20$, as it can be expressed as $10 imes 2$.

C · 15

The correct answer is $15$ because $5 + 5 + 5 = 15$.

A · 9

The correct option is 9 because $7 + 2 = 9$.

A · 100

The correct option is 100 because it has three digits.

C · 15 cm²

The correct option is 15 cm² because the area is calculated as length × width: $5 imes 3 = 15$.

A · Circle

The correct option is a Circle because it is a round shape with no corners.

C · 6

The correct option is 6 because $10 - 4 = 6$.

D · 42

The differences between consecutive terms are 4, 6, 8, 10, so the next difference is 12. Thus $30 + 12 = 42$.

A · $\$25$

One dozen equals 12 eggs. Cost per egg is $\$60 \div 12 = \$5$. So 5 eggs cost $5 \times \$5 = \$25$.

C · 55

The sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$. For $n=10$: $\frac{10 \times 11}{2} = 55$.

D · 5

Order of operations: first multiply $3 \times 4 = 12$. Then $15 - 12 + 2 = 3 + 2 = 5$.

C · 4

The prime numbers between 10 and 20 are 11, 13, 17, and 19. So there are 4 prime numbers.

D · 100 cents

3 quarters = 75 cents, 2 dimes = 20 cents, 1 nickel = 5 cents. Total = 75 + 20 + 5 = 100 cents.

D · Right-angled

This triangle is a right-angled triangle because it satisfies the Pythagorean theorem: $5^2 + 12^2 = 13^2$.

D · 12

The LCM of 4 and 6 is 12, as it is the smallest number that both 4 and 6 divide evenly into.

B · 40 cm²

The area of a rectangle is calculated using the formula: Area = length × width = $10 imes 4 = 40$ cm².

B · $\frac{3}{5}$

$\frac{3}{5}$ is greater than $\frac{1}{2}$ because $3/5 = 0.6$ and $1/2 = 0.5$.

B · 15

Calculating $9 + 6 - 4$ gives $15$, as you first add $9$ and $6$ to get $15$, then subtract $4$.

A · 8

You start with 15 apples and give away 7, so $15 - 7 = 8$ apples remain.

D · 17

A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. $17$ is prime.

C · 180 degrees

The sum of the angles in any triangle is always $180$ degrees.

D · 8

A factor of a number is a whole number that can be divided evenly into that number. $8$ is a factor of $24$ because $24 ÷ 8 = 3$.

A · 97.5 degrees

At 3:15, the hour hand is at $97.5$ degrees from 12 o'clock (3 hours = $90$ degrees + 1/4 of an hour = $7.5$ degrees).

D · 8:5

The ratio of length to width is given by the dimensions of the rectangle, which is $\frac{8}{5}$ or $8:5$.

B · \( \frac{3}{10} \)

The total number of fruits is $10 + 15 + 5 = 30$. The fraction of bananas is $\frac{15}{30} = \frac{1}{2}$.

A · 5:30 PM

Adding 2 hours and 30 minutes to 3:00 PM gives 5:30 PM.

B · 12

The number 12 is a multiple of both 3 and 4, as $12 \div 3 = 4$ and $12 \div 4 = 3$.

C · 5 cm

The perimeter of a rectangle is given by the formula $P = 2(l + w)$. Here, $30 = 2(10 + w)$, leading to $w = 5$ cm.

B · Wednesday

There are 7 days in a week. $100 \div 7 = 14$ weeks and 2 days remainder. Two days after Monday is Wednesday.

B · 12

To find the number of boxes needed, divide the total number of apples by the number of apples per box: $$\frac{120}{8} = 15$$.

B · 4:00 PM

First, calculate the travel time: $$\frac{150 \text{ km}}{60 \text{ km/h}} = 2.5 \text{ hours}$$. Adding 2.5 hours to 2:00 PM gives 4:00 PM.

D · 96 m²

Let the width be $w$ meters. Then, the length is $2w$. The perimeter is given by $$2(w + 2w) = 48$$, which simplifies to $$6w = 48$$, so $w = 8$. Thus, the area is $$w \times 2w = 8 \times 16 = 128 m²$$.

C · 0.6

The total number of balls is $3 + 5 + 2 = 10$. The number of non-blue balls is $3 + 2 = 5$. Therefore, the probability is $$\frac{5}{10} = 0.5$$.

D · 23

Let the integers be $x$, $x+1$, and $x+2$. Their sum is $$x + (x + 1) + (x + 2) = 72$$, which simplifies to $$3x + 3 = 72$$, leading to $$3x = 69$$, hence $x = 23$.

D · 8

The ratio of flour to sugar is $\frac{2}{3}$. For 12 cups of sugar, the amount of flour needed is $$\frac{2}{3} \times 12 = 8$$ cups.

D · 12

The number of girls is $0.4 \times 30 = 12$. Therefore, the number of boys is $30 - 12 = 18$.

A · \frac{9}{12}

If 3 slices are eaten, the number of slices left is $12 - 3 = 9$. The fraction of the pizza left is $$\frac{9}{12} = \frac{3}{4}$$.

C · $15

The total cost for 3 books is $3 \times 15 = 45$. The change from a $50 bill is $50 - 45 = 5$.

A · 60 km/h

Average speed is calculated as total distance divided by total time: $$\frac{240 \text{ km}}{3 \text{ h}} = 80 \text{ km/h}$$.

D · 12 meters

Let the width be $w$ meters. Then the length is $2w$ meters. The area is given by $w \times 2w = 288$. Solving $2w^2 = 288$ gives $w^2 = 144$, so $w = 12$ meters.

C · \( \frac{4}{30} \)

The probability of drawing a red marble first is \( \frac{4}{20} \), and then the probability of drawing a green marble is \( \frac{10}{19} \). Thus, the combined probability is \( \frac{4}{20} \times \frac{10}{19} = \frac{40}{380} = \frac{4}{30} \).

C · 480 square meters

The area of the larger rectangle (including the path) is \( (60 + 4)(40 + 4) = 64 \times 44 = 2816 \) square meters. The area of the original field is \( 60 \times 40 = 2400 \) square meters. Thus, the area of the path is \( 2816 - 2400 = 416 \) square meters.

D · 30

Let the smallest even number be $x$. Then the four numbers are $x, x+2, x+4, x+6$. The equation is $x + (x + 2) + (x + 4) + (x + 6) = 128$. Simplifying gives $4x + 12 = 128$, so $4x = 116$ and $x = 29$. The smallest even number is 30.

D · 55%

Initially, the number of boys is $0.6 \times 240 = 144$. After 20 boys leave, there are $144 - 20 = 124$ boys left. The total number of remaining students is $240 - 20 = 220$. The percentage of boys is \( \frac{124}{220} \times 100 \approx 56.36\% \), which rounds to 55%.