Romanian Master of Mathematics

Description: Romanian Master of Mathematics is a prestigious mathematical competition for high school students in Romania. It is held annually and consists of three rounds: a preliminary round, a semifinal round, and a final round. The competition is open to students of all ages, but the majority of participants are in their final year of high school. The competition is designed to test students' problem-solving skills, mathematical knowledge, and creativity.
Number of Questions: 5
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In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is known as the Pythagorean theorem. If the two shorter sides of a right triangle are 3 and 4 units long, what is the length of the hypotenuse?

  1. 5

  2. 7

  3. 9

  4. 11


Correct Option: A
Explanation:

Using the Pythagorean theorem, we have: (a^2 + b^2 = c^2) where (a) and (b) are the lengths of the shorter sides and (c) is the length of the hypotenuse. Substituting the given values, we get: (3^2 + 4^2 = c^2) (9 + 16 = c^2) (25 = c^2) (c = \sqrt{25}) (c = 5) Therefore, the length of the hypotenuse is 5 units.

A train leaves a station at 10:00 AM and travels at a speed of 60 miles per hour. Another train leaves the same station at 11:00 AM and travels in the same direction at a speed of 75 miles per hour. At what time will the second train overtake the first train?

  1. 12:00 PM

  2. 12:30 PM

  3. 1:00 PM

  4. 1:30 PM


Correct Option: B
Explanation:

To find the time at which the second train will overtake the first train, we need to find the time it takes for the second train to travel the distance between the two trains. The distance between the two trains is equal to the speed of the first train multiplied by the time difference between the two trains. The time difference between the two trains is 1 hour. Therefore, the distance between the two trains is: (60 \text{ mph} \times 1 \text{ hr} = 60 \text{ miles}) The speed of the second train is 75 mph. Therefore, the time it takes for the second train to travel 60 miles is: (60 \text{ miles} \div 75 \text{ mph} = 0.8 \text{ hr}) Converting 0.8 hours to minutes, we get: (0.8 \text{ hr} \times 60 \text{ min/hr} = 48 \text{ min}) Therefore, the second train will overtake the first train at 12:30 PM.

A company has 100 employees. 60% of the employees are male and the rest are female. How many female employees does the company have?

  1. 30

  2. 40

  3. 50

  4. 60


Correct Option: B
Explanation:

The number of male employees is (100 \times 60/100 = 60). Therefore, the number of female employees is (100 - 60 = 40).

A store sells a shirt for \$20. The store offers a 20% discount on the shirt. What is the sale price of the shirt?

  1. \$16

  2. \$17

  3. \$18

  4. \$19


Correct Option: A
Explanation:

The discount is (20 \times \$20/100 = \$4). Therefore, the sale price of the shirt is ($20 - \$4 = \$16).

A farmer has 100 acres of land. He plants corn on 60% of the land, soybeans on 20% of the land, and wheat on the rest of the land. How many acres of land does the farmer plant wheat on?

  1. 10

  2. 20

  3. 30

  4. 40


Correct Option: B
Explanation:

The farmer plants corn on (100 \times 60/100 = 60) acres of land and soybeans on (100 \times 20/100 = 20) acres of land. Therefore, the farmer plants wheat on (100 - 60 - 20 = 20) acres of land.

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