International Team Mathematical Competition

The sum of the first 100 positive integers is (1 + 2 + 3 + ... + 100) = (100 * 101) / 2 = 5050. The sum of the first 100 even integers is (2 + 4 + 6 + ... + 200) = (100 * 202) / 2 = 10100. Therefore, the sum of the first 100 odd integers is 10100 - 5050 = 5050.

The area of a regular hexagon with side length 'a' is given by (3√3 / 2) * a². Substituting a = 6 cm, we get the area as (3√3 / 2) * 6² = 72√3 cm².

There are 4 ways to choose 1 man from 4 men, and 6 ways to choose 1 woman from 6 women. To form a committee of 3 people, we can choose the remaining 1 person from either the men or the women. There are 3 men and 5 women left, so there are 3 + 5 = 8 ways to choose the remaining person. Therefore, the total number of ways to form the committee is 4 * 6 * 8 = 200.

Using the formula (a + b)² = a² + 2ab + b², we can expand the expression as follows: (2 + 3i)² - (2 - 3i)² = (2² + 2 * 2 * 3i + (3i)²) - (2² - 2 * 2 * 3i + (3i)²) = (4 + 12i + 9i²) - (4 - 12i + 9i²) = 4 + 12i + 9i² - 4 + 12i - 9i² = 16.

The farmer plants corn on 40% of the land, which is 0.4 * 120 = 48 acres. He plants soybeans on 30% of the land, which is 0.3 * 120 = 36 acres. Therefore, he plants wheat on 120 - 48 - 36 = 36 acres.

Let the lengths of the two sides be 3x and 4x. Then, the third side has length 42 - 3x - 4x = 42 - 7x. Since the perimeter of the triangle is 42 cm, we have 3x + 4x + 42 - 7x = 42. Solving for x, we get x = 6. Therefore, the length of the third side is 42 - 7x = 42 - 7 * 6 = 14 cm.

The second train travels 75 mph - 60 mph = 15 mph faster than the first train. Therefore, it takes 1 hour for the second train to cover the distance that the first train travels in 1 hour. Since the second train leaves 1 hour after the first train, it will overtake the first train 1 hour after 11:00 AM, which is 1:00 PM.

The area of a rectangle is given by length * width. Substituting the given values, we get the area as 12 meters * 8 meters = 96 square meters.

There are 60% * 100 = 60 men and 40% * 100 = 40 women in the company. 20% of the men leave, which is 0.2 * 60 = 12 men. 30% of the women leave, which is 0.3 * 40 = 12 women. Therefore, the number of remaining employees is 100 - 12 - 12 = 76. The number of remaining women is 40 - 12 = 28. Therefore, the percentage of remaining employees who are women is (28 / 76) * 100 = 44%.

The circumference of a circle is given by 2πr, where r is the radius. Substituting r = 10 cm, we get the circumference as 2π * 10 cm = 20π cm.

There are a total of 10 + 15 + 20 = 45 balls in the bag. The probability of selecting a red ball is the number of red balls divided by the total number of balls, which is 10 / 45 = 1/5.

The farmer plants corn on 40% of the land, which is 0.4 * 100 = 40 acres.

The first train travels for 2 hours, from 10:00 AM to 12:00 PM, at a speed of 60 mph. Therefore, it travels 2 * 60 = 120 miles. The second train travels for 1 hour, from 11:00 AM to 12:00 PM, at a speed of 75 mph. Therefore, it travels 1 * 75 = 75 miles. The distance between the two trains is 120 miles + 75 miles = 195 miles.

The perimeter of a rectangle is given by 2 * (length + width). Substituting the given values, we get the perimeter as 2 * (12 meters + 8 meters) = 2 * 20 meters = 40 meters.

There are 60% * 100 = 60 men and 40% * 100 = 40 women in the company. 20% of the men leave, which is 0.2 * 60 = 12 men. 30% of the women leave, which is 0.3 * 40 = 12 women. Therefore, the number of remaining employees is 100 - 12 - 12 = 76.