Mathematics Olympiads

The sum of all positive integers less than 100 that are divisible by 3 is (3 + 6 + 9 + ... + 96 + 99) = 1665. The sum of all positive integers less than 100 that are divisible by 5 is (5 + 10 + 15 + ... + 90 + 95) = 1015. However, we have counted the numbers that are divisible by both 3 and 5 twice, so we need to subtract them once. The numbers that are divisible by both 3 and 5 are (15, 30, 45, 60, 75, 90), and their sum is 315. Therefore, the sum of all positive integers less than 100 that are divisible by 3 or 5 is 1665 + 1015 - 315 = 2365.

Let the lengths of the two sides be 3x cm and 4x cm. Then, the third side has length (42 - 3x - 4x) cm = (42 - 7x) cm. 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 longest side is 4x cm = 4 * 6 cm = 24 cm.

We can factor the equation as ((x - 2)(x - 3) = 0). Therefore, the solutions are x = 2 and x = 3.

Let E be the set of people who speak English and S be the set of people who speak Spanish. Then, n(E) = 4, n(S) = 3, and n(E ∩ S) = 2. By the principle of inclusion-exclusion, the number of people who speak at least one language is n(E ∪ S) = n(E) + n(S) - n(E ∩ S) = 4 + 3 - 2 = 9.

The vowels in the word "MATHEMATICS" are AEIOU. We can consider them as a single unit. Then, we have 8 letters (including the unit of vowels) to arrange. The number of ways to arrange 8 letters is 8!. However, the vowels can be arranged among themselves in 5! ways. Therefore, the number of ways to arrange the letters in the word "MATHEMATICS" such that the vowels always come together is 8! / 5! = 3780.