China Team Mathematical Olympiad

The sum of the first n positive integers is given by the formula n(n+1)/2. Substituting n = 100, we get 100(101)/2 = 5050.

The area of a circle is given by the formula πr^2. Substituting r = 5, we get π(5)^2 = 25π.

The volume of a cube is given by the formula s^3. Substituting s = 3, we get 3^3 = 27.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1)/(x2 - x1). Substituting the given points, we get (4 - 2)/(3 - 1) = 2.

The equation of a line passing through a point (x1, y1) and having a slope m is given by the formula y - y1 = m(x - x1). Substituting the given point and slope, we get y - 3 = -2(x - 2), which simplifies to y = -2x + 7.

We can solve this equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting a = 1, b = -4, and c = 3, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1) = (4 ± √(16 - 12)) / 2 = (4 ± √4) / 2 = (4 ± 2) / 2. Therefore, the solutions are x = 1 and x = 3.

We can evaluate this expression using the formula for the sum of cubes: a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). Substituting a = 2, b = 3, and c = 4, we get 2^3 + 3^3 + 4^3 = (2 + 3 + 4)(2^2 + 3^2 + 4^2 - 2(3)(4) - 2(3)(4) - 2(4)(2)) = 9(4 + 9 + 16 - 24 - 24 - 16) = 9(100) = 100.

A fair coin has two possible outcomes: head and tail. Since each outcome is equally likely, the probability of getting a head is 1/2.

There are 36 possible outcomes when rolling two dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36 = 1/6.

This sequence is the square of the natural numbers. The next term is the square of 5, which is 25.

The sum of the first 100 prime numbers is 24133.

The largest prime factor of 1001 is 41.

There are 33 positive integers less than 100 that are divisible by 3 or 5.

The value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3 is 3025.

The sum of the first 50 positive even integers is 2550.