Canadian Mathematical Olympiad

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

The area of a circle is given by the formula πr^2. Therefore, the area of a circle with radius 5 cm is π(5)^2 = 25π cm^2.

The volume of a cube is given by the formula s^3, where s is the side length. Therefore, the volume of a cube with side length 4 cm is 4^3 = 64 cm^3.

The equation of a line that passes through two points (x1, y1) and (x2, y2) is given by the formula y - y1 = (y2 - y1)/(x2 - x1) * (x - x1). Substituting the values of the two points, we get y - 3 = (7 - 3)/(5 - 2) * (x - 2). Simplifying this equation, we get y = 2x + 1.

To solve the equation x^2 - 4x + 3 = 0, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting the values of a, b, and c, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1). Simplifying this equation, we get x = (4 ± √(16 - 12)) / 2. Therefore, the solutions to the equation are x = 1 and x = 3.

The derivative of a function f(x) is given by the formula f'(x) = lim_(h->0) [f(x+h) - f(x)] / h. Substituting the values of f(x) and f(x+h), we get f'(x) = lim_(h->0) [(x+h)^3 - 2(x+h)^2 + 3(x+h) - 4 - (x^3 - 2x^2 + 3x - 4)] / h. Simplifying this equation, we get f'(x) = lim_(h->0) [3x^2 + 6xh + 3h^2 - 4x - 4h + 3 - x^3 + 2x^2 - 3x + 4] / h. Dividing both the numerator and denominator by h, we get f'(x) = lim_(h->0) [3x^2 + 6x + 3h - 4 - x^3 + 2x^2 - 3x + 4] / 1. Simplifying this equation further, we get f'(x) = 3x^2 - 4x + 3.

The integral of a function f(x) is given by the formula F(x) = ∫f(x) dx. Substituting the value of f(x), we get F(x) = ∫(2x^3 - 3x^2 + 4x - 5) dx. Integrating each term separately, we get F(x) = (2/4)x^4 - (3/3)x^3 + (4/2)x^2 - 5x + C. Simplifying this equation, we get F(x) = x^4 - x^3 + 2x^2 - 5x + C, where C is the constant of integration.

When a fair coin is tossed, there are two possible outcomes: head or tail. Since the coin is fair, each outcome is equally likely to occur. Therefore, the probability of getting a head is 1/2.

When two fair dice are rolled, there are 36 possible outcomes. The outcomes that result in a sum of 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.

The expected value of a random variable X is given by the formula E(X) = ΣxP(X = x), where x is the value of the random variable and P(X = x) is the probability of X taking the value x. In this case, the random variable X is the sum of two fair dice. The possible values of X are 2, 3, 4, ..., 12. The probability of getting each value is 1/36. Therefore, the expected value of X is E(X) = 2(1/36) + 3(1/36) + 4(1/36) + ... + 12(1/36) = 7.

The median of a set of numbers is the middle value when the numbers are arranged in ascending order. In this case, the numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. The middle value is the 5th value. Therefore, the median of the set of numbers is 10.

The mode of a set of numbers is the value that occurs most frequently. In this case, the value 3 occurs twice, while all other values occur only once. Therefore, the mode of the set of numbers is 3.

The range of a set of numbers is the difference between the largest and smallest values in the set. In this case, the largest value is 19 and the smallest value is 1. Therefore, the range of the set of numbers is 19 - 1 = 18.

The standard deviation of a set of numbers is a measure of how spread out the numbers are. It is calculated by taking the square root of the variance. The variance is calculated by finding the average of the squared differences between each number and the mean. In this case, the mean of the set of numbers is 10. The variance is calculated as follows: [(1 - 10)^2 + (3 - 10)^2 + (5 - 10)^2 + (7 - 10)^2 + (9 - 10)^2 + (11 - 10)^2 + (13 - 10)^2 + (15 - 10)^2 + (17 - 10)^2 + (19 - 10)^2] / 10 = 44.44. The standard deviation is the square root of the variance, which is √44.44 = 6.66.