China Girls' Mathematical Olympiad

To solve the equation x^2 - 4x + 3 = 0, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in the values a = 1, b = -4, and c = 3, we get x = (4 ± √(16 - 12)) / 2 = (4 ± 2) / 2. Therefore, x = 1 or x = 3.

To find the area of a triangle with sides of length a, b, and c, we can use Heron's formula: Area = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter of the triangle. In this case, s = (6 + 8 + 10) / 2 = 12. Plugging in the values a = 6, b = 8, and c = 10, we get Area = √(12(12-6)(12-8)(12-10)) = √(12 * 6 * 4 * 2) = 36 cm^2.

There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Therefore, the probability of randomly selecting a prime number between 1 and 100 is 25 / 100 = 1/4.

To solve the equation 2y + 3 = 7, we can subtract 3 from both sides to get 2y = 4. Then, we can divide both sides by 2 to get y = 2.

The volume of a cube with side length s is given by V = s^3. Plugging in the value s = 5, we get V = 5^3 = 125 cm^3.

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

To solve the equation x^3 - 2x^2 + x - 2 = 0, we can use the Rational Root Theorem to find the possible rational roots. The possible rational roots are ±1, ±2, and ±1/2. Plugging in these values, we find that x = 2 is a root of the equation. Therefore, x = 2 is the correct answer.

The area of a circle with radius r is given by the formula A = πr^2. Plugging in the value r = 10, we get A = π(10)^2 = 100π cm^2.

There are six possible outcomes when rolling a standard six-sided die: 1, 2, 3, 4, 5, and 6. Therefore, the probability of rolling a 6 is 1 / 6.

To solve the equation 3x + 5 = 17, we can subtract 5 from both sides to get 3x = 12. Then, we can divide both sides by 3 to get x = 4.

The volume of a sphere with radius r is given by the formula V = (4/3)πr^3. Plugging in the value r = 7, we get V = (4/3)π(7)^3 = 143π cm^3.

The sum of the first n positive even integers is given by the formula S = n(n+1). Plugging in the value n = 50, we get S = 50(51) = 2550.

To solve the equation 2x - 3 = 5, we can add 3 to both sides to get 2x = 8. Then, we can divide both sides by 2 to get x = 4.

The area of a triangle with base b and height h is given by the formula A = (1/2)bh. Plugging in the values b = 12 and h = 8, we get A = (1/2)(12)(8) = 48 cm^2.

There are 13 red cards in a deck of 52 cards. Therefore, the probability of drawing a red card is 13 / 52 = 1/4.