Iranian 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 3 cm is (3)^3 = 27 cm^3.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1)/(x2 - x1). Therefore, the slope of the line passing through the points (2, 3) and (5, 7) is (7 - 3)/(5 - 2) = 2.

The equation of a line passing through a point (x1, y1) and having a slope of m is given by the formula y - y1 = m(x - x1). Therefore, the equation of the line passing through the point (3, 4) and having a slope of -2 is y - 4 = -2(x - 3), which simplifies to y = -2x + 10.

To solve the equation x^2 - 5x + 6 = 0, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac))/2a. In this case, a = 1, b = -5, and c = 6. Plugging these values into the formula, we get x = (-(-5) ± √((-5)^2 - 4(1)(6)))/2(1) = (5 ± √(25 - 24))/2 = (5 ± 1)/2. Therefore, the solutions to the equation x^2 - 5x + 6 = 0 are x = 2 and x = 3.

We can simplify the expression as follows: 2^3 + 3^3 + 4^3 - 3^2 - 4^2 = (2^3 + 3^3 + 4^3) - (3^2 + 4^2) = (8 + 27 + 64) - (9 + 16) = 99 - 25 = 110.

We can use the divisibility rule for 11 to find the remainder. The rule states that a number is divisible by 11 if the alternating sum of its digits is divisible by 11. In this case, the alternating sum of the digits of 123456789 is (1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9) = 3. Therefore, the remainder when 123456789 is divided by 11 is 3.

We can check each of the options to see if it satisfies the condition. For n = 2, we have n^2 + n + 1 = 2^2 + 2 + 1 = 7, which is a prime number. For n = 3, we have n^2 + n + 1 = 3^2 + 3 + 1 = 13, which is also a prime number. For n = 4, we have n^2 + n + 1 = 4^2 + 4 + 1 = 21, which is not a prime number. For n = 5, we have n^2 + n + 1 = 5^2 + 5 + 1 = 31, which is a prime number. Therefore, the smallest positive integer n such that n^2 + n + 1 is a prime number is 2.

The first 100 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ..., 541. We can use a calculator to find the sum of these numbers, which is 22995.

We can factor 1001 as follows: 1001 = 7 × 11 × 13. Therefore, the largest prime factor of 1001 is 19.

To solve the equation sin(x) = 1/2, we can use the inverse sine function (also known as arcsin). The inverse sine of 1/2 is π/6. Therefore, the solution to the equation sin(x) = 1/2 is x = π/6.

We can use Heron's formula to find the area of the triangle. Heron's formula states that the area of a triangle with sides a, b, and c is given by the formula √(s(s-a)(s-b)(s-c)), where s is the semiperimeter of the triangle. In this case, the semiperimeter of the triangle is (1 + 3 + 5)/2 = 4.5. Plugging this value into Heron's formula, we get the area of the triangle as √(4.5(4.5-1)(4.5-3)(4.5-5)) = √(4.5(3.5)(1.5)(-0.5)) = √(10.125) = 10 square units.

We can use the method of cylindrical shells to find the volume of the solid. The volume of a cylindrical shell is given by the formula 2πrhΔx, where r is the radius of the shell, h is the height of the shell, and Δx is the thickness of the shell. In this case, the radius of the shell is x, the height of the shell is 4 - x^2 - x^2 = 4 - 2x^2, and the thickness of the shell is Δx. Therefore, the volume of the solid is given by the integral ∫2πx(4 - 2x^2)dx from x = 0 to x = 2. Evaluating this integral, we get the volume of the solid as 64π cubic units.

To solve the differential equation dy/dx = x^2 + 1, we can use the method of separation of variables. We can rewrite the equation as dy = (x^2 + 1)dx and then integrate both sides. Integrating the left side gives y = ∫dy = y + C1, where C1 is a constant of integration. Integrating the right side gives x^3/3 + x + C2, where C2 is a constant of integration. Combining these two results, we get the general solution to the differential equation as y = x^3/3 + x + C, where C = C1 + C2 is a constant.