European Girls' Mathematical Olympiad (EGMO)

Using the Pythagorean theorem, we have $a^2 + b^2 = c^2$. Substituting the given values, we get $6^2 + b^2 = 10^2$. Solving for $b$, we get $b = 8$.

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

The volume of a cube is given by the formula $V = a^3$. Substituting the given value, we get $V = 4^3 = 64$.

The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $y - y_1 = (y_2 - y_1)/(x_2 - x_1) * (x - x_1)$. Substituting the given values, we get $y - 3 = (7 - 3)/(5 - 2) * (x - 2)$. Simplifying, we get $y = 2x + 1$.

The slope of a line in the form $Ax + By = C$ is given by the formula $m = -A/B$. Substituting the given values, we get $m = -3/-2 = 3/2$.

We can solve this equation using the quadratic formula: $x = (-b ± √(b^2 - 4ac))/(2a)$. Substituting the given values, we get $x = (-(-5) ± √((-5)^2 - 4(2)(3)))/(2(2))$. Simplifying, we get $x = (5 ± √(25 - 24))/(4) = (5 ± 1)/(4) = 1/2$ or $x = 2$.

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

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

The area of a triangle is given by the formula $A = (1/2)bh$. Substituting the given values, we get $A = (1/2)(10)(8) = 40$.

The volume of a cylinder is given by the formula $V = πr^2h$. Substituting the given values, we get $V = π(3)^2(5) = 45π$.

The equation of a circle with center $(h, k)$ and radius $r$ is given by the formula $(x - h)^2 + (y - k)^2 = r^2$. Substituting the given values, we get $(x - 2)^2 + (y - 3)^2 = 5^2 = 25$.

To solve this equation, we can subtract 5 from both sides: $3x + 5 - 5 = 17 - 5$. Simplifying, we get $3x = 12$. Dividing both sides by 3, we get $x = 4$.

The area of a sector of a circle is given by the formula $A = (θ/360)πr^2$, where θ is the central angle in degrees and r is the radius of the circle. Substituting the given values, we get $A = (60/360)π(10)^2 = (1/6)π(100) = 25π$.

The volume of a sphere is given by the formula $V = (4/3)πr^3$. Substituting the given value, we get $V = (4/3)π(6)^3 = 36π$.

The slope of the line y = 2x + 1 is 2. The slope of a line perpendicular to this line is the negative reciprocal of 2, which is -1/2. Using the point-slope form of a line, we can write the equation of the line as $y - 4 = (-1/2)(x - 3)$. Simplifying, we get $y = -(1/2)x + 7$.