Simplify the following expression:
1/(1 + loga (bc)) + 1/(1 + logb (ac)) + 1/(1 + logc (ac))
Note that a, b, c are greater than 0 but none is equal to 1.
Suppose the binary operation ☆ is defined so that
x ☆ y = (x + y) / (1 + xy).
Simplify the expression
{ … [(2 ☆ 3) ☆ 4] ☆ … } ☆ 2021
If in the following image AC is the diameter of the circle where AB = 4cm and BC = 1cm. Find the x?
The gothic window design below has arc BC centered at A and arc AC centered at B. Semicircles D and E have a radius of 1 and the circle F is tangent to the arcs and the semicircles. What is the radius of circle F?
Let ABC be a triangle with AB < AC < BC. Let the incentre and incircle of triangle ABC be I and ω, respectively. Let X be the point on the line BC different from C such that the line through X parallel to AC is tangent to ω. Similarly, let Y be the point on the line BC different from B such that the line through Y parallel to AB is tangent to ω. Let AI intersect the circumcircle of triangle ABC again at P ≠ A. Let K and L be the midpoints of AC and AB, respectively. Prove that ∠KIL + ∠YPX = 180°.
