ABC is right-angled at A. The angle bisector from A meets BC at D, so that DAB = 45°. If CD = 1 and BD = AD + 1, find the lengths of AC and AD.
An ant, located in a square field, is 13 meters from one of the corner posts of the field, 17 meters from the corner post diagonally opposite that one, and 20 meters from a third corner post. Find the area of the field. Assume the land is flat.
Let a, b, and c be positive real numbers such that abc = 1. Show that a2 + b2 + c2 a3 + b3 + c3.
Let f(x) = sin(x) + sin(x°), with domain the real numbers. Is f a periodic function?
(Note: sin(x) is the sine of a real number, x, (or, equivalently, the sine of x radians), while sin(x°) is the sine of x degrees.)
Show that, for each n 3, n! can be represented as the sum of n distinct divisors of itself. (For example, 3! = 1 + 2 + 3.)
A point P is chosen at random inside an equilateral triangle of side length 1. Find the expected value of the sum of the (perpendicular) distances from P to the three sides of the triangle.
Find all powers of 2 such that, after deleting the first digit, another power of 2 remains. (For example, 25 = 32. On deleting the initial 3, we are left with 2 = 21.) Numbers are written in standard decimal notation, with no leading zeroes.
A triangle has two acute angles, A and B. Show that the triangle is right-angled if, and only if, sin2A + sin2B = sin(A + B).
Let n be a positive integer. Prove that n! + 1 is composite for infinitely many values of n.
Back to top