Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

How many factors of 2⁴ × 3⁵ × 10⁴ are perfect squares which are greater than 1?

What is the largest positive integer N such that (N² + 7N + 12)/(N² − N − 12) is also a positive integer?

Let a₁, a₂, ... be integers such that a₁ − a₂ + a₃ − a₄ + ⋯ + (−1)ⁿ⁻¹aₙ = n, for n ≥ 1. Then a₅₁ + a₅₂ + ⋯ + a₁₀₂₃ equals

In 2010, a library contained a total of 11500 books in two categories - fiction and non-fiction. In 2015, the library contained a total of 12760 books in these two categories. During this period, there was 10% increase in the fiction category while there was 12% increase in the n…

In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is

Let f be a function such that f(mn) = f(m) × f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals

Let a, b, x, y be real numbers such that a² + b² = 25, x² + y² = 169 and ax + by = 65. If k = ay − bx, then

Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is 3/2 times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is