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

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

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

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

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

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

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

The real root of the equation 2^(6x) + 2^(3x+2) − 21 = 0 is

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…