For how many distinct real values of x does the equation below hold true? (Consider a > 0.) x²·logₐ(16)/logₐ(32) − logₐ(64)/logₐ(32) − x = 0

There are five dustbins along a circular path at different places. Ramesh takes multiple rounds of the path every morning, always at the same speed. He noticed that it took him a different number of steps to walk between any two consecutive dustbins. Ramesh also noticed that star…

A farmer has a quadrilateral parcel of land with a perimeter of 700 feet. Two opposite angles of that parcel of land are right angles, while the remaining two are not. The farmer wants to do organic farming on that parcel of land. The cost of organic farming is Rs. 400 per square…

An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running par…

Consider the quadratic function f(x) = ax² + bx + a having two irrational roots, with a and b being two positive integers, such that a, b ≤ 9. If all such permissible pairs (a, b) are equally likely, what is the probability that a + b is greater than 9?

A solid trophy, consisting of two parts, has been designed in the following manner: the bottom part is a frustum of a cone with the bottom radius 30 cm, the top radius 20 cm, and height 40 cm, while the top part is a hemisphere with radius 20 cm. Moreover, the flat surface of the…

Eight employees of an organization have been rated on a scale of 1 to 50 for their performance. All ratings are integers. The overall average rating of the eight employees is 30. While the five employees with the highest ratings average 38, the five employees with the lowest rati…

If a, b, and c are all positive integers, with 4a > b, then which of the following conditions is BOTH NECESSARY AND SUFFICIENT for the expression ∛[(3)^a (21)^(3a−b) (49)^(2b+c)] to be a positive integer?

Arun selected an integer x between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer x and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an x are possible?