What is the minimum number of weights which enable us to weigh any integer number of grams of gold from 1 to 100 on a standard balance with two pans? (Weights can be placed only on the left pan.)

A class consists of 30 students. Each of them has registered for 5 courses. Each course instructor conducts an exam out of 200 marks. The average percentage marks of all 30 students across all courses they have registered for, is 80%. Two of them apply for revaluation in a course…

If one of the lines given by the equation \(2x^2 + axy + 3y^2 = 0\) coincides with one of those given by \(2x^2 + bxy - 3y^2 = 0\), and the other lines represented by them are perpendicular, then \(a^2 + b^2\) is ______

There are two taps, T1 and T2, at the bottom of a water tank, either or both of which may be opened to empty the water tank, each at a constant rate. If T1 is opened keeping T2 closed, the water tank (initially full) becomes empty in half an hour. If both T1 and T2 are kept open,…

If a function \(f(a) = \max(a, 0)\), then the smallest integer value of \(x\) for which the equation \(f(x - 3) + 2f(x + 1) = 8\) holds true is ______

There are 5 parallel lines on the plane. On the same plane, there are \(n\) other lines which are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of \(n\)?

In a class, 60% and 68% of students passed their Physics and Mathematics examinations respectively. Then at least ______ percentage of students passed both their Physics and Mathematics examinations.

It is given that the sequence \(\{x_n\}\) satisfies \(x_1 = 0\), \(x_{n+1} = x_n + 1 + 2\sqrt{1 + x_n}\) for \(n = 1, 2, \dots\). Then \(x_{31}\) is ______

Suppose that a real-valued function \(f(x)\) of real numbers satisfies \(f(x + xy) = f(x) + f(xy)\) for all real \(x, y\), and that \(f(2020) = 1\). Compute \(f(2021)\).