ISI Entrance Paper 2015 – from Indian Statistical Institute’s B.Stat Entrance

this is a work in progress. post problems, solutions and correction in the comment section

1. Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distint points. Show that the circle passing through these three points also passes through (0,1). Discussion
2. Find all such Natural number n such that 7 divides $5^n + 1$ Discussion
3. Find all functions f, such that |f(x) – f(y)| = 2|x-y|. Discussion
4. Say $0 < a_1 < a_2 < ... < a_n$ be n real numbers. Show that the equation $frac{a_1}{a_1 - x } + frac{a_2}{a_2 - x} + ... + frac{a_n}{a_n - x} = 2015$ has n real solutions.
5. Consider the set S = {1, 2, 3, …, j}. In a subset P of S, Max P be the maximum element of that subset. Show that the sum of all Max P (over all subsets of the set) is $(j-1)2^j + 1$
6. There are three unit circles each of which is tangential to the other two. A triangle is drawn such that each side of the triangle is tangential to exactly two of the circle. Find the length of sides of this triangle.
7. Let (m_1< m_2 < ldots m_{k-1}< m_k) be (k) distinct positive integers such that their reciprocals are in arithmetic progression. 1.Show that (k< m_1 + 2). 2. Give an example of such a sequence of length (k) for any positive integer (k).
8. Let $P(x) = x^7 + x^6 + b_5 x^5 + b_4 x^4 + ... + b_0$ and $Q(x) = x^5 + c_4 x^4 + c_3 x^3 + ... + c_0$ , P(i) = Q(i), i= 1,2,3 …, 6 . Show that there exists a negative integer r such that P(r) = Q(r) .