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ISI Entrance Paper 2014 – from Indian Statistical Institute’s B.Stat Entrance

1. In a class there are 100 student. We define $\mathbf { A_i}$ as the number of friends of $\mathbf { i^{th} }$ student and $\mathbf { C_i }$ as the number of students who has at least i friends. Prove that $\mathbf { \sum_1^{100} A_i = \sum_0^{99} C_i }$
2. Let PQR be a triangle. Take a point A on or inside the triangle. Let f(x, y) = ax + by + c. Show that $\mathbf { f(A) \le max { f(P), f(Q) , f(R)} }$
Solution
3. Let $\mathbf { y = x^4 + ax^3 + bx^2 + cx +d , a,b,c,d,e \mathbb{R}}$. it is given that the functions cuts the x axis at least 3 distinct points. Then show that it either cuts the x axis at 4 distinct point or 3 distinct point and at any one of these three points we have a maxima or minima.
Solution
4. Let f(x) and g(x) be twice differentiable non decreasing functions such that f”(x) = g(x) and g”(x) = f(x) and f(x) . g(x) is a linear function. Then show that f(x) = g(x) = 0.
5. Prove that sum of any 12 consecutive integers cannot be perfect square. Give an example where sum of 11 consecutive integers is a perfect square
Solution
6. $\mathbf { A = {(x, y) , x = u + v , y = v , u^2 + v^2 \le 1} }$ . Then what is the maximum length of a line segment enclosed in this area
7. Let f(x) be a non decreasing function defined on the domain $\mathbf {[0, \infty) }$ . Then show that if $\mathbf { 0\le x < y < z < \infty , (z-x) \int_y^z f(u) du \ge (z-y) \int_x^z f(u) du }$
8. n (> 1) lotus leafs are arranged in a circle. A frog jumps from a particular leaf by the following rule: It always moves clockwise. From staring point it skips 1 leaf and jumps to the next. Then it skips 2 leaves and jumps to the following. That is in the 3rd jump it skips 3 leaves and 4th jump it skips 4 leaves and so on. In this manner it keeps moving round and round the circle of leaves. It may go to one leaf more than once. If it reaches each leaf at least once then n (the number of leaves) cannot be odd.
Solution

(These problems are collected from student feedback. Courtesy Krishnendu Bhowmick)