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Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let $x_1, x_2, \cdots , x_n $ be positive reals with $x_1+x_2+\cdots+x_n=1 $. Then show that $ \sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1} $

**Problem 2: **

Consider three positive real numbers $a,b$ and $c$. Show that there cannot exist two distinct positive integers $m$ and $n$ such that both $\mathbf{a^m+b^m=c^m}$ and $\mathbf{ a^n+b^n=c^n} $ hold.

**Problem 3: **

Let $\mathbb{R} $ denote the set of real numbers. Suppose a function $f:R \rightarrow R$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R} $. Show that

(i) $f$ is one-one,

(ii) $f$ cannot be strictly decreasing, and

(iii) if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R} $.

**Problem 4: **

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $ and $f''(x)\le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

**Problem 5: **

$ABCD$ is a trapezium such that $\mathbf{AB\parallel DC} $ and $ \mathbf{\frac{AB}{DC}=\alpha} $ >1. Suppose $P$ and $Q$ are points on $AC$ and $BD$ respectively, such that $\mathbf{\frac{AP}{AC}=\frac{BQ}{BD}=\frac{\alpha -1}{\alpha+1}} $

Prove that $PQCD$ is a parallelogram.

**Problem 6: **

Let $\mathbf{\alpha } $ be a complex number such that both $\mathbf{ \alpha } $ and $\mathbf{\alpha+1 } $ have modulus $1$. If for a positive integer $n$, $ \mathbf{ 1+\alpha } $ is an $n$-th root of unity, then show that $\mathbf{ \alpha } $ is also an $n$-th root of unity and $n$ is a multiple of $6$.

**Problem 7:**

** **(i) Show that there cannot exists three prime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$.

(ii) Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

**Problem 8: **

Let $\mathbf{I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} , dx , n=1,2,3,4} $ . Arrange $\mathbf{I_1, I_2, I_3, I_4 } $ in increasing order of magnitude. Justify your answer.

**Problem 9: **

Consider all non-empty subsets of the set $\mathbf{{1,2\cdots,n}}$. For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as $\mathbf{S_n} $. For example, $ \mathbf{S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3} } $

1. Show that $\mathbf{S_n=\frac1n+\left(1+\frac1n\right)S_{n-1} }$.

2. Hence or otherwise, deduce that $\mathbf{S_n=n} $.

**Problem 10: **

Show that the triangle whose angles satisfy the equality $ \mathbf{\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2} $ is right angled.

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problem 8 is based on basic inequalities

Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies f(x)≥0,f′(x)≤0 and f”(x)≤f(x), for all x\ge 0. Show that f′(0)≥−2–√.

How to proceed with no. 4?

Can you help me solving the 19 number question ??...

Consider the L-shaped diagram below ..that's one