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ISI Entrance Paper 2011 – from Indian Statistical Institute’s B.Stat Entrance

1. Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that $\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}$
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.
3. Let $\mathbb{R}$ denote the set of real numbers. Suppose a function f:R-> 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 $x \in \mathbb{R}$.
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 $f'(0) \ge -\sqrt2$.
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.
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.
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.
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.
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, $\mathbf{S_3=\frac11+\frac12+\frac13+\frac1{1\cdot 2}+\frac1{1\cdot 3}+\frac1{2\cdot 3} +\frac1{1\cdot 2\cdot 3} }$
(i) Show that $\mathbf{S_n=\frac1n+\left(1+\frac1n\right)S_{n-1} }$.
(ii) Hence or otherwise, deduce that $\mathbf{S_n=n}$.
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.