- Let \(x_1, x_2, \cdots , x_n \) be positive reals with \(x_1+x_2+\cdots+x_n=1 \). Then show that \(\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1} \)
- 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.
- 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 \(x \in \mathbb{R} \). - 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 \(f'(0) \ge -\sqrt2\).
- 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. - 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.
- (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. - 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.
- 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 \(\mathbf{S_n=n} \). - 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?