Consider three positive real numbers a,b and c. Show that there cannot exist two distinct positive integers m and n such that both and hold.

Let denote the set of real numbers. Suppose a function f:R-> R satisfies f(f(f(x)))=x for all . 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 .

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

ABCD is a trapezium such that and >1. Suppose P and Q are points on AC and BD respectively, such that
Prove that PQCD is a parallelogram.

Let be a complex number such that both and have modulus 1. If for a positive integer n, is an n-th root of unity, then show that 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 . Arrange in increasing order of magnitude. Justify your answer.

Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as . For example,
(i) Show that .
(ii) Hence or otherwise, deduce that .

Show that the triangle whose angles satisfy the equality is right angled.

When I originally commented I appear to have clicked the -Notify me when new comments
are added- checkbox and now each time a comment is added I
receive 4 emails with the same comment. Is there an easy method you can remove me from that service?

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–√.

When I originally commented I appear to have clicked the -Notify me when new comments

are added- checkbox and now each time a comment is added I

receive 4 emails with the same comment. Is there an easy method you can remove me from that service?

Kudos!

Do you wish to be removed from ‘comment notification’ service or from all updates from this site

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?