 This post contains Chennai Mathematical Institute, CMI, 2015 Objective, and Subjective Problems and Solutions. Please contribute problems and solutions in the comments.

## Objective

1. For all finite word strings comprising A and B only, A string is arranged by dictionary order. eg. ABAA
2.  For any arbitrary string w, with another string y<w, there cannot always exist a string x, w<x<y
3. There is an infinite set of strings a1,a2… such that ai<a(i+1) for all i.
4. There are fewer than 50 strings less than AABBABBA
• Ten people are seated in a circle. One person contributes five hundred rupees. Every person contributes the average of the money contributed by his two neighbors.
1. What is the sum contributed by all the ten?
1. >5000
2. < 5000
3. .=5000
4. Cannot say.
2. 2. What is maximum contribution by an individual?
1. 500
2. =500
3. none
• There are 4 bins and 4 balls. Let $P(E_i)$ be the probability of first n balls falling into distinct bins.
Find

1. $P(E_4)$
2. $P(E_4|E_3)$
3. $P(E_4|E_2)$
4. $P(E_3|E_4)$
• Let $f(x) = \sin^{-1} (\sin (\pi x))$. Find
1. f(2.7).
2. f'(2.7)
3. integral from 0 to 2.5 of f(x)dx
4. value of x for which f'(x) does not exist
• In some country number plates are formed by 2 digits and 3 vowels. It is called confusing if it has both digit 0 and vowel o.
1. How many such number plates exist?
2. How many are not confusing
• A number is called magical if a and b are not coprime to n, a+b is also not coprime to n. For example, 2 is magical as all even numbers are not coprime to 2. Find whether the following numbers are magical
1. 129
2. 128
3. 127
4. 100
• a) In the expansion of $(1+ \sqrt 2)^10 = \sum_0^10 C_i (\sqrt 2)^i$, the term with maximum value is
b) If $(1+\sqrt 2)^n = p_n+q_n \sqrt 2$ , where $p_n$ and $q_n$ are integers, $\lim_{ n to \infty} \frac {p_n}{q_n} ^{10}$ is

## Subjective

1. In a circle, AB be the diameter.. X is an external point. Using straight edge construct a perpendicular to AB from X
1. If X is inside the circle then how can this be done
Discussion
2. a be a positive integer from set {2, 3, 4, … 9999}. Show that there are exactly two positive integers in that set such that 10000 divides a*a-1.
1. Put $n^2 - 1$ in place of 9999. How many positive integers a exists such that $n^2$ divides a(a-1)
Discussion
3. P(x) is a polynomial. Show that $\displaystyle { \lim_{t to \infty} \frac{P(t)} {e^t} }$ exists. Also show that the limit does not depend on the polynomial.
4. We define function $\displaystyle { f(x) = \frac {e^{\frac{-1}{x}}}{x}}$ when x< 0; f(x) = 0 if x=0 and $\displaystyle { f(x) = \frac {e^{\frac{-1}{x}}}{x}}$ when x > 0 . Show that the function is continuous and differentiable. Find limit at x =0
5. p,q,r any real number such that $p^2 + q^2 + r^2 = 1$
1. Show that $3*(p^2 q + p^2 r) + 2(r^3 +q^3) \le 2$
2. Suppose $f(p,q,r) = 3(p^2 q + p^2 r ) + 2(r^3 +q^3)$ .  At what values (p,q, r) does f(p,q,r) maximizes and minimizes?
6. Let g(n) is GCD of (2n+9) and $6n^2+11n-2$ then then find greatest value of g(n)