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Sunday 27 May 2012

C.M.I. ENTRANCE 2012

CHENNAI MATHEMATICAL INSTITUTE
B.SC. MATH ENTRANCE 2012

ANSWER FIVE 6 MARK QUESTIONS AND SEVEN OUT 10 MARK QUESTIONS.

6 mark questions

  1. Find the number of real solutions of x = 99 sin \(\pi \) x
  2. Find \(\displaystyle\lim_{x\to\infty}\dfrac{x^{100} ln(x)}{e^x tan^{-1}(\frac{\pi}{3} + sin x)}\)
  3. (part A)Suppose there are k students and n identical chocolates. The chocolates are to be distributed one by one to the students (with each student having equal probability of receiving each chocolate). Find the probability of a particular student getting at least one chocolate.
    (part B) Suppose the number of ways of distributing the k chocolates to n students be \(\dbinom{n+k-1}{k}\). Find the probability of a particular student getting at least one chocolate.
  4. Show that \(\dfrac{ln 12}{ln 18}\) is an irrational number.
  5. Give an example of a polynomial with real coefficients such that \(P(\sqrt{2} + i)=0\). Further given an example of a polynomial with rational coefficients such that \(P(\sqrt{2} + i)=0\).
  6. Say f(1) = 2; f(2) = 3, f(3) = 1; then show that f'(x) = 0 for some x (given that f is a continuously differentiable function defined on all real numbers).
10 mark questions
  1. (part A) Suppose a plane has 2n points; n red points and n blue points. One blue point and one red point is joined by a line segment. Like this n line segments are drawn by pairing a red and a blue point. Prove that each such scheme of pairing segments will have two segments which do not intersect each other.
    (part B) Suppose the position of the n red points are given. Prove that we can put n blue points in such a way that there are two segments (produced in the manner described in part A) which do not intersect each other.
  2. (part A) Let ABCD be any quadrilateral. E, F, G and H be the mid points of the sides AB, BC, CD and DA respectively. Prove that EFGH is a parallelogram whose area is half of the quadrilateral ABCD.
    (part B) Suppose the coordinates of E, F, G, H are given:  E (0,0) , F(0, -1), G (1, -1) , H (1, 0). Find all points A in the first quadrant such that E, F, G, H be the midpoints of quadrilateral ABCD.
  3. Let f be a function whose domain and co-domain be non negative natural numbers such that f(f(f(n))) < f(n+1). Prove that:
    (a) If f(n) = 0 then n = 0.
    (b) f(n) < n+1
    (c) If f(x) < n then x<n
    Using the above prove that f is an identity function, that is f(n) = n.
  4. Consider a sequence \(c_{n+2} = a c_{n+1} + b c_n\) for \(n \ge 0\) where \(c_0 = 0\). If gcd(b, k) = 1 then show that k divides n for infinitely many n.
  5. Find out the value of \(x^{2012} + \dfrac{1}{x^{2012}}\) when \(x + \dfrac{1}{x} = \dfrac{\sqrt{5} + 1}{2}\).
    Hint
    (a) Show that \(|{r + \dfrac{1}{r}}|\ge 2\) for all real r.
    (b) Prove that \(\sin \dfrac{\pi}{5} < \cos \dfrac{2\pi}{5} < \sin \dfrac{3\pi}{5}\).
  6. A polynomial P(x) takes values \(prime^{positive number}\) for every positive integer n,then show that p(x) is a constant polynomial.
    If such a polynomial exist then show that there also exists a polynomial g(x)= \(prime^l\) where l is a fixed number.
  7. Consider a set A = {1, 2, ... , n}. Suppose \(A_1 , A_2 , ... , A_k \) be subsets of set A such that any two of them consists at least one common element. Show that the greatest value of k is \(2^{n-1}\). Further, show that if they any two of them have a common element but intersection of all of them is a null set then the greatest value of k is \(2^{n-1}\).
  8. Suppose \(\displaystyle x = \sum_{i=1}^{10} \dfrac{1}{10 \sqrt 3} . \dfrac{1}{1+ (\dfrac{i}{10 \sqrt 3})^2}\) and \(\displaystyle y = \sum_{i=0}^9 \dfrac{1}{10 \sqrt 3} . \dfrac{1}{1+ (\dfrac{i}{10 \sqrt 3})^2}\)
    1.  Show that \(x < \dfrac{\pi}{6} < y \) 
    2. \(\dfrac{x+y}{2} < \dfrac{\pi}{6} \)

[b]OTHERS PLEASE CONTRIBUTE THE REST OF THE QUESTIONS (AND SOLUTIONS). WE ARE TRYING ON OUR END TO DO THE SAME[/b]

Monday 14 May 2012

Solutions to I.S.I. 2012 Subjective (B.Stat, B.Math)

Q7. Consider two circles with radii a, and b and centers at (b, 0), (a, 0) respectively with b<a. Let the crescent shaped region M has a third circle which at any position is tangential to both the inner circle and the outer circle. Find the locus of center c of the third circle as it traverses through the region M (remaining tangential to both the circle.


Discussion:



Join AC and BC. AC passes through, \(T_1\) the point of tangency of the smaller circle with the circle with center at (a, 0) and BC when extended touches \(T_2\) which is the other point of the tangency.

Assume the radius of the moving (and growing circle) to be r at a particular instance. Then AC = a+r and BC = b-r.

Then AC+BC = a+b which is a constant for any position of C. Hence C is a point whose some of distances from two fixed points at any instant is a constant. This is the locus definition of an ellipse with foci at (a, 0) and (b, 0).





Q8. Let S = {1, 2, ... , n}. Let \(f_1 , f_2 , ... \)  be functions from S to S (one-one and onto). For any function f, call D, subset of S, to be invariant if for all x in D, f(x)  is also in D. Note that for any function the null set and the entire set are 'invariant' sets. Let deg(f) be the number of invariant subsets for a function.
 a) Prove that there exists a function with deg(f)=2.
 b) For a particular value of k prove that there exist a function with deg(f) = \(2^k\)


Discussion:


(a)

Consider the function defined piecewise as f(x) = x - 1 is \(x \ne 1\) and f(x) = n if x = 1

Of course null set and the entire sets are invariant subsets. We prove that there are no other invariant subsets.

Suppose D =  {\(a_1 , a_2 , ... , a_k \)} be an invariant subset with at least one element.

Since we are working with natural numbers only, it is possible to arrange the elements in ascending order (there is a least element by well ordering principle).

Suppose after rearrangement D = {\(b_1 , b_2 , ... , b_k \)} where \(b_1\) is the least element of the set

If \(b_1 \ne 1\) then \(f(b_1) = b_1 -1\) is not inside D as \(b_1\) is the smallest element in D. Hence D is no more an invariant subset which is contrary to our initial assumption.

This \(b_1\) must equal to 1.

As D is invariant subset \(f(b_1) = n \) must belong to D. Again f(n) = n-1 is also in D and so on. Thus all the elements from 1 to n are in D making D=S.

Hence we have proved that degree of this function is 2.

(b)

For a natural number 'k' to find a function with deg(f) = \(2^k\) define the function piecewise as

f(x) = x for \(1\le x \le k-1\)
      = n for x=k
      = x-1 for the rest of elements in 'n'

To construct an invariant subset the 'k-1' elements which are identically mapped, and the entirety of the 'k to n' elements considered as a unit must be considered. Thus there are total k-1 + 1 elements with which subsets are to be constructed. There are \(2^k\) subsets possible.

Tuesday 8 May 2012

USAJMO 2012 questions

  1. Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S are concyclic (in other words these four points lie on a circle).
  2. Find all integers \(n \ge 3 \) such that among any n positive real numbers \( a_1 , a_2 , ... , a_n \) with
    max
    \((a_1 , a_2 , ... , a_n) \le n\) min \(( a_1 , a_2 , ... , a_n)\) there exist three that are the side lengths of an acute triangle.
  3. Let a, b, c be positive real numbers. Prove that \(\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2)\).
  4. Let \(\alpha\) be an irrational number with \(0 < \alpha < 1\), and draw a circle in the plane whose circumference has length 1. Given any integer \(n \ge 3 \), define a sequence of points \(P_1 , P_2 , ... , P_n \) as follows. First select any point \(P_1\) on the circle, and for \( 2 \le k \le n \) define \(P_k\) as the point on the circle for which the length of the arc \(P_{k-1} P_k\) is \(\alpha\), when travelling counterclockwise around the circle from \(P_{k-1} \) to \(P_k\). Suppose that \(P_a\) and \(P_b\) are the nearest adjacent points on either side of \(P_n\). Prove that \(a+b \le n\).
  5. For distinct positive integers a, b < 2012, define f(a, b) to be the number of integers k with \(1\le k < 2012\) such that the remainder when ak divided by 2012 is greater than that of bk divided by 2012. Let S be the minimum value of f(a, b), where a and b range over all pairs of distinct positive integers less than 2012. Determine S.
  6. Let P be a point in the plane of triangle ABC, and \(\gamma\) be a line passing through P. Let A', B', C'  be the points where reflections of the lines PA, PB, PC with respect to \(\gamma\) intersect lines BC, AC, AB, respectively. Prove that A', B' and C' are collinear.