4. Let X = {1, 2, 3, ... , 10}. Find the number of pairs {A, B} such that A ⊆ X, B ⊆ X, A ≠ B and A∩B = {5, 7, 8}. Solution: First we put 5, 7, 8 in each of A and B. We are left out with 7 elements of […]
4. Let X = {1, 2, 3, ... , 10}. Find the number of pairs {A, B} such that A ⊆ X, B ⊆ X, A ≠ B and A∩B = {5, 7, 8}. Solution: First we put 5, 7, 8 in each of A and B. We are left out with 7 elements of […]
3. Let a and b are positive real numbers such that a+b = 1. Prove that \( (a^a b^b + a^b b^a \le 1)\) Solution: We use the weighted A.M.-G.M. inequality which states that: \( \frac {w_1 a_1 + w_2 a_2 }{w_1 + w_2} \ge ({a_1}^{w_1} {a_2}^{w_2})^{\frac{1}{w_1 + w_2}} \) First we put \( w_1 […]
2. Let a, b, c be positive integers such that a divides $ (b^5)$ , b divides $(c^5)$ and c divides $ (a^5)$. Prove that abc divides $((a+b+c)^{31})$. Solution: A general term of the expansion of $((a+b+c)^{31})$ is $(\frac {31!}{p!q!r!} a^p b^q c^r)$ where p+q+r = 31 (by multinomial theorem; this may reasoned as following: […]
1. Let ABCD be a unit square. Draw a quadrant of a circle with A as the center and B, D as the end points of the arc. Similarly draw a quadrant of a circle with B as the center and A, C as the end points of the arc. Inscribe a circle Γ touching the […]
The Problem Suppose ABC is any triangle. D, E, F are points on BC, CA, AB respectively such that $latex (\frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB})$. Prove that the centroids of triangles ABC and DEF coincide. A little Complex Number Let A, B, C be points on the Complex plane with complex coordinates a, b, c. […]
353 is a prime number. So is 7919 (in fact it is the 1000th prime). There are 25 primes between 1 and 100. From 1 to 1000 there are 168 of them. It is difficult to check whether a number is prime or not. One simple method is to try and divide the number with […]
Given: AB is the diameter of a circle with center O. C be any point on the circle. OC. is joined. Let Q be the midpoint of OC. AQ produced meet the circle at E. CD be perpendicular to diameter AB. ED and CB are joined. R.T.P. : CM = MB Construction: AC and BD […]
CHENNAI MATHEMATICAL INSTITUTE B.SC. MATH ENTRANCE 2012ANSWER FIVE 6 MARK QUESTIONS AND SEVEN OUT 10 MARK QUESTIONS.6 mark questions Find the number of real solutions of $latex x = 99 \sin (\pi ) x $ Find $latex {\displaystyle\lim_{xto\infty}\dfrac{x^{100} \ln(x)}{e^x \tan^{-1}(\frac{\pi}{3} + \sin x)}}$ (part A)Suppose there are k students and n identical chocolates. The chocolates […]
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 […]