## quinta-feira, 3 de outubro de 2013

### Ashani Dasgupta's Math Blog

EXCELLENT BLOG MATHEMATICS
http://cheentasutra.wordpress.com/

# Crease of a square paper

A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.
Discussion:
Assuming the side of the square is ‘s’. Let a part of the crease be ‘x’ (hence the remaining part is ‘s-x’). We apply Pythagoras Theorem we solve for x:
$x^2 + \frac {s^2 }{4} = (s-x)^2$ implies $x = \frac {3s}{8}$ and $s-x = \frac {5s}{8}$
Hence the ratio is 5:3.
Standard

# RMO 1990

1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least two of them will always be of the same size (radius). Prove that there are at least three balls which lie in the same box have the same colour and have the same size (radius).
2. For all positive real numbers a, b, c prove that $\frac{a}{b+c} + \frac{b}{a+c}+ \frac{c}{a+b} \ge \frac{3}{2}$
3. A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.

### Discussion

4. Find the remainder when $2^{1990}$ is divided by 1990.
5. P is any point inside a triangle ABC. The perimeter of the triangle AB +BC +CA=2s. Prove that s < AP + BP + CP < 2s.
6. N is a 50 digit number (in a decimal scale). All digits except the $26^{th}$ digit (from the left) are 1.If N is divisible by 13, find the $26^{th}$ digit.
7. A census-man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said “We do not mind giving you the sum of the ages of any two ladies you may choose”. There upon the census man said, “In that case, please give me the sum of the ages of every possible pair of you”. They gave the sums as follows: 30, 33, 41, 58, 66, 69. The census-man took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?
8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equilateral.
Standard

# INMO 2012

1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =$\sqrt {2 + \sqrt {2} }$ and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.
2. Let $p_1 and $q_1 be two sets of prime numbers such that $p_4 - p_1 = 8$  and $q_4 - q_1 = 8$ . Suppose $p_1>5$ and $q_1>5$ . Prove that 30 divides $p_1 - q_1$.
3. Define a sequence $$ n∈N of functions as $f_0(x )=1, f_1(x )=x$, $(f_n(x))^2 - 1 = f_{n-1} (x) f_{n+1} (x)$, for $n \ge 1$ . Prove that each $f_n(x )$ is a polynomial with integer coefficients.
4. Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.
5. Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.
6. Let $f :Z \mapsto Z$be a function satisfying $f(0) \neq 0$ , $f(1)=0$ and
1. $f(xy) + f(x)f(y) = f(x) + f(y)$,
2. $(f(x-y) - f(0) ) f(x) f(y) = 0$ for all x , $y \in Z$ simultaneously.
1. Find the set of all possible values of the function f.
2. If $f(10) \neq 0$ and $f(2) = 0$, find the set of all integers n such that $f(n) \neq 0$ .