Natural Numbers

Find all natural numbers n such that the number n(n + 1)(n + 2)(n + 3) has exactly three di fferent prime  divisors.

age of child

In a family there are four children of different ages, each age being a positive integer not less than 2 and not greater than 16. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?

ordered triples

How many ordered triples (a, b, c) of positive integers are there such that none of a, b, c exceeds 2010 and each of a, b, c divides a + b + c?

nice triple

Call a triple (a, b, c) of positive integers a nice triple if a, b, c forms a non-decreasing arithmetic progression, gcd(b, a) = gcd(b, c) = 1 and the product abc is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.

31 points

In our school netball league a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game. After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister's team has won 5 games, drawn 2 and lost 3. How many points has her team gained?

29

A swimming club has three categories of members: junior, senior, veteran. The ratio of junior to senior members is 3:2 and the ratio of senior members to veterans is 5:2 . find the total number of members in the swimming club.

2024

Ulysses, Sanga, Isitaa and Radhe have their 12th, 14th, 15th and 15th birthdays today. In what year will their ages first total 100?

numbers 2

1.  N = 3 x 7 x 11 x 15 x 19 x ........... x 2003 x 2007 x 2011. Find the last three digits of N.
2.  When a positive integer n is divided by 5, 7, 9, 11 the remainders are 1, 2, 3, 4 respectively. Find the minimum value of n.

numbers 1

1. Find the number of positive integer n, such that the remainder is 7 when 2007 is divided by n.
2. What is the remainder of (123456789 x 123456789 x 123456789 x 123456789) when it is divided by 8?

locus

On shifting the origin to a point P, the axes remaining parallel to the original axes,
 the equation ax + by +c = 0 is transformed to ax + by + c + k = 0. 
Then the locus of P is

AM & HM

The harmonic mean of two positive integers is 2006. Find the greatest possible value of their
arithmetic mean.

3 digit numbers

The numbers 246, 462, and 624 are all divisible by 6. What is the greatest possible common
divisor of the three-digit numbers abc, bca, and cab when a, b, and c are all different?

Greatest Number

Find the greatest 9-digit number whose digits’ product is 9!.

Primes

Find all positive integers n for which 3n − 4, 4n − 5, and 5n − 3 are all prime.

weekend trivia

A piggy bank contains 4 copper coins and 3 silver coins. A second piggy bank contains 6 copper coins and 4 silver coins. A bank is chosen randomly and a coin is taken out of it. What is  the probability that it is a copper coin?

KVPY SA

Suppose ABCDEFGHIJ is a ten digit number, where the digits are all distinct. Moreover, A > B > C satisfy   A + B + C = 9,
 D > E > F are consecutive even digits and G > H > I > J are consecutive odd digits. Then A = ( 8 )

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more on basics

1         Given a semicircle of radius 1, let a be the side of an equilateral triangle which is inscribed in the semicircle with its vertices on the boundary of the semicircle ( boundary includes the bounding diameter also). Then the set of possible values of a is  

2.        A ray of light originating at the vertex A of the square ABCD passes through the vertex B after getting reflected by BC, CD and DA in that order. If  x is the angle of the initial position of the ray with AB then sin x =

            

           more problems visit www.scribd.com/hmookherjee

RMO INMO 111123456

1. Boddhi places some counters onto the squares of an 8 by 8 chessboard so that there is at most one counter in each of the 64 squares. Determine, with justification, the maximum number that he can place without having five or more counters in the same row, or in the same column, or on either of the two long diagonals.
2. Two circles S and T touch at X. They have a common tangent which meets S at A and T at B. The points A and B are different. Let AP be a diameter of S. Prove that B, X and P lie on a straight line.
3. Find all real numbers x, y and z which satisfy the simultaneous equations x^2 − 4y + 7 = 0, y^2 − 6z + 14 = 0 and               z^2 − 2x − 7 = 0.
4. Find all positive integers n such that 12 n −119 and 75 n −539 are both perfect squares.
5. A triangle has sides of length at most 2, 3 and 4 respectively. Determine, with proof, the maximum possible area of the triangle.
6. Let ABC be a triangle. Let S be the circle through B tangent to CA at A and let T be the circle through C tangent to AB at A. The circles S and T intersect at A and D. Let E be the point where the line AD meets the circle ABC. Prove that D is the midpoint of AE.

<<< bmoc 201213  >>>

RMO, INMO 111

Determine all functions f : R -->> R, where R is the set of all real numbers, satisfying the following 2 conditions:
(i) There exists a real number M such that for every real number x, f(x) < M is satisfied ,
(ii) For every pair of real numbers x and y, f(xf(y)) + yf(x) = xf(y) + f(xy) is satisfi ed .

<<< 2011 APMO Prob >>>

RMO, INMO 11

Into each box of a 2012 x 2012 square grid, a real number greater than or equal to 0 and less than or equal to 1 is inserted. Consider splitting the grid into 2 non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to 1,
no matter how the grid is split into 2 such rectangles.
Determine the maximum possible value for the sum of all the 2012 x 2012 numbers inserted into the boxes.
<<< 2012 APMO PROBLEMS >>>

RMO, INMO 1

Let P be a point in the interior of a triangle ABC, and let D, E, F be the point of intersection of the line AP and the side BC of the triangle, of the line BP and the side CA, and of the line CP and the side AB, respectively. 
Prove that the area of the triangle ABC must be 6 if the area of each of the triangles PFA, PDB and PEC is 1.
<<<2012 APMO PROBLEMS>>>

Olympiad Aspirants

Avratanu is trying to open a lock whose code is a sequence that is three letters long, with
each of the letters being one of A, B or C, possibly repeated. The lock has three buttons,
labeled A, B and C. When the most recent 3 button-presses form the code, the lock opens.
What is the minimum number of total button presses Avratanu needs to guarantee opening the

lock?

hey Bunny can / can't get her maxima

Asmi, the bunny is hopping on the positive integers. First, she is told
a positive integer n. Then, Asmi chooses positive integers a, d and hops on
all of the spaces a, a + d, a + 2d, ........, a + 2013d. However, Asmi must make
these choices so that the number of every space that she hops on is less than n and relatively
prime to n.
A positive integer n is called bunny-unfriendly if, when given that n, Asmi is unable
to find positive integers a, d that allow her to perform the hops she wants. Find the maximum
bunny-unfriendly integer, or prove that no such maximum exists.

for exclusively for OLYMPIAD aspirants

max , min

find the minimum and maximum value of
(sin x - cos x - 1)(sin x + cos x + 1) for all real x

quadratic polynomials

The quadratic polynomial p(x) has the following properties:
p(x) is non-negative for all real numbers, p(1) = 0 & p(2) = 2.
compute p(0) + p(3)

linear functions

let f be a linear function such that f(6) - f(2) = 12,
then compute f(12) - f(2) = 

PERIOD OF THE FUNCTIONS

Find the fundamental period of the function
f(x) = cos (tan x + cot x) . cos (tan x - cot x) 

NUMBER OF SOLUTIONS

f(f(x)) =x/2, f(x) = 2 x (2-x), x belongs to the closed intervals of 0 to 2  

Graphs & Functions

1. Y = sec x + (1/x),   - π/2 < x < π/2

2. Y = tan x + (1/x⁴), - π/2 < x < π/2

Draw the graph of the following functions:-

1>> y = x + (x / mod x)

2>> y = sqrt (x) . sin(1/x)

CMI ENTRANCE EXAM 2013 QUESTIONS WITH SOLUTIONS

cmi 2013