Prove or disprove

Let f : R ---> R be a real function. Prove or disprove each of the following statements.

(a) If  f is continuous and range (f ) = R   then f is monotonic.

(b) If  f  is monotonic and range(f ) = R then f  is continuous.

            (c) If f  is monotonic and f  is continuous then range(f ) = R .

ABCD = B! + C! + D!

Compute the largest base-10 integer AB C D, with A > 0,
such that ABCD = B! + C! + D!.

the smallest possible value of x + yz

Positive integers x, y, z satisfy (xy + z) = 160.
Compute the smallest possible value of x + yz.

least positive integer greater than 2013 that cannot be written as the sum of two palindromes

A palindrome is a positive integer, not ending in 0, that reads the same forwards and backwards.
For example, 35253; 171; 44; and 2 are all palindromes, but 17 and 1210 are not.
Compute the least positive integer greater than 2013 that cannot be written as
the sum of two palindromes.

x.y is a multiple of 1584

Let x be the smallest positive integer such that 1584 .x is a perfect cube, and let y be the smallest
positive integer such that x.y is a multiple of 1584. Compute y.

all coefficients of f are divisible by 5

Let f be a polynomial of degree 2 with integer coefficients.
Suppose that f(k) is divisible by 5 for every integer k.
Prove that all coefficients of f are divisible by 5.

at most two points in common

Two different ellipses are given. One focus of the first ellipse coincides with one focus
of the second ellipse. Prove that the ellipses have at most two points in common.

f : R → R

Find all continuous functions f : R → R such that f(x) − f(y) is rational for
all reals x and y such that x − y is rational

Can we obtain the number 2010 from the number 1

Can we obtain the number 2010 from the number 1 by applying any combination of
the functions sine, cosine, tangent, cotangent, arcsine, arccosine, arctangent and arccotangent?

Vedansi has her birthday bash on 31/08, a piece of Black Forest Cake

Vedansi has her birthday bash on 31/08, a piece of Black Forest Cake.
She chooses a positive number and cut the piece into two, in the ratio 1 : p.
She can then choose any piece and cut it in the same way. Is it possible for her
to obtain, after a finite number of cuts, two piles of pieces each containing half
the original amount of cheese, if
(A)   p is an irrational;
(B) p is a rational, not equals to 1

Is it possible to divide the lines

Is it possible to divide the lines in the plane into pairs of perpendicular lines so that
every line belongs to exactly one pair?

10 @ 9 @ 8 @ 7 @ 6 @ 5 @ 4 @ 3 @ 2

Brackets are to be inserted into the expression 10 @ 9 @ 8 @ 7 @ 6 @ 5 @ 4 @ 3 @ 2 

so that the resulting number is an integer. ( @ means the symbol of division );

(A)  Determine the maximum value of this integer.

(B) Determine the minimum value of this integer.

                                                                                                       [ 44800, 7 ]

three digits are even and three digits are odd

How many different six-digit numbers are there whose three digits are even and three digits are odd?

one or two of them can get no objects

Seven different objects must be divided among three people. In how many ways can it be done if
 one or two of them can get no objects.

[ 2187 ]

natural numbers m & n

There are natural numbers m & n. Find all the fractions m/n whose denominator is smaller than the numerator by 16, the fraction itself is smaller than the sum of the trebled inverse and 2, and the numerator is not greater than 30.

all irreducible fractions

Find the sum of all irreducible fractions between 10 & 20 with denominator of 3.        

[ 300 ]

adding each unordered pair of distinct numbers from S are all different

Let S be a subset of {1, 2, 3,............, 9}, such that the sums formed by adding each
unordered pair of distinct numbers from S are all different.
For example, the subset {1, 2, 3, 5} has this property,
but {1, 2, 3, 4, 5} does not, since the pairs {1, 4} and {2, 3} have the same sum, namely 5.
What is the maximum number of elements that S can contain?

Let S be a set of n points in the plane

Let S be a set of n points in the plane such that any two points of S are at least 1
unit apart. Prove there is a subset T of S with at least n/7 points such that any two
points of T are at least √3 units apart

Consider a standard twelve-hour clock

Consider a standard twelve-hour clock whose hour and minute hands move continuously.
Let m be an integer, with m belongs to {1,2,3,..........,720}. At precisely m minutes after 12:00,
the angle made by the hour hand and minute hand is exactly 1 degree. Determine all possible
values of m.

all ordered triples (x, y, z) of real numbers

Find all ordered triples (x, y, z) of real numbers which satisfy the following system of equations
xy = z - x - y
yz = x - y - z
zx = y - z -x

all functions f defined on the set of rational numbers

Determine all functions f defined on the set of rational numbers that take rational values
for which f(2f(x) + f(y)) = 2x + y for each x and y.

ABCD is a convex quadrilateral for which AB is the longest side

ABCD is a convex quadrilateral for which AB is the longest side. Points M and N are located on
sides AB and BC respectively, so that each of the segments AN and CM divides the quadrilateral
into two parts of equal area. Prove that the segment MN bisects the diagonal BD.

3^a +7^b is a perfect sqaure

Find all ordered pairs (a, b) such that a and b are integers and 3^a + 7^b is a
perfect square

A passing is defined when one skater passes another one

Three speed skaters have a friendly race on a skating oval. They all start from
the same point and skate in the same direction, but with di fferent speeds that
they maintain throughout the race. The slowest skater does 1 lap a minute, the
fastest one does 3.14 laps a minute, and the middle one does L laps a minute for
some 1 < L < 3.14. The race ends at the moment when all three skaters again
come together to the same point on the oval (which may diff er from the starting
4 point.) Find how many di fferent choices for L are there such that 117 passings
occur before the end of the race. (Note:  A passing is defi ned when one skater passes
another one. The beginning and the end of the race when all three skaters are at
together are not counted as a passing.)

A, B, P be three points on a circle

Let A, B, P be three points on a circle. Prove that if a and b are the distances from P to
the tangents at A and B and c is the distance from P to the chord AB, then c^2= ab.

Pixie-Vedansi has divided a square up into finitely many white and red rectangles

Pixie-Vedansi has divided a square up into finitely many white and red rectangles, each with sides
parallel to the sides of the square. Within each white rectangle, she writes down its width
divided by its height. Within each red rectangle, she writes down its height divided by
its width. Finally, she calculates x, the sum of these numbers. If the total area of the
white rectangles equals the total area of the red rectangles, what is the smallest possible
value of x?

70-digit numbers n

Consider 70-digit numbers n, with the property that each of the digits 1, 2, 3, . . . , 7
appears in the decimal expansion of n ten times (and 8, 9, and 0 do not appear).
Show that no number of this form can divide another number of this form

One mapping is selected at random

One mapping is selected at random from all mapping of the set S = {1, 2, 3,....., n} into itself.
if the probability that the mapping is Injective (One-One) is 3/32;
Then find the number of possible divisors of n.

FG is as large as possible and a five digit number is made

In the equation A + B + C + D + E = FG, where FG being the two digit number
whose value is (10F + G) and letters A, B, C, D, E, F & G each represents different
digits. if FG is as large as possible and a five digit number is made using letters
A, B, C, D, E, F & G  (repetition is not allowed) then
(i) find the probability that number divisible by 5;
(ii) find the probability that number divisible by 3;
(i) find the probability that number divisible by 4; 

cos(cos(cos(cos(cos(cos(cos(cos x)))))))

Let f(x) = cos(cos(cos(cos(cos(cos(cos(cos x))))))), and suppose that the number a
satisfies the equation a = cos a. Express f'(a) as a polynomial in a.

((n!)!)!

Find the largest number n such that (2004!)! is divisible by ((n!)!)!

the sum of the ten numbers

Ten positive integers are arranged around a circle. Each number is one more than the
greatest common divisor of its two neighbors.
What is the sum of the ten numbers?

the moth achieve its objective

 A moth starts at vertex A of a certain cube and is trying to get to vertex B, which is
opposite A, in five or fewer “steps,” where a step consists in traveling along an edge
from one vertex to another. The moth will stop as soon as it reaches B.
How many ways can the moth achieve its objective?

f(x) = f '(x)f "(x)

A non-zero polynomial f(x) with real coefficients has the property that f(x) = f '(x)f "(x).
What is the leading coefficient of f(x)?   (1/18)

Subsets is all in one color

Given Six three element subsets of a finite set X, show that it is possible to color the elements of X  in two colors such that none of the given subsets is all in one color.

every k-element subset of {1, 2, 3, ......., 50}

find the smallest integer k such that every k-element subset of {1, 2, 3, ......., 50} contains two distinct elements a , b such that  (a + b) divides ab.

don't contain two consecutive numbers

Find the number of non-empty subsets of {1, 2, 3, .............., n} which don't contain two consecutive numbers.