MathDB

1

{A,B,C} is a partition of {1,2,...3n} and |A|=|B|=|C|=n

n

be a positive integer and

\{A,B,C\}

a partition of

\{1,2,\ldots,3n\}

such that

|A|=|B|=|C|=n

. Prove that there exist

x \in A

,

y \in B

,

z \in C

such that one of

x,y,z

is the sum of the other two.

8

1

a+b+c>r+r_a+r_b+r_c

Let

ABC

be a triangle, and let

r, r_1, r_2, r_3

denoted its inradius and the exradii opposite the vertices

A,B,C

, respectively. Suppose

a>r_1, b>r_2, c>r_3

. Prove that (a) triangle

ABC

is acute, (b)

a+b+c>r+r_1+r_2+r_3

.

5

1

integers of form 81x+100y

On the real number line, paint red all points that correspond to integers of the form

81x+100y

, where

x

and

y

are positive integers. Paint the remaining integer point blue. Find a point

P

on the line such that, for every integer point

T

, the reflection of

T

with respect to

P

is an integer point of a different colour than

T

.

2

1

find all triples (a,b,c)

Find all triples

(a,b,c)

of positive integers such that (i)

a \leq b \leq c

; (ii)

\text{gcd}(a,b,c)=1

; and (iii)

a^3+b^3+c^3

is divisible by each of the numbers

a^2b, b^2c, c^2a

.

1

A"B"C" is equilateral

Let

A',B',C'

be the midpoints of the sides

BC, CA, AB

, respectively, of an acute non-isosceles triangle

ABC

, and let

D,E,F

be the feet of the altitudes through the vertices

A,B,C

on these sides respectively. Consider the arc

DA'

of the nine point circle of triangle

ABC

lying outside the triangle. Let the point of trisection of this arc closer to

A'

be

A''

. Define analogously the points

B''

(on arc

EB'

) and

C''

(on arc

FC'

). Show that triangle

A''B''C''

is equilateral.

6

1

Number of regions formed by zigzags

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find

z_n

, the maximum number of regions into which

n

zig-zags can divide the plane. For example,

z_1=2,z_2=12

(see the diagram). Of these

z_n

regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in

n

of degree not exceeding

2

. [asy] draw((30,0)--(-70,0), Arrow); draw((30,0)--(-20,-40)); draw((-20,-40)--(80,-40), Arrow); draw((0,-60)--(-40,20), dashed, Arrow); draw((0,-60)--(0,15), dashed); draw((0,15)--(40,-65),dashed, Arrow); [/asy]

10

1

prove that 4p-3 is a square

Let

n

be a positive integer greater than

1

, and let

p

be a prime such that

n

divides

p-1

and

p

divides

n^3-1

. Prove that

4p-3

is a square.

4

1

one of the line is parallel to the euler line

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

3

1

Find f if f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)

Find all functions

f: \mathbb R \to \mathbb R

such that for all reals

x

and

y

,

f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).

7

1

Easy

p

is a polynomial with integer coefficients and for every natural

n

we have

p(n)>n

.

x_k

is a sequence that:

x_1=1, x_{i+1}=p(x_i)

for every

N

one of

x_i

is divisible by

N.

Prove that

p(x)=x+1

2003 India IMO Training Camp

Subcontests

{A,B,C} is a partition of {1,2,...3n} and |A|=|B|=|C|=n

a+b+c>r+r_a+r_b+r_c

integers of form 81x+100y

find all triples (a,b,c)

A"B"C" is equilateral

Number of regions formed by zigzags

prove that 4p-3 is a square

one of the line is parallel to the euler line

Find f if f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)

Easy

2003 India IMO Training Camp

Subcontests

{A,B,C} is a partition of {1,2,...3n} and |A|=|B|=|C|=n

a+b+c&gt;r+r_a+r_b+r_c

integers of form 81x+100y

find all triples (a,b,c)

A&quot;B&quot;C&quot; is equilateral

Number of regions formed by zigzags

prove that 4p-3 is a square

one of the line is parallel to the euler line

Find f if f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y)

Easy

a+b+c>r+r_a+r_b+r_c

A"B"C" is equilateral