MathDB

1

India TST : Day4 :Problem 2

n\ge 2

with the following property:
There exists three distinct primes

p,q,r

such that whenever

a_1,a_2,a_3,\cdots,a_n

are

n

distinct positive integers with the property that at least one of

p,q,r

divides

a_j - a_k \ \forall 1\le j\le k\le n

, one of

p,q,r

divides all of these differences.

10

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India TST : Day4 :Problem 1

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

12

1

India TST : Day4 :Problem 3

Let

G

be a simple graph with vertex set V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\} .

j

and j\plus{}1 are connected by an edge for

0\le j\le n

. Let

A

be a subset of

V

and

G(A)

be the induced subgraph associated with

A

. Let

O(G(A))

be number of components of

G(A)

having an odd number of vertices. Let T(p,r)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\} for

r\le p \le 2r

. Prove That |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}.

9

1

India TST : Day3 :Problem 3

Let f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k and g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1} be two polynomials with real coefficients. Let g(x) have 0,2^{n\plus{}1} as two of its roots. Prove That

f(x)

has a positive root less than

2^n

.

8

1

India TST : Day3 :Problem 2

Let

n

be a natural number

\ge 2

which divides 3^n\plus{}4^n.Prove That

7\mid n

.

7

1

India TST : Day3 :Problem 1

Let

P

be any point in the interior of a

\triangle ABC

.Prove That \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}.

5

1

India TST:Day 2 Problem 2

Let

f(x)

and

g(y)

be two monic polynomials of degree=

n

having complex coefficients. We know that there exist complex numbers

a_i,b_i,c_i \forall 1\le i \le n

, such that f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}. Prove that there exists

a,b,c\in\mathbb{C}

such that f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c.

6

1

India TST:Day 2 Problem 3

Prove The Following identity: \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}. The Second term on left hand side is to be regarded zero for j=0.

4

1

India TST:Day2 Problem 1

Let

\gamma

be circumcircle of

\triangle ABC

.Let

R_a

be radius of circle touching

AB,AC

&

\gamma

internally.Define

R_b,R_c

similarly. Prove That \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}.

3

1

India TST 2009 DAY 1:PROBLEM 3

Let

a,b

be two distinct odd natural numbers.Define a Sequence

{ < a_n > }_{n\ge 0}

like following: a_1 \equal{} a \\ a_2 \equal{} b \\ a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2}). Prove that there exists a natural number

N

such that a_n \equal{} gcd(a,b) \forall n\ge N.

2

1

India TST 2009 DAY 1:PROBLEM 2

Let us consider a simle graph with vertex set

V

. All ordered pair

(a,b)

of integers with gcd(a,b) \equal{} 1, are elements of V.

(a,b)

is connected to (a,b \plus{} kab) by an edge and to (a \plus{} kab,b) by another edge for all integer k. Prove that for all

(a,b)\in V

, there exists a path fromm

(1,1)

to

(a,b)

.

1

India TST 2009 DAY 1:PROBLEM 1

Let

ABC

be a triangle with

\angle A = 60^{\circ}

.Prove that if

T

is point of contact of Incircle And Nine-Point Circle, Then

AT = r

,

r

being inradius.

2009 India IMO Training Camp

Subcontests

India TST : Day4 :Problem 2

India TST : Day4 :Problem 1

India TST : Day4 :Problem 3

India TST : Day3 :Problem 3

India TST : Day3 :Problem 2

India TST : Day3 :Problem 1

India TST:Day 2 Problem 2

India TST:Day 2 Problem 3

India TST:Day2 Problem 1

India TST 2009 DAY 1:PROBLEM 3

India TST 2009 DAY 1:PROBLEM 2

India TST 2009 DAY 1:PROBLEM 1