MathDB

1

French pre-TST-2004/2005 #8

f

be a function from the set

Q

of the rational numbers onto itself such that

f(x+y)=f(x)+f(y)+2547

for all rational numbers

x,y

. Moreover

f(2004) = 2547

. Determine

f(2547).

Pierre.

7

1

French pre-TST-2004/2005 #7

Prove that a prime of the form

2^{2^n}+1

cannot be the difference of two fifth powers of two positive integers. Pierre.

6

1

French pre-TST-2004/2005 #6

On each unit square of a

9 \times 9

square, there is a bettle. Simultaneously, at the whistle, each bettle moves from its unit square to another one which has only a common vertex with the original one (thus in diagonal). Some bettles can go to the same unit square. Determine the minimum number of empty unit squares after the moves. Pierre.

5

1

French pre-TST-2004/2005 #5

Let

I

be the incenter of the triangle

ABC

. Let

A_1,A_2

be two distinct points on the line

BC

, let

B_1,B_2

be two distinct points on the line

CA

, and let

C_1,C_2

be two distinct points on the line

BA

such that

AI = A_1I = A_2I

and

BI = B_1I = B_2I

and

CI = C_1I = C_2I

. Prove that

A_1A_2+B_1B_2+C_1C_2 = p

where

p

denotes the perimeter of

ABC.

Pierre.

4

1

French pre-TST-2004/2005 #4

Let

x,y,z

be positive real numbers such that

x^2+y^2+z^2 = 25.

Find the minimum of

\frac {xy} z + \frac {yz} x + \frac {zx} y .

Pierre.

3

1

French pre-TST-2004/2005 #3

Two players write alternatively some integers on the blackboard. The rules are the following : - The first player write

1

. - At each of the other turns, the player has to write

a+1

or

2a

where

a

is any number already wrote in the blackboard and

2a \leq 1000.

- One cannot write a number which has already been written, and no number is erased. - The player who writes

1000

is the winner. Determine which player has a winning strategy. Pierre.

2

1

French pre-TST-2004/2005 #2

Let

\omega (n)

denote the number of prime divisors of the integer

n>1

. Find the least integer

k

such that the inequality

2^{\omega (n) } \leq k \cdot n^{\frac 1 4}

holds for all

n > 1.

Pierre.

1

French pre-TST-2004/2005 #1

Let

I

be the incenter of the triangle

ABC

, et let

A',B',C'

be the symmetric of

I

with respect to the lines

BC,CA,AB

respectively. It is known that

B

belongs to the circumcircle of

A'B'C'

. Find

\widehat {ABC}

. Pierre.

2005 France Pre-TST

Subcontests

French pre-TST-2004/2005 #8

French pre-TST-2004/2005 #7

French pre-TST-2004/2005 #6

French pre-TST-2004/2005 #5

French pre-TST-2004/2005 #4

French pre-TST-2004/2005 #3

French pre-TST-2004/2005 #2

French pre-TST-2004/2005 #1