MathDB

1

VIII Brazilian Olympic Revenge, 2009 - Problem 6

a, n \in \mathbb{Z}^{*}_{+}

.

a

is defined inductively in the base

n

-recursive. We first write

a

in the base

n

, e.g., as a sum of terms of the form

k_tn^t

, with

0 \le k_t < n

. For each exponent

t

, we write

t

in the base

n

-recursive, until all the numbers in the representation are less than

n

. For instance,

1309 = 3^6 + 2.3^5 + 1.3^4 + 1.3^2 + 1.3 + 1

= 3^{2.3} + 2.3^{3+2} + 1.3^{3+1} + 1.3^2 + 1

Let

x_1 \in \mathbb{Z}

arbitrary. We define

x_n

recursively, as following: if

x_{n-1} > 0

, we write

x_{n-1}

in the base

n

-recursive and we replace all the numbers

n

for

n+1

(even the exponents!), so we obtain the successor of

x_n

. If

x_{n-1} = 0

, then

x_n = 0

.
Example:

x_1 = 2^{2^{2} + 2 + 1} + 2^{2+1} + 2 + 1

\Rightarrow x_2 = 3^{3^{3} + 3 + 1} + 3^{3+1} + 3

\Rightarrow x_3 = 4^{4^{4} + 4 + 1} + 4^{4+1} + 3

\Rightarrow x_4 = 5^{5^{5} + 5 + 1} + 5^{5+1} + 2

\Rightarrow x_5 = 6^{6^{6} + 6 + 1} + 6^{6+1} + 1

\Rightarrow x_6 = 7^{7^{7} + 7 + 1} + 7^{7+1}

\Rightarrow x_7 = 8^{8^{8} + 8 + 1} + 7.8^8 + 7.8^7 + 7.8^6 + ... + 7

.

.

.

Prove that

\exists N : x_N = 0

.

5

1

VIII Brazilian Olympic Revenge, 2009 - Problem 5

Thin and Fat eat a pizza of

2n

pieces. Each piece contains a distinct amount of olives between

1

and

2n

. Thin eats the first piece, and the two players alternately eat a piece neighbor of an eaten piece. However, neither Thin nor Fat like olives, so they will choose pieces that minimizes the total amount of olives they eat. For each arrangement

\sigma

of the olives, let

s(\sigma)

the minimal amount of olives that Thin can eat, considering that both play in the best way possible. Let

S(n)

the maximum of

s(\sigma)

, considering all arrangements.

a)

Prove that

n^2-1+\lfloor \frac{n}{2} \rfloor \le S(n) \le n^2+\lfloor \frac{n}{2} \rfloor

b)

Prove that

S(n)=n^2-1+\frac{n}{2}

for each even n.

4

1

VIII Brazilian Olympic Revenge, 2009 - Problem 4

Let

d_i(k)

the number of divisors of

k

greater than

i

. Let

f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)

. Find all

n \in N

such that

f(n)

is a perfect square.

1

VIII Brazilian Olympic Revenge, 2009 - Problem 1

Given a scalene triangle

ABC

with circuncenter

O

and circumscribed circle

\Gamma

. Let

D, E ,F

the midpoints of

BC, AC, AB

. Let

M=OE \cap AD

,

N=OF \cap AD

and

P=CM \cap BN

. Let

X=AO \cap PE

,

Y=AP \cap OF

. Let

r

the tangent of

\Gamma

through

A

. Prove that

r, EF, XY

are concurrent.

3

1

VIII Brazilian Olympic Revenge, 2009 - Problem 3

Let

ABC

to be a triangle with incenter

I

.

\omega_{A}

,

\omega_{B}

and

\omega_{C}

are the incircles of the triangles

BIC

,

CIA

and

AIB

, repectively. After all,

T

is the tangent point between

\omega_{A}

and

BC

. Prove that the other internal common tangent to

\omega_{B}

and

\omega_{C}

passes through the point

T

.

2

1

VIII Brazilian Olympic Revenge, 2009 - Problem 2

Prove that

\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}

2009 Olympic Revenge

Subcontests

VIII Brazilian Olympic Revenge, 2009 - Problem 6

VIII Brazilian Olympic Revenge, 2009 - Problem 5

VIII Brazilian Olympic Revenge, 2009 - Problem 4

VIII Brazilian Olympic Revenge, 2009 - Problem 1

VIII Brazilian Olympic Revenge, 2009 - Problem 3

VIII Brazilian Olympic Revenge, 2009 - Problem 2