MathDB

1

2015 Brazilian Olympic Revenge Problem 5

A_1 A_2 A_3

, let

a_i

denote the side opposite to

A_i

, where indices are taken modulo 3. Let

D_1 \in a_1

. For

D_i \in A_i

, let

\omega_i

be the incircle of the triangle formed by lines

a_i, a_{i+1}, A_iD_i

, and

D_{i+1} \in a_{i+1}

with

A_{i+1} D_{i+1}

tangent to

\omega_i

. Show that the set

\{D_i: i \in \mathbb{N}\}

is finite.

4

1

2015 Brazilian Olympic Revenge Problem 4

Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer

k

is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point

A

, can move to any point

P

such that

|AP| \le k

, as long as

P

is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let

f(k)

denote the least number of rounds the Devil takes to win. Prove that

0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).

Note:

a(x) = b(x) (1+o(1))

if

\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1

.

3

1

2015 Brazilian Olympic Revenge Problem 3

For every

n \in \mathbb{N}

, there exist integers

k

such that

n | k

and

k

contains only zeroes and ones in its decimal representation. Let

f(n)

denote the least possible number of ones in any such

k

. Determine whether there exists a constant

C

such that

f(n) < C

for all

n \in \mathbb{N}

.

2

1

2015 Brazilian Olympic Revenge Problem 2

Given

v = (a,b,c,d) \in \mathbb{N}^4

, let

\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)

and

\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))

for

k > 1

. Define

f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}

and

\max(v) = \max\{a,b,c,d\}.

Show that

f(v) < 1000\log \max(v)

for all sufficiently large

v

and

f(v) > 0.001 \log \max (v)

for infinitely many

v

.

1

2015 Brazilian Olympic Revenge Problem 1

For

n \in \mathbb{N}

, let

P(n)

denote the product of distinct prime factors of

n

, with

P(1) = 1

. Show that for any

a_0 \in \mathbb{N}

, if we define a sequence

a_{k+1} = a_k + P(a_k)

for

k \ge 0

, there exists some

k \in \mathbb{N}

with

a_k/P(a_k) = 2015

.

2015 Olympic Revenge

Subcontests

2015 Brazilian Olympic Revenge Problem 5

2015 Brazilian Olympic Revenge Problem 4

2015 Brazilian Olympic Revenge Problem 3

2015 Brazilian Olympic Revenge Problem 2

2015 Brazilian Olympic Revenge Problem 1