MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2005 Chile National Olympiad
2005 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
1
1
Hide problems
distance covered by ant in a unit square > = \sqrt2
In the center of the square of side
1
1
1
shown in the figure is an ant. At one point the ant starts walking until it touches the left side
(
a
)
(a)
(
a
)
, then continues walking until it reaches the bottom side
(
b
)
(b)
(
b
)
, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of
2
2
2
. [asy] unitsize(2 cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);label("
a
a
a
", (0,0.5), W); label("
b
b
b
", (0.5,0), S); dot((0.5,0.5)); [/asy]
7
1
Hide problems
triominos cover a m x n board
Consider a
2
×
2
2\times2
2
×
2
square with one corner removed from
1
×
1
1\times1
1
×
1
, leaving a shape in the form of
L
L
L
. [asy] unitsize(0.5 cm);draw((1,0)--(1,2)--(0,2)--(0,0)--(2,0)--(2,1)--(0,1)); [/asy] We will call this figure triomino. Determine all values of
m
,
n
m, n
m
,
n
such that a rectangle of
m
×
n
m\times n
m
×
n
can be exactly covered with triominos.
6
1
Hide problems
100 tickets to win a million pesos
A box contains
100
100
100
tickets. Each ticket has a real number written on it. There are no restrictions on the type of number except that they are all different (they can be integers, rational, positive, negative, irrational, large or small). Of course there is one ticket that has the highest number and that is the winner. The game consists of drawing a ticket at random, looking at it and deciding whether to keep it or not. If we choose to keep him, it is verified if he was the oldest, in which case we win a million pesos (if we don't win, the game is over). If we don't think it's the biggest, we can discard it and draw another one, repeating the process until we like one or we run out of tickets. Going back to choose a previously discarded ticket is prohibited. Find a game strategy that gives at least a
25
%
25\%
25%
chance of winning.
5
1
Hide problems
g(nm) = g(n) + g(m) + g(n)g(m), g(n^2 + 1) = (g(n) + 1)^2, g(1) = 0
Compute
g
(
2005
)
g(2005)
g
(
2005
)
where
g
g
g
is a function defined on the natural numbers that has the following properties: i)
g
(
1
)
=
0
g(1) = 0
g
(
1
)
=
0
ii)
g
(
n
m
)
=
g
(
n
)
+
g
(
m
)
+
g
(
n
)
g
(
m
)
g(nm) = g(n) + g(m) + g(n)g(m)
g
(
nm
)
=
g
(
n
)
+
g
(
m
)
+
g
(
n
)
g
(
m
)
for any pair of integers
n
,
m
n, m
n
,
m
. iii)
g
(
n
2
+
1
)
=
(
g
(
n
)
+
1
)
2
g(n^2 + 1) = (g(n) + 1)^2
g
(
n
2
+
1
)
=
(
g
(
n
)
+
1
)
2
for every integer
n
n
n
.
4
1
Hide problems
f(1)+f(2)+...+f(2005) if f(a) is largest integer <= \sqrt[4]{a}
Let
f
(
a
)
f(a)
f
(
a
)
be the largest integer less than or equal to the fourth root of "
a
a
a
". Calculate
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
2005
)
.
f(1)+f(2)+...+f(2005).
f
(
1
)
+
f
(
2
)
+
...
+
f
(
2005
)
.
3
1
Hide problems
max n such that Fibonacci number f_N<f_1+f_2+...+f_{2004}+f_{2005}
The Fibonacci numbers
f
n
f_n
f
n
are defined for each natural number
n
n
n
as follows:
f
0
=
f
1
=
1
f_0=f_1=1
f
0
=
f
1
=
1
and for
n
n
n
greater than or equal to
2
2
2
, by recurrence:
f
n
=
f
n
−
1
+
f
n
−
2
f_n=f_{n-1}+f_{n-2}
f
n
=
f
n
−
1
+
f
n
−
2
Let
S
=
f
1
+
f
2
+
.
.
.
+
f
2004
+
f
2005
S=f_1+f_2+...+f_{2004}+f_{2005}
S
=
f
1
+
f
2
+
...
+
f
2004
+
f
2005
. Calculate the largest value of
N
N
N
, such that the Fibonacci number
f
N
f_N
f
N
satisfies
f
N
<
S
f_N<S
f
N
<
S
2
1
Hide problems
p divides m if m/n= sum 1/(i-1)
Let
p
p
p
be a prime number greater than
2
2
2
and let
m
,
n
m, n
m
,
n
be integers such that:
m
n
=
1
+
1
2
+
1
3
+
.
.
.
+
1
p
−
1
.
\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.
n
m
=
1
+
2
1
+
3
1
+
...
+
p
−
1
1
.
Prove that
p
p
p
divides
m
m
m
.