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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2008 Chile National Olympiad
2008 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
5
1
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trucker cross the desert if total supply of gasoline at station is 180 ;iters
When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of
800
800
800
km between two stations consecutive service. Your truck can only store
50
50
50
liters of benzene, and has a yield of
10
10
10
km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of
50
50
50
liters you can travel
100
100
100
km, leave
30
30
30
liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel.a) Can the trucker cross the desert if at this service station the total supply is
140
140
140
liters? b) Can the trucker cross the desert if the total supply of gasoline at the station is
180
180
180
liters?
6
1
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\pi + \sqrt2 transcendental if it is known that \pi is transcendental
It is known that the number
π
\pi
π
is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number
π
+
2
\pi + \sqrt2
π
+
2
.
4
1
Hide problems
3 colors are available to paint the plane
Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length
1
1
1
whose endpoints have the same color.
3
1
Hide problems
functional equation with strictly increasing function and f((x+y)/2)
Determine all strictly increasing functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all
x
≠
y
x \ne y
x
=
y
to hold
2
[
f
(
y
)
−
f
(
x
+
y
2
)
]
f
(
x
)
−
f
(
y
)
=
f
(
x
)
−
f
(
y
)
2
[
f
(
x
+
y
2
)
−
f
(
x
)
]
\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}
f
(
x
)
−
f
(
y
)
2
[
f
(
y
)
−
f
(
2
x
+
y
)
]
=
2
[
f
(
2
x
+
y
)
−
f
(
x
)
]
f
(
x
)
−
f
(
y
)
1
1
Hide problems
3-digit number of buses wanted
Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of
13
13
13
, and the second digit of them is the average of the other two. What are the bus lines that go to Alberto's house?
2
1
Hide problems
existence of a triangle with sides PA \sqrt2, PB,PC Chile 2008 L2 P2
Let
A
B
C
ABC
A
BC
be right isosceles triangle with right angle in
A
A
A
. Given a point
P
P
P
inside the triangle, denote by
a
,
b
a, b
a
,
b
and
c
c
c
the lengths of
P
A
,
P
B
PA, PB
P
A
,
PB
and
P
C
PC
PC
, respectively. Prove that there is a triangle whose sides have a length of
a
2
,
b
a\sqrt2 , b
a
2
,
b
and
c
c
c
.