MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1999 Chile National Olympiad
1999 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
2
1
Hide problems
TS _|_ BC wanted, altitudes related (Chile NMO 1999 P2)
In an acute triangle
A
B
C
ABC
A
BC
, let
A
K
‾
,
B
L
‾
,
C
M
‾
\overline {AK}, \overline {BL}, \overline {CM}
A
K
,
B
L
,
CM
be the altitudes of the triangle concurrent at the point
H
H
H
and let
P
P
P
the midpoint of
A
H
‾
\overline {AH}
A
H
. Let's define
S
=
B
H
‾
∩
M
K
‾
S = \overline {BH} \cap \overline {MK}
S
=
B
H
∩
M
K
and
T
=
L
P
‾
∩
A
B
‾
T = \overline {LP} \cap \overline {AB}
T
=
L
P
∩
A
B
. Show that
T
S
‾
⊥
B
C
‾
\overline {TS} \perp \overline {BC}
TS
⊥
BC
7
1
Hide problems
f (n + f (n)) = 1, f (1998) = 2 (Chile NMO 1999 P7)
Let
f
f
f
be a function defined on the set of positive integers , and with values in the same set, which satisfies:
∙
\bullet
∙
f
(
n
+
f
(
n
)
)
=
1
f (n + f (n)) = 1
f
(
n
+
f
(
n
))
=
1
for all
n
≥
1
n\ge 1
n
≥
1
.
∙
\bullet
∙
f
(
1998
)
=
2
f (1998) = 2
f
(
1998
)
=
2
Find the lowest possible value of the sum
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
1999
)
f (1) + f (2) +... + f (1999)
f
(
1
)
+
f
(
2
)
+
...
+
f
(
1999
)
, and find the formula of
f
f
f
for which this minimum is satisfied,
1
1
Hide problems
n=a^2+b^2=c^2+d^2, a-c=7, d-b=13 (Chile NMO 1999 P1)
Pedrito's lucky number is
34117
34117
34117
. His friend Ramanujan points out that
34117
=
16
6
2
+
8
1
2
=
15
9
2
+
9
4
2
34117 = 166^2 + 81^2 = 159^2 + 94^2
34117
=
16
6
2
+
8
1
2
=
15
9
2
+
9
4
2
and
166
−
159
=
7
166-159 = 7
166
−
159
=
7
,
94
−
81
=
13
94- 81 = 13
94
−
81
=
13
. Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is
7
7
7
and in the difference between the seconds it gives
13
13
13
. Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.
3
1
Hide problems
1000 red squares in 1999x1999 board (Chile NMO 1999 P3)
It is possible to paint with the colors red and blue the squares of a grid board
1999
×
1999
1999\times 1999
1999
×
1999
, so that in each of the
1999
1999
1999
rows, in each of the
1999
1999
1999
columns and each of the the
2
2
2
diagonals are exactly
1000
1000
1000
squares painted red?
4
1
Hide problems
X, O in nxn board, not in adjacent squares (Chile NMO 1999 P4)
Given a
n
×
n
n \times n
n
×
n
grid board . How many ways can an
X
X
X
and an
O
O
O
be placed in such a way that they are not in adjacent squares?
5
1
Hide problems
x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0 (Chile NMO 1999 P5)
Consider the numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
that satisfy:
∙
\bullet
∙
x
i
∈
{
−
1
,
1
}
x_i \in \{-1,1\}
x
i
∈
{
−
1
,
1
}
, with
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2,...,n
i
=
1
,
2
,
...
,
n
∙
\bullet
∙
x
1
x
2
x
3
x
4
+
x
2
x
3
x
4
x
5
+
.
.
.
+
x
n
x
1
x
2
x
3
=
0
x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0
x
1
x
2
x
3
x
4
+
x
2
x
3
x
4
x
5
+
...
+
x
n
x
1
x
2
x
3
=
0
Prove that
n
n
n
is a multiple of
4
4
4
.
6
1
Hide problems
infinite triangles with 2 same sides and 3 angles,non congruent Chile 1999 L2 p6
Prove that there are infinite pairs of non-congruent triangles that have the same angles and two of their equal sides. Develop an algorithm or rule to obtain these pairs of triangles and indicate at least one pair that satisfies the asserted.