MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2024 Chile National Olympiad.
2024 Chile National Olympiad.
Part of
Chile National Olympiad
Subcontests
(6)
6
1
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Can you find another prime?
Let
133
…
33
133\ldots 33
133
…
33
be a number with
k
≥
2
k \geq 2
k
≥
2
digits, which we assume is prime. Prove that
k
(
k
+
2
)
k(k + 2)
k
(
k
+
2
)
is a multiple of 24. (For example, 133...33 is a prime number when
k
=
16
k = 16
k
=
16
5
1
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Subcases.....
You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is
S
S
S
. Find all possible values of
S
S
S
that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.
4
1
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Circle is the solution????
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers that satisfy the system
(
x
+
1
)
(
x
2
+
1
)
=
y
3
+
1
(x + 1)(x^2 + 1) = y^3 + 1
(
x
+
1
)
(
x
2
+
1
)
=
y
3
+
1
(
y
+
1
)
(
y
2
+
1
)
=
x
3
+
1
(y + 1)(y^2 + 1) = x^3 + 1
(
y
+
1
)
(
y
2
+
1
)
=
x
3
+
1
3
1
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Finally radical axis in National
Let
A
D
AD
A
D
and
B
E
BE
BE
be altitudes of triangle
△
A
B
C
\triangle ABC
△
A
BC
that meet at the orthocenter
H
H
H
. The midpoints of segments
A
B
AB
A
B
and
C
H
CH
C
H
are
X
X
X
and
Y
Y
Y
, respectively. Prove that the line
X
Y
XY
X
Y
is perpendicular to line
D
E
DE
D
E
.
2
1
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What happens if d is negative?
On a table, there are many coins and a container with two coins. Vale and Diego play the following game, where Vale starts and then Diego plays, alternating turns. If at the beginning of a turn the container contains
n
n
n
coins, the player can add a number
d
d
d
of coins, where
d
d
d
divides exactly into
n
n
n
and
d
<
n
d < n
d
<
n
. The first player to complete at least 2024 coins in the container wins. Prove that there exists a strategy for Vale to win, no matter the decisions made by Diego.
1
1
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Antoher funcion.....
Let
f
(
x
)
=
10
0
x
10
0
x
+
10
f(x) = \frac{100^x}{100^x + 10}
f
(
x
)
=
10
0
x
+
10
10
0
x
. Determine the value of:
f
(
1
2024
)
−
f
(
2
2024
)
+
f
(
3
2024
)
−
f
(
4
2024
)
+
…
−
f
(
2022
2024
)
+
f
(
2023
2024
)
f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right)
f
(
2024
1
)
−
f
(
2024
2
)
+
f
(
2024
3
)
−
f
(
2024
4
)
+
…
−
f
(
2024
2022
)
+
f
(
2024
2023
)