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National and Regional Contests
Chile Contests
Chile Junior Math Olympiad
2023 Chile Junior Math Olympiad
2023 Chile Junior Math Olympiad
Part of
Chile Junior Math Olympiad
Subcontests
(4)
4
1
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triangle inscribed in triangle, equal ratios - 2023 Chile NMO L1 P4
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle with side
1
1
1
. The points
P
P
P
,
Q
Q
Q
,
R
R
R
are chosen on the sides of the segments
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
respectively in such a way that
A
P
P
B
=
B
Q
Q
C
=
C
R
R
A
=
2
5
.
\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RA}=\frac25.
PB
A
P
=
QC
BQ
=
R
A
CR
=
5
2
.
Find the area of triangle
P
Q
R
PQR
PQR
. https://cdn.artofproblemsolving.com/attachments/8/4/6184d66bd3ae23db29a93eeef241c46ae0ad44.png
5
1
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1600 bananas among 100 monkeys - 2023 Chile NMO L1 P5
1600
1600
1600
bananas are distributed among
100
100
100
monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.
3
1
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9 triangles of equal area, by 8 points, equilateral - 2023 Chile NMO L1 P3
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle with side
1
1
1
. Four points are marked
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
P
4
P_4
P
4
on side
A
C
AC
A
C
and four points
Q
1
Q_1
Q
1
,
Q
2
Q_2
Q
2
,
Q
3
Q_3
Q
3
,
Q
4
Q_4
Q
4
on side
A
B
AB
A
B
(see figure) of such a way to generate
9
9
9
triangles of equal area. Find the length of segment
A
P
4
AP_4
A
P
4
. https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.pngPS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3
1
1
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3-digit numbers wanted - 2023 Chile NMO L1 P1
Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.