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2003 JBMO Shortlist
2003 JBMO Shortlist
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JBMO ShortLists
Subcontests
(6)
6
1
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inequality with areas defined by parallel to sides of ABC
Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into
6
6
6
parts with the marked areas as in the figure. Show that
a
α
+
b
β
+
c
γ
≥
3
2
\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}
α
a
+
β
b
+
γ
c
≥
2
3
https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png
5
1
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semicircle tangent to isosceles triangle
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. A semi-circle of diameter
[
E
F
]
[EF]
[
EF
]
with
E
,
F
∈
[
B
C
]
E, F \in [BC]
E
,
F
∈
[
BC
]
, is tangent to the sides
A
B
,
A
C
AB,AC
A
B
,
A
C
in
M
,
N
M, N
M
,
N
respectively and
A
E
AE
A
E
intersects the semicircle at
P
P
P
. Prove that
P
F
PF
PF
passes through the midpoint of
[
M
N
]
[MN]
[
MN
]
.
4
1
Hide problems
3 equal circles with a common points, orthocenter
Three equal circles have a common point
M
M
M
and intersect in pairs at points
A
,
B
,
C
A, B, C
A
,
B
,
C
. Prove that that
M
M
M
is the orthocenter of triangle
A
B
C
ABC
A
BC
.
3
1
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G,B,C,A' concyclic iff GA ⊥ GC (AC=CA')
Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
, and
A
′
A'
A
′
the symmetric of
A
A
A
wrt
C
C
C
. Show that
G
,
B
,
C
,
A
′
G, B, C, A'
G
,
B
,
C
,
A
′
are concyclic if and only if
G
A
⊥
G
C
GA \perp GC
G
A
⊥
GC
.
2
1
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triangle with area = perimeter
Is there a triangle with
12
c
m
2
12 \, cm^2
12
c
m
2
area and
12
12
12
cm perimeter?
1
1
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diagonals divide quadrilateral into prime areas
Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?