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Problems
Contests
International Contests
Iranian Geometry Olympiad
2014 Iranian Geometry Olympiad (junior)
2014 Iranian Geometry Olympiad (junior)
Part of
Iranian Geometry Olympiad
Subcontests
(5)
P2
1
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ratio of triangles concerning touchpoints of incircle and perpendiculars
The inscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
touches
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
at
D
,
E
D,E
D
,
E
and
F
F
F
respectively. Denote the perpendicular foots from
F
,
E
F, E
F
,
E
to
B
C
BC
BC
by
K
,
L
K, L
K
,
L
respectively. Let the second intersection of these perpendiculars with the incircle be
M
,
N
M, N
M
,
N
respectively. Show that
S
△
B
M
D
S
△
C
N
D
=
D
K
D
L
\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}
S
△
CN
D
S
△
BM
D
=
D
L
DK
by Mahdi Etesami Fard
P3
1
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constant ratio in 2 inscribed 93-gons with parallel sides
Each of Mahdi and Morteza has drawn an inscribed
93
93
93
-gon. Denote the first one by
A
1
A
2
…
A
93
A_1A_2…A_{93}
A
1
A
2
…
A
93
and the second by
B
1
B
2
…
B
93
B_1B_2…B_{93}
B
1
B
2
…
B
93
. It is known that
A
i
A
i
+
1
/
/
B
i
B
i
+
1
A_iA_{i+1} // B_iB_{i+1}
A
i
A
i
+
1
//
B
i
B
i
+
1
for
1
≤
i
≤
93
1 \le i \le 93
1
≤
i
≤
93
(
A
93
=
A
1
,
B
93
=
B
1
A_{93} = A_1, B_{93} = B_1
A
93
=
A
1
,
B
93
=
B
1
). Show that
A
i
A
i
+
1
B
i
B
i
+
1
\frac{A_iA_{i+1} }{ B_iB_{i+1}}
B
i
B
i
+
1
A
i
A
i
+
1
is a constant number independent of
i
i
i
.by Morteza Saghafian
P4
1
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equal angles when <C = <A + 90^o
In a triangle ABC we have
∠
C
=
∠
A
+
9
0
o
\angle C = \angle A + 90^o
∠
C
=
∠
A
+
9
0
o
. The point
D
D
D
on the continuation of
B
C
BC
BC
is given such that
A
C
=
A
D
AC = AD
A
C
=
A
D
. A point
E
E
E
in the side of
B
C
BC
BC
in which
A
A
A
doesn’t lie is chosen such that
∠
E
B
C
=
∠
A
,
∠
E
D
C
=
1
2
∠
A
\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A
∠
EBC
=
∠
A
,
∠
E
D
C
=
2
1
∠
A
. Prove that
∠
C
E
D
=
∠
A
B
C
\angle CED = \angle ABC
∠
CE
D
=
∠
A
BC
.by Morteza Saghafian
P5
1
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inequality with midpoint of arc with equal angles
Two points
X
,
Y
X, Y
X
,
Y
lie on the arc
B
C
BC
BC
of the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
(this arc does not contain
A
A
A
) such that
∠
B
A
X
=
∠
C
A
Y
\angle BAX = \angle CAY
∠
B
A
X
=
∠
C
A
Y
. Let
M
M
M
denotes the midpoint of the chord
A
X
AX
A
X
. Show that
B
M
+
C
M
>
A
Y
BM +CM > AY
BM
+
CM
>
A
Y
.by Mahan Tajrobekar
P1
1
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very easy!
ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.