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Saint Petersburg Mathematical Olympiad
2021 Saint Petersburg Mathematical Olympiad
3
Strange pentagon geo
Strange pentagon geo
Source: St Petersburg 2021 10.3
December 23, 2021
geometry
pentagon
Problem Statement
Given a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, points
A
1
,
B
1
,
C
1
,
D
1
,
E
1
A_1, B_1, C_1, D_1, E_1
A
1
,
B
1
,
C
1
,
D
1
,
E
1
are such that
A
A
1
⊥
B
E
,
B
B
1
⊥
A
C
,
C
C
1
⊥
B
D
,
D
D
1
⊥
C
E
,
E
E
1
⊥
D
A
.
AA_1 \perp BE, BB_1 \perp AC, CC_1 \perp BD, DD_1 \perp CE, EE_1 \perp DA.
A
A
1
⊥
BE
,
B
B
1
⊥
A
C
,
C
C
1
⊥
B
D
,
D
D
1
⊥
CE
,
E
E
1
⊥
D
A
.
In addition,
A
E
1
=
A
B
1
,
B
C
1
=
B
A
1
,
C
B
1
=
C
D
1
AE_1 = AB_1, BC_1 = BA_1, CB_1 = CD_1
A
E
1
=
A
B
1
,
B
C
1
=
B
A
1
,
C
B
1
=
C
D
1
and
D
C
1
=
D
E
1
DC_1 = DE_1
D
C
1
=
D
E
1
. Prove that
E
D
1
=
E
A
1
ED_1 = EA_1
E
D
1
=
E
A
1
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