MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2016 Harvard-MIT Mathematics Tournament
24
24
Part of
2016 Harvard-MIT Mathematics Tournament
Problems
(1)
2016 Guts #24
Source:
12/24/2016
Let
Δ
A
1
B
1
C
\Delta A_1B_1C
Δ
A
1
B
1
C
be a triangle with
∠
A
1
B
1
C
=
9
0
∘
\angle A_1B_1C = 90^{\circ}
∠
A
1
B
1
C
=
9
0
∘
and
C
A
1
C
B
1
=
5
+
2
\frac{CA_1}{CB_1} = \sqrt{5}+2
C
B
1
C
A
1
=
5
+
2
. For any
i
≥
2
i \ge 2
i
≥
2
, define
A
i
A_i
A
i
to be the point on the line
A
1
C
A_1C
A
1
C
such that
A
i
B
i
−
1
⊥
A
1
C
A_iB_{i-1} \perp A_1C
A
i
B
i
−
1
⊥
A
1
C
and define
B
i
B_i
B
i
to be the point on the line
B
1
C
B_1C
B
1
C
such that
A
i
B
i
⊥
B
1
C
A_iB_i \perp B_1C
A
i
B
i
⊥
B
1
C
. Let
Γ
1
\Gamma_1
Γ
1
be the incircle of
Δ
A
1
B
1
C
\Delta A_1B_1C
Δ
A
1
B
1
C
and for
i
≥
2
i \ge 2
i
≥
2
,
Γ
i
\Gamma_i
Γ
i
be the circle tangent to
Γ
i
−
1
,
A
1
C
,
B
1
C
\Gamma_{i-1}, A_1C, B_1C
Γ
i
−
1
,
A
1
C
,
B
1
C
which is smaller than
Γ
i
−
1
\Gamma_{i-1}
Γ
i
−
1
.How many integers
k
k
k
are there such that the line
A
1
B
2016
A_1B_{2016}
A
1
B
2016
intersects
Γ
k
\Gamma_{k}
Γ
k
?