MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2013 HMIC
2013 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(5)
1
1
Hide problems
2013 HMIC p1
Let
S
S
S
be a set of size
n
n
n
, and
k
k
k
be a positive integer. For each
1
≤
i
≤
k
n
1 \le i \le kn
1
≤
i
≤
kn
, there is a subset
S
i
⊂
S
S_i \subset S
S
i
⊂
S
such that
∣
S
i
∣
=
2
|S_i| = 2
∣
S
i
∣
=
2
. Furthermore, for each
e
∈
S
e \in S
e
∈
S
, there are exactly
2
k
2k
2
k
values of
i
i
i
such that
e
∈
S
i
e \in S_i
e
∈
S
i
. Show that it is possible to choose one element from
S
i
S_i
S
i
for each
1
≤
i
≤
k
n
1 \le i \le kn
1
≤
i
≤
kn
such that every element of
S
S
S
is chosen exactly
k
k
k
times.
4
1
Hide problems
2013 HMIC p4 countable subset
A subset
U
⊂
R
U \subset R
U
⊂
R
is open if for any
x
∈
U
x \in U
x
∈
U
, there exist real numbers
a
,
b
a, b
a
,
b
such that
x
∈
(
a
,
b
)
⊂
U
x \in (a, b) \subset U
x
∈
(
a
,
b
)
⊂
U
. Suppose
S
⊂
R
S \subset R
S
⊂
R
has the property that any open set intersecting
(
0
,
1
)
(0, 1)
(
0
,
1
)
also intersects
S
S
S
. Let
T
T
T
be a countable collection of open sets containing
S
S
S
. Prove that the intersection of all of the sets of
T
T
T
is not a countable subset of
R
R
R
. (A set
Γ
\Gamma
Γ
is countable if there exists a bijective function
f
:
Γ
→
Z
f : \Gamma \to Z
f
:
Γ
→
Z
.)
3
1
Hide problems
2013 HMIC p3 geometry
Triangle
A
B
C
ABC
A
BC
is inscribed in a circle
ω
\omega
ω
such that
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
and
∠
B
=
7
5
o
\angle B = 75^o
∠
B
=
7
5
o
. Let the bisector of angle
A
A
A
meet
B
C
BC
BC
and
ω
\omega
ω
at
E
E
E
and
D
D
D
, respectively. Let the reflections of
A
A
A
across
D
D
D
and
C
C
C
be
D
′
D'
D
′
and
C
′
C'
C
′
, respectively. If the tangent to
ω
\omega
ω
at
A
A
A
meets line
B
C
BC
BC
at
P
P
P
, and the circumcircle of
A
P
D
′
APD'
A
P
D
′
meets line
A
C
AC
A
C
at
F
≠
A
F \ne A
F
=
A
, prove that the circumcircle of
C
′
F
E
C'FE
C
′
FE
is tangent to
B
C
BC
BC
at
E
E
E
.
2
1
Hide problems
2013 HMIC p2 (x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that, for all real numbers
x
,
y
,
x, y,
x
,
y
,
(
x
−
y
)
(
f
(
x
)
−
f
(
y
)
)
=
f
(
x
−
f
(
y
)
)
f
(
f
(
x
)
−
y
)
.
(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).
(
x
−
y
)
(
f
(
x
)
−
f
(
y
))
=
f
(
x
−
f
(
y
))
f
(
f
(
x
)
−
y
)
.
5
1
Hide problems
2013 HMIC #5
I'd really appreciate help on this.(a) Given a set
X
X
X
of points in the plane, let
f
X
(
n
)
f_{X}(n)
f
X
(
n
)
be the largest possible area of a polygon with at most
n
n
n
vertices, all of which are points of
X
X
X
. Prove that if
m
,
n
m, n
m
,
n
are integers with
m
≥
n
>
2
m \geq n > 2
m
≥
n
>
2
then
f
X
(
m
)
+
f
X
(
n
)
≥
f
X
(
m
+
1
)
+
f
X
(
n
−
1
)
f_{X}(m) + f_{X}(n) \geq f_{X}(m + 1) + f_{X}(n - 1)
f
X
(
m
)
+
f
X
(
n
)
≥
f
X
(
m
+
1
)
+
f
X
(
n
−
1
)
.(b) Let
P
0
P_0
P
0
be a
1
×
2
1 \times 2
1
×
2
rectangle (including its interior) and inductively define the polygon
P
i
P_i
P
i
to be the result of folding
P
i
−
1
P_{i-1}
P
i
−
1
over some line that cuts
P
i
−
1
P_{i-1}
P
i
−
1
into two connected parts. The diameter of a polygon
P
i
P_i
P
i
is the maximum distance between two points of
P
i
P_i
P
i
. Determine the smallest possible diameter of
P
2013
P_{2013}
P
2013
.