MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2016 HMIC
2016 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(5)
5
1
Hide problems
Sums of perfect powers lack arithmetic progressions
Let
S
=
{
a
1
,
…
,
a
n
}
S = \{a_1, \ldots, a_n \}
S
=
{
a
1
,
…
,
a
n
}
be a finite set of positive integers of size
n
≥
1
n \ge 1
n
≥
1
, and let
T
T
T
be the set of all positive integers that can be expressed as sums of perfect powers (including
1
1
1
) of distinct numbers in
S
S
S
, meaning
T
=
{
∑
i
=
1
n
a
i
e
i
∣
e
1
,
e
2
,
…
,
e
n
≥
0
}
.
T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}.
T
=
{
i
=
1
∑
n
a
i
e
i
∣
e
1
,
e
2
,
…
,
e
n
≥
0
}
.
Show that there is a positive integer
N
N
N
(only depending on
n
n
n
) such that
T
T
T
contains no arithmetic progression of length
N
N
N
.Yang Liu
4
1
Hide problems
Generalization of SL 2002 A3
Let
P
P
P
be an odd-degree integer-coefficient polynomial. Suppose that
x
P
(
x
)
=
y
P
(
y
)
xP(x)=yP(y)
x
P
(
x
)
=
y
P
(
y
)
for infinitely many pairs
x
,
y
x,y
x
,
y
of integers with
x
≠
y
x\ne y
x
=
y
. Prove that the equation
P
(
x
)
=
0
P(x)=0
P
(
x
)
=
0
has an integer root.Victor Wang
3
1
Hide problems
Functional Inequality on N
Denote by
N
\mathbb{N}
N
the positive integers. Let
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
be a function such that, for any
w
,
x
,
y
,
z
∈
N
w,x,y,z \in \mathbb{N}
w
,
x
,
y
,
z
∈
N
,
f
(
f
(
f
(
z
)
)
)
f
(
w
x
f
(
y
f
(
z
)
)
)
=
z
2
f
(
x
f
(
y
)
)
f
(
w
)
.
f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w).
f
(
f
(
f
(
z
)))
f
(
w
x
f
(
y
f
(
z
)))
=
z
2
f
(
x
f
(
y
))
f
(
w
)
.
Show that
f
(
n
!
)
≥
n
!
f(n!) \ge n!
f
(
n
!)
≥
n
!
for every positive integer
n
n
n
.Pakawut Jiradilok
2
1
Hide problems
Geometry warmup: internally tangent circles
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcenter
O
O
O
, orthocenter
H
H
H
, and circumcircle
Ω
\Omega
Ω
. Let
M
M
M
be the midpoint of
A
H
AH
A
H
and
N
N
N
the midpoint of
B
H
BH
B
H
. Assume the points
M
M
M
,
N
N
N
,
O
O
O
,
H
H
H
are distinct and lie on a circle
ω
\omega
ω
. Prove that the circles
ω
\omega
ω
and
Ω
\Omega
Ω
are internally tangent to each other.Dhroova Aiylam and Evan Chen
1
1
Hide problems
Thesus wanders the plane
Theseus starts at the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the plane. If Theseus is standing at the point
(
x
,
y
)
(x, y)
(
x
,
y
)
in the plane, he can step one unit to the north to point
(
x
,
y
+
1
)
(x, y+1)
(
x
,
y
+
1
)
, one unit to the west to point
(
x
−
1
,
y
)
(x-1, y)
(
x
−
1
,
y
)
, one unit to the south to point
(
x
,
y
−
1
)
(x, y-1)
(
x
,
y
−
1
)
, or one unit to the east to point
(
x
+
1
,
y
)
(x+1, y)
(
x
+
1
,
y
)
. After a sequence of more than two such moves, starting with a step one unit to the south (to point
(
0
,
−
1
)
(0, -1)
(
0
,
−
1
)
), Theseus finds himself back at the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
. He never visited any point other than
(
0
,
0
)
(0, 0)
(
0
,
0
)
more than once, and never visited the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
except at the start and end of this sequence of moves.Let
X
X
X
be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let
Y
Y
Y
be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that
∣
X
−
Y
∣
=
1
|X - Y| = 1
∣
X
−
Y
∣
=
1
.Mitchell Lee