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Harvard-MIT Mathematics Tournament
2015 HMIC
2015 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(5)
4
1
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2015 HMIC #4: Subsets
Prove that there exists a positive integer
N
N
N
such that for any positive integer
n
≥
N
n \ge N
n
≥
N
, there are at least
2015
2015
2015
non-empty subsets
S
S
S
of
{
n
2
+
1
,
n
2
+
2
,
…
,
n
2
+
3
n
}
\{ n^2 + 1, n^2 + 2, \dots, n^2 + 3n \}
{
n
2
+
1
,
n
2
+
2
,
…
,
n
2
+
3
n
}
with the property that the product of the elements of
S
S
S
is a perfect square.
3
1
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2015 HMIC #3: Linear Algebra!?!?
Let
M
M
M
be a
2014
×
2014
2014\times 2014
2014
×
2014
invertible matrix, and let
F
(
M
)
\mathcal{F}(M)
F
(
M
)
denote the set of matrices whose rows are a permutation of the rows of
M
M
M
. Find the number of matrices
F
∈
F
(
M
)
F\in\mathcal{F}(M)
F
∈
F
(
M
)
such that
det
(
M
+
F
)
≠
0
\det(M + F) \ne 0
det
(
M
+
F
)
=
0
.
1
1
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2015 HMIC #1: Multiplicative number theory
Let
S
S
S
be the set of positive integers
n
n
n
such that the inequality
ϕ
(
n
)
⋅
τ
(
n
)
≥
n
3
3
\phi(n) \cdot \tau(n) \geq \sqrt{\frac{n^3}{3}}
ϕ
(
n
)
⋅
τ
(
n
)
≥
3
n
3
holds, where
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
is the number of positive integers
k
≤
n
k \le n
k
≤
n
that are relatively prime to
n
n
n
, and
τ
(
n
)
\tau(n)
τ
(
n
)
is the number of positive divisors of
n
n
n
. Prove that
S
S
S
is finite.
2
1
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2015 HMIC #2: another Farey-style problem
Let
m
,
n
m,n
m
,
n
be positive integers with
m
≥
n
m \ge n
m
≥
n
. Let
S
S
S
be the set of pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of relatively prime positive integers such that
a
,
b
≤
m
a,b \le m
a
,
b
≤
m
and
a
+
b
>
m
a+b > m
a
+
b
>
m
.For each pair
(
a
,
b
)
∈
S
(a,b)\in S
(
a
,
b
)
∈
S
, consider the nonnegative integer solution
(
u
,
v
)
(u,v)
(
u
,
v
)
to the equation
a
u
−
b
v
=
n
au - bv = n
a
u
−
b
v
=
n
chosen with
v
≥
0
v \ge 0
v
≥
0
minimal, and let
I
(
a
,
b
)
I(a,b)
I
(
a
,
b
)
denote the (open) interval
(
v
/
a
,
u
/
b
)
(v/a, u/b)
(
v
/
a
,
u
/
b
)
.Prove that
I
(
a
,
b
)
⊆
(
0
,
1
)
I(a,b) \subseteq (0,1)
I
(
a
,
b
)
⊆
(
0
,
1
)
for every
(
a
,
b
)
∈
S
(a,b)\in S
(
a
,
b
)
∈
S
, and that any fixed irrational number
α
∈
(
0
,
1
)
\alpha\in(0,1)
α
∈
(
0
,
1
)
lies in
I
(
a
,
b
)
I(a,b)
I
(
a
,
b
)
for exactly
n
n
n
distinct pairs
(
a
,
b
)
∈
S
(a,b)\in S
(
a
,
b
)
∈
S
.Victor Wang, inspired by 2013 ISL N7
5
1
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2015 HMIC #5: (cyclotomic) Diophantine
Let
ω
=
e
2
π
i
/
5
\omega = e^{2\pi i /5}
ω
=
e
2
πi
/5
be a primitive fifth root of unity. Prove that there do not exist integers
a
,
b
,
c
,
d
,
k
a, b, c, d, k
a
,
b
,
c
,
d
,
k
with
k
>
1
k > 1
k
>
1
such that
(
a
+
b
ω
+
c
ω
2
+
d
ω
3
)
k
=
1
+
ω
.
(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.
(
a
+
bω
+
c
ω
2
+
d
ω
3
)
k
=
1
+
ω
.
Carl Lian