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Harvard-MIT Mathematics Tournament
2021 HMIC
2021 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(4)
5
1
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Like IMO 2014/2 but with number theory
In an
n
×
n
n \times n
n
×
n
square grid,
n
n
n
squares are marked so that every rectangle composed of exactly
n
n
n
grid squares contains at least one marked square. Determine all possible values of
n
n
n
.
4
1
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Averaging regular tetrahedra
Let
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
,
B
1
B
2
B
3
B
4
B_1B_2B_3B_4
B
1
B
2
B
3
B
4
, and
C
1
C
2
C
3
C
4
C_1C_2C_3C_4
C
1
C
2
C
3
C
4
be three regular tetrahedra in
3
3
3
-dimensional space, no two of which are congruent. Suppose that, for each
i
∈
{
1
,
2
,
3
,
4
}
i\in \{1,2,3,4\}
i
∈
{
1
,
2
,
3
,
4
}
,
C
i
C_i
C
i
is the midpoint of the line segment
A
i
B
i
A_iB_i
A
i
B
i
. Determine whether the four lines
A
1
B
1
A_1B_1
A
1
B
1
,
A
2
B
2
A_2B_2
A
2
B
2
,
A
3
B
3
A_3B_3
A
3
B
3
, and
A
4
B
4
A_4B_4
A
4
B
4
must concur.
3
1
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Sum of complex numbers can be small
Let
A
A
A
be a set of
n
≥
2
n\ge2
n
≥
2
positive integers, and let
f
(
x
)
=
∑
a
∈
A
x
a
\textstyle f(x)=\sum_{a\in A}x^a
f
(
x
)
=
∑
a
∈
A
x
a
. Prove that there exists a complex number
z
z
z
with
∣
z
∣
=
1
\lvert z\rvert=1
∣
z
∣
=
1
and
∣
f
(
z
)
∣
=
n
−
2
\lvert f(z)\rvert=\sqrt{n-2}
∣
f
(
z
)∣
=
n
−
2
.
1
1
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Swapping people on a circle
2021
2021
2021
people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up
1000
1000
1000
positions to the left of their original position.