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Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2019 HMIC
2019 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(5)
5
1
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Good functions with parameter $\alpha$
Let
p
=
2017
p = 2017
p
=
2017
be a prime and
F
p
\mathbb{F}_p
F
p
be the integers modulo
p
p
p
. A function
f
:
Z
→
F
p
f: \mathbb{Z}\rightarrow\mathbb{F}_p
f
:
Z
→
F
p
is called good if there is
α
∈
F
p
\alpha\in\mathbb{F}_p
α
∈
F
p
with
α
≢
0
(
m
o
d
p
)
\alpha\not\equiv 0\pmod{p}
α
≡
0
(
mod
p
)
such that
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
α
y
f
(
x
−
y
)
(
m
o
d
p
)
f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
α
y
f
(
x
−
y
)
(
mod
p
)
for all
x
,
y
∈
Z
x, y\in\mathbb{Z}
x
,
y
∈
Z
. How many good functions are there that are periodic with minimal period
2016
2016
2016
?Ashwin Sah
4
1
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Cactus graphs
A cactus is a finite simple connected graph where no two cycles share an edge. Show that in a nonempty cactus, there must exist a vertex which is part of at most one cycle.Kevin Yang
3
1
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Four points at large distance
Do there exist four points
P
i
=
(
x
i
,
y
i
)
∈
R
2
(
1
≤
i
≤
4
)
P_i = (x_i, y_i) \in \mathbb{R}^2\ (1\leq i \leq 4)
P
i
=
(
x
i
,
y
i
)
∈
R
2
(
1
≤
i
≤
4
)
on the plane such that:[*] for all
i
=
1
,
2
,
3
,
4
i = 1,2,3,4
i
=
1
,
2
,
3
,
4
, the inequality
x
i
4
+
y
i
4
≤
x
i
3
+
y
i
3
x_i^4 + y_i^4 \le x_i^3+ y_i^3
x
i
4
+
y
i
4
≤
x
i
3
+
y
i
3
holds, and [*] for all
i
≠
j
i \neq j
i
=
j
, the distance between
P
i
P_i
P
i
and
P
j
P_j
P
j
is greater than
1
1
1
?Pakawut Jiradilok
2
1
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Guessing a permutation
Annie has a permutation
(
a
1
,
a
2
,
…
,
a
2019
)
(a_1, a_2, \dots ,a_{2019})
(
a
1
,
a
2
,
…
,
a
2019
)
of
S
=
{
1
,
2
,
…
,
2019
}
S=\{1,2,\dots,2019\}
S
=
{
1
,
2
,
…
,
2019
}
, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an
n
n
n
-tuple
(
y
1
,
y
2
,
…
,
y
2019
)
(y_1, y_2, \dots, y_{2019})
(
y
1
,
y
2
,
…
,
y
2019
)
of integers in
S
S
S
, and then Annie gives the number of indices
i
∈
S
i\in S
i
∈
S
such that
a
i
=
y
i
a_i=y_i
a
i
=
y
i
. (a) Show that Yannick can always guess Annie's permutation with at most
1200000
1200000
1200000
guesses. (b) Show that Yannick can always guess Annie's permutation with at most
24000
24000
24000
guesses.Yannick Yao
1
1
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Euler line intersects (BIC) twice
Let
A
B
C
ABC
A
BC
be an acute scalene triangle with incenter
I
I
I
. Show that the circumcircle of
B
I
C
BIC
B
I
C
intersects the Euler line of
A
B
C
ABC
A
BC
in two distinct points.(Recall that the Euler line of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.)Andrew Gu