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National and Regional Contests
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USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2023 HMIC
2023 HMIC
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(5)
P5
1
Hide problems
infinite sequence of positive integers
Let
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
be an infinite sequence of positive integers such that, for all positive integers
m
m
m
and
n
,
n,
n
,
we have that
a
m
+
n
a_{m+n}
a
m
+
n
divides
a
m
a
n
−
1.
a_ma_n-1.
a
m
a
n
−
1.
Prove that there exists an integer
C
C
C
such that, for all positive integers
k
>
C
,
k>C,
k
>
C
,
we have
a
k
=
1.
a_k=1.
a
k
=
1.
P4
1
Hide problems
numbers on a blackboard
Let
n
>
1
n>1
n
>
1
be a positive integer. Claire writes
n
n
n
distinct positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \dots, x_n
x
1
,
x
2
,
…
,
x
n
in a row on a blackboard. In a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
m
o
v
e
<
/
s
p
a
n
>
,
<span class='latex-italic'>move</span>,
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
m
o
v
e
<
/
s
p
an
>
,
William can erase a number
x
x
x
and replace it with either
1
x
\tfrac{1}{x}
x
1
or
x
+
1
x+1
x
+
1
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.[*]Prove that there exists a positive constant
A
,
A,
A
,
independent of
n
,
n,
n
,
such that William can always reach his goal in at most
A
n
log
n
An \log n
A
n
lo
g
n
moves. [*]Prove that there exists a positive constant
B
,
B,
B
,
independent of
n
,
n,
n
,
such that Claire can choose the initial numbers such that William cannot attain his goal in less than
B
n
log
n
Bn \log n
B
n
lo
g
n
moves.
P3
1
Hide problems
invert your expectations pt.2
Triangle
A
B
C
ABC
A
BC
has incircle
ω
\omega
ω
and
A
A
A
-excircle
ω
A
.
\omega_A.
ω
A
.
Circle
γ
B
\gamma_B
γ
B
passes through
B
B
B
and is externally tangent to
ω
\omega
ω
and
ω
A
.
\omega_A.
ω
A
.
Circle
γ
C
\gamma_C
γ
C
passes through
C
C
C
and is externally tangent to
ω
\omega
ω
and
ω
A
.
\omega_A.
ω
A
.
If
γ
B
\gamma_B
γ
B
intersects line
B
C
BC
BC
again at
D
,
D,
D
,
and
γ
C
\gamma_C
γ
C
intersects line
B
C
BC
BC
again at
E
,
E,
E
,
prove that
B
D
=
E
C
.
BD=EC.
B
D
=
EC
.
P2
1
Hide problems
Mundane Primes
A prime number
p
p
p
is mundane if there exist positive integers
a
a
a
and
b
b
b
less than
p
2
\tfrac{p}{2}
2
p
such that
a
b
−
1
p
\tfrac{ab-1}{p}
p
ab
−
1
is a positive integer. Find, with proof, all prime numbers that are not mundane.
P1
1
Hide problems
functional equations over positive rationals make me big sad
Let
Q
+
\mathbb{Q}^{+}
Q
+
denote the set of positive rational numbers. Find, with proof, all functions
f
:
Q
+
→
Q
+
f:\mathbb{Q}^+ \to \mathbb{Q}^+
f
:
Q
+
→
Q
+
such that, for all positive rational numbers
x
x
x
and
y
,
y,
y
,
we have
f
(
x
)
=
f
(
x
+
y
)
+
f
(
x
+
x
2
f
(
y
)
)
.
f(x)=f(x+y)+f(x+x^2f(y)).
f
(
x
)
=
f
(
x
+
y
)
+
f
(
x
+
x
2
f
(
y
))
.