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2023 UMD Math Competition Part II
2023 UMD Math Competition Part II
Part of
UMD Math Competition
Subcontests
(5)
1
1
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Indian raga music
An Indian raga has two kinds of notes: a short note, which lasts for
1
1
1
beat and a long note, which lasts for
2
2
2
beats. For example, there are
3
3
3
ragas which are
3
3
3
beats long;
3
3
3
short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long?
4
1
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Triangle ratios
Assume every side length of a triangle
A
B
C
ABC
A
BC
is more than
2
2
2
and two of its angles are given by
∠
A
B
C
=
5
7
∘
\angle ABC = 57^\circ
∠
A
BC
=
5
7
∘
and
A
C
B
=
6
3
∘
ACB = 63^\circ
A
CB
=
6
3
∘
. Point
P
P
P
is chosen on side
B
C
BC
BC
with
B
P
:
P
C
=
2
:
1
BP:PC = 2:1
BP
:
PC
=
2
:
1
. Points
M
,
N
M,N
M
,
N
are chosen on sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively so that
B
M
=
2
BM = 2
BM
=
2
and
C
N
=
1
CN = 1
CN
=
1
. Let
Q
Q
Q
be the point on segment
M
N
MN
MN
for which
M
Q
:
Q
N
=
2
:
1
MQ:QN = 2:1
MQ
:
QN
=
2
:
1
. Find the value of
P
Q
PQ
PQ
. Your answer must be in simplest form.
3
1
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Divisibility of binomials
Let
p
p
p
be a prime, and
n
>
p
n > p
n
>
p
be an integer. Prove that
(
n
+
p
−
1
p
)
−
(
n
p
)
\binom{n+p-1}{p} - \binom{n}{p}
(
p
n
+
p
−
1
)
−
(
p
n
)
is divisible by
n
n
n
.
2
1
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House visitors go to neighbor
Let
n
≥
2
n \ge 2
n
≥
2
be an integer. There are
n
n
n
houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of
n
n
n
(with full justification) for which we can design a town with
n
n
n
houses where every house is visited.
5
1
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AM-GM Jensen Bait
Let
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
≤
1
0 \le a_1 \le a_2 \le \dots \le a_n \le 1
0
≤
a
1
≤
a
2
≤
⋯
≤
a
n
≤
1
be
n
n
n
real numbers with
n
≥
2
n \ge 2
n
≥
2
. Assume
a
1
+
a
2
+
⋯
+
a
n
≥
n
−
1
a_1 + a_2 + \dots + a_n \ge n-1
a
1
+
a
2
+
⋯
+
a
n
≥
n
−
1
. Prove that
a
2
a
3
…
a
n
≥
(
1
−
1
n
)
n
−
1
a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1}
a
2
a
3
…
a
n
≥
(
1
−
n
1
)
n
−
1