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National and Regional Contests
USA Contests
MAA AMC
USAMO
2010 USAMO
2010 USAMO
Part of
USAMO
Subcontests
(4)
3
1
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Super Product: 2010 USAMO #3
The 2010 positive numbers
a
1
,
a
2
,
…
,
a
2010
a_1, a_2, \ldots , a_{2010}
a
1
,
a
2
,
…
,
a
2010
satisfy the inequality
a
i
a
j
≤
i
+
j
a_ia_j \le i+j
a
i
a
j
≤
i
+
j
for all distinct indices
i
,
j
i, j
i
,
j
. Determine, with proof, the largest possible value of the product
a
1
a
2
…
a
2010
a_1a_2\ldots a_{2010}
a
1
a
2
…
a
2010
.
6
1
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Blackboard erasures: 2010 USAMO #6
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer
k
k
k
at most one of the pairs
(
k
,
k
)
(k, k)
(
k
,
k
)
and
(
−
k
,
−
k
)
(-k, -k)
(
−
k
,
−
k
)
is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number
N
N
N
of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
5
1
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2010 USAMO #5 - Egyptian Fractions and Odd Primes
Let
q
=
3
p
−
5
2
q = \frac{3p-5}{2}
q
=
2
3
p
−
5
where
p
p
p
is an odd prime, and let
S
q
=
1
2
⋅
3
⋅
4
+
1
5
⋅
6
⋅
7
+
⋯
+
1
q
(
q
+
1
)
(
q
+
2
)
S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}
S
q
=
2
⋅
3
⋅
4
1
+
5
⋅
6
⋅
7
1
+
⋯
+
q
(
q
+
1
)
(
q
+
2
)
1
Prove that if
1
p
−
2
S
q
=
m
n
\frac{1}{p}-2S_q = \frac{m}{n}
p
1
−
2
S
q
=
n
m
for integers
m
m
m
and
n
n
n
, then
m
−
n
m - n
m
−
n
is divisible by
p
p
p
.
2
1
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'10 USAMO #2: Students in a Circle
There are
n
n
n
students standing in a circle, one behind the other. The students have heights
h
1
<
h
2
<
⋯
<
h
n
h_1<h_2<\dots <h_n
h
1
<
h
2
<
⋯
<
h
n
. If a student with height
h
k
h_k
h
k
is standing directly behind a student with height
h
k
−
2
h_{k-2}
h
k
−
2
or less, the two students are permitted to switch places. Prove that it is not possible to make more than
(
n
3
)
\binom{n}{3}
(
3
n
)
such switches before reaching a position in which no further switches are possible.