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Contests
National and Regional Contests
USA Contests
MAA AMC
USAMO
1975 USAMO
1975 USAMO
Part of
USAMO
Subcontests
(5)
5
1
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Expected Cards Until Second Ace
A deck of
n
n
n
playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is (n\plus{}1)/2.
4
1
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Maximizing AP*PB
Two given circles intersect in two points
P
P
P
and
Q
Q
Q
. Show how to construct a segment
A
B
AB
A
B
passing through
P
P
P
and terminating on the circles such that
A
P
⋅
P
B
AP \cdot PB
A
P
⋅
PB
is a maximum.
3
1
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Polynomial of Degree n
If
P
(
x
)
P(x)
P
(
x
)
denotes a polynomial of degree
n
n
n
such that P(k)\equal{}\frac{k}{k\plus{}1} for k\equal{}0,1,2,\ldots,n, determine P(n\plus{}1).
2
1
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Four Points in Space
Let
A
,
B
,
C
,
A,B,C,
A
,
B
,
C
,
and
D
D
D
denote four points in space and
A
B
AB
A
B
the distance between
A
A
A
and
B
B
B
, and so on. Show that AC^2\plus{}BD^2\plus{}AD^2\plus{}BC^2 \ge AB^2\plus{}CD^2.
1
1
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Prove Fraction is Integer
(a) Prove that [5x]\plus{}[5y] \ge [3x\plus{}y] \plus{} [3y\plus{}x], where
x
,
y
≥
0
x,y \ge 0
x
,
y
≥
0
and
denotes the greatest integer
≤
u
\le u
≤
u
(e.g., [\sqrt{2}]\equal{}1). (b) Using (a) or otherwise, prove that \frac{(5m)!(5n)!}{m!n!(3m\plus{}n)!(3n\plus{}m)!} is integral for all positive integral
m
m
m
and
n
n
n
.