MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
USAMO
2006 USAMO
2006 USAMO
Part of
USAMO
Subcontests
(6)
6
1
Hide problems
Darij is going to solve Feng's Geometry Problem - Pwn it!
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral, and let
E
E
E
and
F
F
F
be points on sides
A
D
AD
A
D
and
B
C
BC
BC
, respectively, such that
A
E
E
D
=
B
F
F
C
\frac{AE}{ED} = \frac{BF}{FC}
E
D
A
E
=
FC
BF
. Ray
F
E
FE
FE
meets rays
B
A
BA
B
A
and
C
D
CD
C
D
at
S
S
S
and
T
T
T
, respectively. Prove that the circumcircles of triangles
S
A
E
SAE
S
A
E
,
S
B
F
SBF
SBF
,
T
C
F
TCF
TCF
, and
T
D
E
TDE
T
D
E
pass through a common point.
5
1
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Frog jump analysis
A mathematical frog jumps along the number line. The frog starts at
1
1
1
, and jumps according to the following rule: if the frog is at integer
n
n
n
, then it can jump either to
n
+
1
n+1
n
+
1
or to
n
+
2
m
n
+
1
n + 2^{m_n+1}
n
+
2
m
n
+
1
where
2
m
n
2^{m_n}
2
m
n
is the largest power of
2
2
2
that is a factor of
n
.
n.
n
.
Show that if
k
≥
2
k \geq 2
k
≥
2
is a positive integer and
i
i
i
is a nonnegative integer, then the minimum number of jumps needed to reach
2
i
k
2^ik
2
i
k
is greater than the minimum number of jumps needed to reach
2
i
.
2^i.
2
i
.
4
1
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USAMO sum equals product - original problems please
Find all positive integers
n
n
n
such that there are
k
≥
2
k \geq 2
k
≥
2
positive rational numbers
a
1
,
a
2
,
…
,
a
k
a_1, a_2, \ldots, a_k
a
1
,
a
2
,
…
,
a
k
satisfying
a
1
+
a
2
+
…
+
a
k
=
a
1
⋅
a
2
⋯
a
k
=
n
.
a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.
a
1
+
a
2
+
…
+
a
k
=
a
1
⋅
a
2
⋯
a
k
=
n
.
3
1
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IMO's Hojoo Lee vs. USAMO's Harazi
For integral
m
m
m
, let
p
(
m
)
p(m)
p
(
m
)
be the greatest prime divisor of
m
.
m.
m
.
By convention, we set
p
(
±
1
)
=
1
p(\pm 1) = 1
p
(
±
1
)
=
1
and
p
(
0
)
=
∞
.
p(0) = \infty.
p
(
0
)
=
∞.
Find all polynomials
f
f
f
with integer coefficients such that the sequence
{
p
(
f
(
n
2
)
)
−
2
n
}
n
≥
0
\{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0}
{
p
(
f
(
n
2
)
)
−
2
n
}
n
≥
0
is bounded above. (In particular, this requires
f
(
n
2
)
≠
0
f \left (n^2 \right ) \neq 0
f
(
n
2
)
=
0
for
n
≥
0.
n \geq 0.
n
≥
0.
)
2
1
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Every subset of size k has sum at most N/2
For a given positive integer
k
k
k
find, in terms of
k
k
k
, the minimum value of
N
N
N
for which there is a set of
2
k
+
1
2k + 1
2
k
+
1
distinct positive integers that has sum greater than
N
N
N
but every subset of size
k
k
k
has sum at most
N
2
.
\tfrac{N}{2}.
2
N
.
1
1
Hide problems
USAMO Inequality chain for getting started on the contest
Let
p
p
p
be a prime number and let
s
s
s
be an integer with
0
<
s
<
p
.
0 < s < p.
0
<
s
<
p
.
Prove that there exist integers
m
m
m
and
n
n
n
with
0
<
m
<
n
<
p
0 < m < n < p
0
<
m
<
n
<
p
and
{
s
m
p
}
<
{
s
n
p
}
<
s
p
\left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p}
{
p
s
m
}
<
{
p
s
n
}
<
p
s
if and only if
s
s
s
is not a divisor of
p
−
1
p-1
p
−
1
. Note: For
x
x
x
a real number, let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denote the greatest integer less than or equal to
x
x
x
, and let
{
x
}
=
x
−
⌊
x
⌋
\{x\} = x - \lfloor x \rfloor
{
x
}
=
x
−
⌊
x
⌋
denote the fractional part of x.