USA
USAMO
2006
Day 1
1 Let p be a prime number and let s be an integer with 0 < s < p. Prove that there exist
integers m and n with 0 < m < n < p and{
sm
p
}
<
{
sn
p
}
<
s
p
if and only if s is not a divisor of p− 1.
Note: For x a real number, let bxc denote the greatest integer less than or equal to x, and let
{x} = x− bxc denote the fractional part of x.
2 For a given positive integer k find, in terms of k, the minimum value of N for which there is
a set of 2k+1 distinct positive integers that has sum greater than N but every subset of size
k has sum at most N2 .
3 For integral m, let p(m) be the greatest prime divisor of m. By convention, we set p(±1) = 1
and p(0) =∞. Find all polynomials f with integer coefficients such that the sequence
{p (f (n2))− 2n}n≥0
is bounded above. (In particular, this requires f
(
n2
) 6= 0 for n ≥ 0.)
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
USA
USAMO
2006
Day 2
1 Find all positive integers n such that there are k ≥ 2 positive rational numbers a1, a2, . . . , ak
satisfying a1 + a2 + . . .+ ak = a1 · a2 · · · ak = n.
2 A mathematical frog jumps along the number line. The frog starts at 1, and jumps according
to the following rule: if the frog is at integer n, then it can jump either to n+1 or to n+2mn+1
where 2mn is the largest power of 2 that is a factor of n. Show that if k ≥ 2 is a positive
integer and i is a nonnegative integer, then the minimum number of jumps needed to reach
2ik is greater than the minimum number of jumps needed to reach 2i.
3 Let ABCD be a quadrilateral, and let E and F be points on sides AD and BC, respectively,
such that AEED =
BF
FC . Ray FE meets rays BA and CD at S and T, respectively. Prove that
the circumcircles of triangles SAE, SBF, TCF, and TDE pass through a common point.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
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