USA
USAMO
1997
Day 1 - 01 May 1997
1 Let p1, p2, p3, . . . be the prime numbers listed in increasing order, and let x0 be a real number
between 0 and 1. For positive integer k, define
xk =
0 if xk−1 = 0,{
pk
xk−1
}
if xk−1 6= 0,
where {x} denotes the fractional part of x. (The fractional part of x is given by x − bxc
where bxc is the greatest integer less than or equal to x.) Find, with proof, all x0 satisfying
0 < x0 < 1 for which the sequence x0, x1, x2, . . . eventually becomes 0.
2 Let ABC be a triangle. Take points D, E, F on the perpendicular bisectors of BC, CA, AB
respectively. Show that the lines through A, B, C perpendicular to EF , FD, DE respectively
are concurrent. [hide=”Remark”]So, I got it down to where the perpendiculars are altitudes
of triangle ABC, but is it okay to just say, ”Since the they are altitudes of triagnle ABC,
they are concurrent at the orthocenter of ABC”? or do I have to prove that the altitudes are
concurrent?
3 Prove that for any integer n, there exists a unique polynomialQ with coefficients in {0, 1, . . . , 9}
such that Q(−2) = Q(−5) = n.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS Math Olympiad Resources Page Page 1
USA
USAMO
1997
Day 2 - 02 May 1997
4 To clip a convex n-gon means to choose a pair of consecutive sides AB,BC and to replace
them by the three segments AM,MN , and NC, where M is the midpoint of AB and N
is the midpoint of BC. In other words, one cuts off the triangle MBN to obtain a convex
(n+ 1)-gon. A regular hexagon P6 of area 1 is clipped to obtain a heptagon P7. Then P7 is
clipped (in one of the seven possible ways) to obtain an octagon P8, and so on. Prove that
no matter how the clippings are done, the area of Pn is greater than 13 , for all n ≥ 6.
5 Prove that, for all positive real numbers a, b, c, the inequality
1
a3 + b3 + abc
+
1
b3 + c3 + abc
+
1
c3 + a3 + abc
≤ 1
abc
holds.
Alternative formulation: If a, b, c are positive reals, then prove that∑ 1
a3+b3+abc
≤ 1abc .
Hereby, the
∑
sign stands for cyclic summation over the letters a, b, c.
6 Suppose the sequence of nonnegative integers a1, a2, . . . , a1997 satisfies
ai + aj ≤ ai+j ≤ ai + aj + 1
for all i, j ≥ 1 with i+ j ≤ 1997. Show that there exists a real number x such that an = bnxc
(the greatest integer ≤ nx) for all 1 ≤ n ≤ 1997.
http://www.artofproblemsolving.com/
This file was downloaded from the AoPS Math Olympiad Resources Page Page 2
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