The 23rd United States of America Mathematics Olympiad
1994年第23届美国数学奥林匹克 
 a_{1}, a_{2}, a_{3}, ... are positive integers such that a_{n} > a_{n1} + 1. Put b_{n} = a_{1} + a_{2} + ... + a_{n}. Show that there is always a square in the range b_{n}, b_{n}+1, b_{n}+2, ... , b_{n+1}1 .
 The sequence a_{1}, a_{2}, ... , a_{99} has a_{1} = a_{3} = a_{5} = ... = a_{97} = 1, a_{2} = a_{4} = a_{6} = ... = a_{98} = 2, and a_{99} = 3. We interpret subscripts greater than 99 by subtracting 99, so that a_{100} means a_{1} etc. An allowed move is to change the value of any one of the a_{n} to another member of {1, 2, 3} different from its two neighbors, a_{n1} and a_{n+1}. Is there a sequence of allowed moves which results in a_{m} = a_{m+2} = ... = a_{m+96} = 1, a_{m+1} = a_{m+3} = ... = a_{m+95} = 2, a_{m+97} = 3, a_{n+98} = 2 for some m ? [ So if m = 1, we have just interchanged the values of a_{98} and a_{99} ]
 The hexagon ABCDEF has the following properties: (1) its vertices lie on a circle; (2) AB = CD = EF; and (3) the diagonals AD, BE, CF meet at a point. Let X be the intersection of AD and CE. Show that CX/XE = (AC/CE)^{2} .
 x_{i} is a infinite sequence of positive reals such that for all n, x_{1} + x_{2} + ... + x_{n} ≥ √n. Show that x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2} > (1 + 1/2 + 1/3 + ... + 1/n) / 4 for all n .
 X is a set of n positive integers with sum s and product p. Show for any integer N ≥ s, ∑( parity(Y) (N  sum(Y))Cs ) = p, where aCb is the binomial coefficient a!/(b! (ab)! ), the sum is taken over all subsets Y of X, parity(Y) = 1 if Y is empty or has an even number of elements, 1 if Y has an odd number of elements, and sum(Y) is the sum of the elements in Y .

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