Subcontests
(6)Sequence
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=⌊nx⌋ (the greatest integer ≤nx) for all 1≤n≤1997. Clipping Polygons
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 31, for all n≥6. Polynomial
Prove that for any integer n, there exists a unique polynomial Q with coefficients in {0,1,…,9} such that Q(−2)=Q(−5)=n. Decreasing primes
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
x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \$$.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases}
where {x} denotes the fractional part of x. (The fractional part of x is given by x−⌊x⌋ where ⌊x⌋ 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.