APMO

Question

For a positive integer m denote by S(m) and P(m) the sum and product, respectively, of the digits of m. Show that for each positive integer n, there exist positive integers a_1, a_2, , a_n satisfying the following conditions: S(a_1)<S(a_2)<<S(a_n) and S(a_i)=P(a_i+1) (i=1,2, , n) . (We let a_n+1=a_1.) (Problem Committee of the Japan Mathematical Olympiad Foundation)

Accepted Answer

Let k be a sufficiently large positive integer. Choose for each i=2,3, , n, a_i to be a positive integer among whose digits the number 2 appears exactly k+i-2 times and the number 1 appears exactly 2^k+i-1-2(k+i-2) times, and nothing else. Then, we have S(a_i)=2^k+i-1 and P(a_i)=2^k+i-2 for each i, 2 i n. Then, we let a_1 be a positive integer among whose digits the number 2 appears exactly k+n-1 times and the number 1 appears exactly 2^k-2(k+n-1) times, and nothing else. Then, we see that a_1 satisfies S(a_1)=2^k and P(a_1)=2^k+n-1. Such a choice of a_1 is possible if we take k to be large enough to satisfy 2^k>2(k+n-1) and we see that the numbers a_1, , a_n chosen this way satisfy the given requirements.