MathDB

3

Part of 1992 China National Olympiad

Problems(2)

China Mathematical Olympiad 1992 problem3

Source: China Mathematical Olympiad 1992 problem3

9/30/2013
Given a 9×99\times 9 grid, we assign either +1+1 or −1-1 to each square on the grid. We define an adjustment as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have 8181 products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to 11 through finite adjustments.
combinatorics unsolvedcombinatorics
China Mathematical Olympiad 1992 problem6

Source: China Mathematical Olympiad 1992 problem6

9/30/2013
Let sequence {a1,a2,… }\{a_1,a_2,\dots \} with integer terms satisfy the following conditions: 1) an+1=3an−3an−1+an−2,n=2,3,…a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots ; 2) 2a1=a0+a2−22a_1=a_0+a_2-2 ; 3) for arbitrary natural number mm, there exist mm consecutive terms ak,ak−1,…,ak+m−1a_k, a_{k-1}, \dots ,a_{k+m-1} among the sequence such that all such mm terms are perfect squares. Prove that all terms of the sequence {a1,a2,… }\{a_1,a_2,\dots \} are perfect squares.
number theory unsolvednumber theory