1994年IMO中国国家队选拔考试试题

1994

Day 11Find all sets comprising of 4natural numbers such that the product of any 3numbers in the set leaves a remainder of 1when divided by the remaining number.2An n by n grid,where every square contains a number,is called an n -code if the numbers in every row and column form an arithmetic progression.If it is su?cient to know the numbers in certain squares of an n -code to obtain the numbers in the entire grid,call these squares a key.

a.)Find the smallest s ∈N such that any s squares in an n ?code (n ≥4)form a key.

b.)Find the smallest t ∈N such that any t squares along the diagonals of an n -code (n ≥4)form a key.3Find the smallest n ∈N such that if any 5vertices of a regular n -gon are colored red,there exists a line of symmetry l of the n -gon such that every red point is re?ected across l to a non-red point.https://www.360docs.net/doc/8315080589.html,/This ?le was downloaded from the AoPS ?MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/

1994

Day 21Given 5n real numbers r i ,s i ,t i ,u i ,v i ≥1(1≤i ≤n ),let R =1n n i =1r i ,S =1n n

i =1s i ,T =1n n i =1t i ,U =1n n i =1u i ,V =1n n i =1v i .Prove that n i =1r i s i t i u i v i +1r i s i t i u i v i ?1≥ RST UV +1RST UV ?1 n .2Given distinct prime numbers p and q and a natural number n ≥3,?nd all a ∈Z such that the polynomial f (x )=x n +ax n ?1+pq can be factored into 2integral polynomials of degree at least 1.3For any 2convex polygons S and T ,if all the vertices of S are vertices of T ,call S a sub-polygon of T .

I.Prove that for an odd number n ≥5,there exists m sub-polygons of a convex n -gon such that they do not share any edges,and every edge and diagonal of the n -gon are edges of the m sub-polygons.

II.Find the smallest possible value of m .https://www.360docs.net/doc/8315080589.html,/This ?le was downloaded from the AoPS ?MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/