Category Archives: problem

2009-16 Commutative ring

Let k>1 be a fixed integer. Let π be a fixed nonidentity permutation of {1,2,…,k}. Let I be an ideal of a ring R such that for any nonzero element a of R, aI≠0 and Ia≠0 hold.

Prove that if \(a_1 a_2\ldots a_k=a_{\pi(1)} a_{\pi(2)} \ldots a_{\pi(k)}\)  for any elements \(a_1, a_2,\ldots,a_k \in I\), then R is commutative.

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2009-14 New notion on the convexity

Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.

Prove that  a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.

(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)

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2009-12 Colorful sum

Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.

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2009-9 min or max

Suppose that * is an associative and commutative binary operation on the set of rational numbers such that 

  1. 0*0=0
  2. (a+c)*(b+c)=(a*b)+c for all rational numbers a,b,c.

Prove that either

  1. a*b=max(a,b) for all rational numbers a,b, or
  2. a*b=min(a,b) for all rational number a,b.
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