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# INTERNATIONL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2006

INTERNATIONL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2006. Mathematics Contest in Taiwan, Exploration Problems.

Answer the following 5 questions, and show your detailed solution in the answer sheet. Write down the question number in each paper. Each question is worth 8 points. Time limit: 60 minutes.

1. The solution to each clue of this crossnumber is a two-digit number. None of

these numbers begins with zero. Complete the crossnumber, stating the order in which you solved the clues and explaining why there is only one solution.

Clues Across

1. A square number

3. A multiple of 11

Clues Down

1. A multiple of 7

2. A cube number

2. Notice that 2^2 + 2^2 = 2^3 , so two squares can sum to give a cube; however, the two squares here are equal (to 4).

(a) Find two unequal squares whose sum is a cube.

(b) Show that there are infinitely many pairs of unequal squares whose sum is

equal to a cube.