# 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.

3. Is it possible to find a number 11…11 that is divisible by 19?

4. The menu in the school cafeteria never changes. It consists of 10 different dishes. Peter decides to make his school lunch different everyday (at least 1 dish). For each lunch, he may eat any number of dished, but no two are identical.
(a) What is the maximum numbers of days Peter can do so?
(b) What is the total number of dishes Peter has consumed during this period?

5. A sequence of shapes is made as follows.

(1) Shape S1 is a shaded square of side 1 unit.
(2) Shape S2 is made by dividing S1 into 9 equal squares and removing four
of these, so that only the central and corner squares remain.
(3) Shape S3 is made by applying the process in (2) to each of the squares of
S2.
(4) Shape S4 is made by applying the process in (2) to each of the squares of
S3. And so on.

(a) Find the area and perimeter of shape S3, giving your answers as fractions.
(b) Find the least value of k for which the shape Sk has the area less than 1/30
and also has perimeter greater than 30.