International Youth Mathematics Contest 2007 HEMIC
Individual Competition
Instructions:
- Write down your name, team name and candidate number on the answer sheet.
- Write down all answers on the answer sheet.
- Answer all 15 problems. Problems are in ascending order of level of difficulty. Only NUMERICAL answers are needed.
- Each problem is worth 6 points and the total is 90 points.
- or problems involving more than one answer, points are given only when ALL answers are correct.
- Take ? = 3.14 if necessary.
- No calculator or calculating device is allowed.
- Answer the problems with pencil, blue or black ball pen.
- All materials will be collected at the end of the competition.
1. The product of two three-digit numbers abc and cba is 396396, where a > c. Find the value of abc!
2. In a right-angled triangle ACD, the area of shaded region is 10 cm^2, as shown in the figure below. AD = 5 cm, AB = BC, DE = EC. Find the length of AB!
INTERNATIONL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2006
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 = 23 , 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.
Note: ^ means square
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