Kaohsiung First Invitational World Youth Mathematics Intercity Competition
Individual Contest
Time allowed: 2 hours
Part I Write your answer to each of the twelve questions on the answer sheet. Working need not be shown. Each question carries 5 marks.
1. Find the remainder when the number
122333444455555666666777777788888888999999999
is divided by 9.
2. Find the sum of a, b, c and d in the following figure.
b_ c_
a_
d_
3. Among the numbers 12, 22, ..., 19992, how many of them have odd numbers as their tens-digits?
4. The height of a building is 60 m. At a certain moment during daytime, it casts a shadow of length 40 m. If a vertical pole of length
2 m is erected on the roof of the building, find the length of the shadow of the pole at the same moment.
5. Calculate 19992 -19982 + 19972 - 19962 + ... + 32 - 22 + 12.
7. The following figure shows an equilateral triangle of side 1. The three circles touch each other and the sides of the triangle. Find the radius of the circle.
8. Let A, B and C be positive integers. The sum of 160 and the square of A is equal to the sum of 5 and the square of B. The sum of 320 and the square of A is equal to the sum of 5 and the square of C. Find A.
9. Let x be a two-digit number. Denote by f(x) the sum of x and its digits minus the product of its digits; for example, if x = 32, then f(x) = 32 + 3 + 2 – 3 ∞ 2 = 31. Find the value of x which gives the largest possible value for f(x).
10. The figure below shows a triangle ABC. The perpendicular sides AB and AC have lengths 15 and 8 respectively. D and F are points on AB. E and G are points on AC. The segments CD, DE, EF and FG divide triangle ABC into five triangles of equal areas. The length of only one of these segments is integral. What is this length?
A
B
C G
F
E
11. How many squares (excluding those with the shaded part inside) are there in the following figure?
12. There are two committees A and B. Committee A had 13 members while committee B had 6 members. Each member is paid $6000 per day for attending the first 30 days of meetings, and $9000 per day thereafter. Committee B met twice as many days as Committee A, and the
expenditure on attendance were the same for the two committees. If the total expenditure on attendance for these two committees was over $3000000, how much was it?
Part II Show your steps for each the three questions in the spaces
provided on the answer sheet. Each question carries 20 marks. Partial solutions are allowed.
1. The figure below shows a cubical wire framework of side 1. An ant starts from A to crawl along the sides of the framework. If it does not repeat any part of its path and returns to A finally, what is the longest possible length of path it has travelled? Justify your claim.
2. In the following figure, AC = 8CE and BC = 4BD. If AD = 164 and BE = 52, find AB.
C B
D
A E
3. When a particular six-digit number is multiplied by 2, 3, 4, 5, 6
