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Invitational World Youth Mathematics Intercity
Competition 1999
Individual Contest
Section A.
In this section, there are 12 questions. Fill in the correct answer in the space provided at the
end of each question. Each correct answer is worth 5 points.
1
. Find the remainder when 122333444455555666666777777788888888999999999 is divided by
9
.
2
. Find the sum of the angles a, b, c and d in the following figure.
a
c
b
d
2 2
2
. How many of the numbers 1 , 2 , . . . , 1999 have odd numbers as their tens-digits?
3
4
.
The height of a building is 60 metres. At a certain moment during daytime, it casts a shadow of
length 40 metres. If a vertical pole of length 2 metres is erected on the roof of the building, find
the length of the shadow of the pole at the same moment.
2 2 2 2 2 2 2
. Calculate 1999 −1998 +1997 −1996 +⋯+ 3 − 2 +1 .
5
6
. Among all four-digits numbers with 3 as their thousands-digits, how many have exactly two
identical digits?
7
. The diagram below shows an equilateral triangle of side 1. The three circles touch each other
and the sides of the triangle. Find the radii of the circles.
8
9
. Let
the square of
Find
a
,
b
and
c
b
be positive integers. The sum of 160 and the square of a is equal the sum of 5 and
. The sum of 320 and the square of is equal to the sum of 5 and the square of
a
b.
a
.
. Let x be a two-digit number. Denote by f (x) the sum of x and its digits minus the product of
its digits. Find the value of x which gives the largest possible value for f (x) .
