ABACUS International Math Challenge
for
7th and 8th graders
September, 2004
C.433. Sam is writing at-least-2-digit
positive whole numbers on a piece of paper. After how many numbers written
on the paper can we be sure that there are three of them that start with
the same digit and also end with an identical digit. (The first and last
digits do not have to be the same.)
C.434. Three girls' birth years are 1986,
1989, and 1992. Their last names are Kiss, Hunter, and Goldbach. Their first
names are Olga, Maria, and Renata. It happened one year that they realized
that everybody is the same age as the number of letters in their own full
names (first + last name). What year did this happen and what are their
names?
C.435. ABC is a 3-digit number in base
10. A, B, and C are positive , and we know that (AxB-C)/(B-C)=0. How many
such 3-digit numbers are there?
C.436. We are coloring the edges of a cube
one-by-one. At the beginning none of the edges is colored, and in every
step we may color only an edge that is not in contact with a colored edge.
How many edges can you color this way the most?
C.437. How many 3-digit numbers are there
in which the number of even digits is even?
C.438. Take all those 9-digit multiples
of 1125 in which every digit is different. How many of them have even digits
in the odd places (1st, 3rd, 5th, 7th, and 9th digits)?
C.439. One train is 100 meters, another
train is 200 meters long. If each of them travels with a constant speed
towards each other, they can pass each other completely in 5 seconds. With
the same speeds, traveling in the same direction, one can pass the other
in 15 seconds. (In this case passing means that it takes the front of the
faster train 15 seconds to get from the back of the slower train to the
front of the slower train.) What are the speeds of the two trains?
C.440. An ant is crawling on the edges
of a cube, starting from one vertex. How many edges can it go through the
most if it can go on every edge only once?