ABACUS International Math Challenge
for
5th and 6th graders
February, 2000
B.169. What is the value of the following
sum? 1/2+1/3+1/4+1/5+1/6+1/7+2/3+2/4+2/5+2/6+2/7+3/4+3/5+3/6+3/7+4/5+4/6+4/7+5/6+5/7+6/7
B.170. Find the smallest 3-digit number
from which you cannot create a prime number by changing one of its digits.
B.171. How many different ways can you
get 2000 as the sum of positive whole numbers? (You may use the same number
more than once in the same sum.)
B.172. Can you write the numbers 0, 1,
2, 3, ..., 8, 9 on the circumference of a circle so that the sum of any
three consecutive numbers is less than 16 but more than 11?
B.173. Microland has a population of 100
people. Some of them are liars (who always lie), and the others are true
(who always tell the truth). There are three groups on the island: Sun-admirers,
Moon-admirers, and Earth-admirers. Everybody on the island belongs to exactly
one group. In a survey everybody had to answer all three of the following
questions:
1) Are you a Sun-admirer?
2) Are you a Moon-admirer?
3) Are you an Earth-admirer?
The number of "Yes" answers for the first question was 60,
for the second question it was 40, and for the third question it was 30.
How many liars live on this island?
B.174. Out of Morse signals (. and _) you
want to make 2000 different signal-series using as little total number of
signals as possible. How many signals will the longest series have?
by Bognár Ferencné, Hungary
B.175. Let's call a positive whole number
"lucky" if its digits can be divided into two groups so that the
sum of the digits in each group is the same. For example 34175 is LUCKY
because 3+7=4+1+5. Find the smallest 5-digit lucky number that has a lucky
neighbor.
B.176. Nobody knows whether the equation
x/y+y/z+z/x=4 has any positive whole number solutions or not. Find such
positive whole numbers for which the value of x/y+y/z+z/x is close to 4.
Please, send your solutions to: