ABACUS International Math Challenge
for
5th and 6th graders
November, 2004
B.449. One year two deadlines of a school's
monthly math challenge fell on a Monday in October and in November. The
date of the day in November is 5 times the date of the day in October. What
days were these two deadlines?
B.450. There are a total of 2004 red, blue,
white and yellow balls in a box. We pick a few of them randomly. In order
to make sure that we have at least one ball from each color, we have to
pick at least 1661 balls. How many of the same color balls could there be
in the box the most?
B.451. Each of two explorers could carry
with him enough food for 12 days, and can walk 30 km a day. 300 km inwards
from the edge of the desert, where they are, is a spot where they want to
post a flag. Can they do it so that both could get back to the edge of the
desert alive?
B.452. There is a 1530 cm long ribbon.
We want to cut it up into 30 cm and 24 cm pieces, so that we would not waste
any of the ribbon, but have at least one of each size of pieces. How many
of each size should we cut?
B.453. How many 3-digit numbers have at
least 2 identical digits?
B.454. You can see the crossection of a
vase on the diagram. What is its area if the side of a little square on
the diagram is 2 cm? (Each curved line is a half or a quarter of a circle.)
B.455. Starting from the origin, a spider
is crawling on the lines of a (vertical-horizontal) square grid. What we
can see from the outside is that first it goes to the right by 1 unit, then
it goes up 2 units, then it goes left 3 units, then it goes down 4 units,
then it goes right 5 units, up 6 units, left 7 units down 8 units, and so
on. After every left turn, it goes one more unit in that direction than
it went in the previous direction. Where is the spider after crawling the
2004th section?
B.456. Is it true that if we pick any 30
numbers out of the first 50 positive whole numbers then among the 30 numbers
there will always be two numbers such that one of them is twice the other?
Please, send your solutions to: