ABACUS International Math Challenge
for
5th and 6th graders
March, 2008
B.633. Sam and Kate each have a little
garden. Sam's is a square garden with an area of 2304 sq.dm, Kate's is a
rectangular garden with an area of 2279 sq.dm. The sides of each garden
measured in decimeters are whole numbers. What are the dimensions of the
gardens if they have an equal perimeter?
B.634. Hannah and Barbara both have nice
calculators but none of them allow the girls to calculate 1x2x3x4x5x6x7x8x9x10x11x12x13x14,
so they have to use paper and pencil. Hannah got 87178290120, and Barbara
got 87178290200. Can you decide who is wrong without calculating the product?
B.635. Using metal wires, Tom wants to
make all those rectangular solids with prime edges (measured in centimeters),
and with a volume no more than 30 cubic cm. How long of a wire does he need?
B.636. a, b, c, and d are positive whole
numbers. We know that ab=10, cd=11 and a-b+c-d=13. What is a+b+c+d?
B.637. We know the following about two
fractions:
a) The value of each is 1/2.
b) In one fraction the numerator and the denominator are both 2-digit
numbers, and in the other fraction the numerator and the denominator are
both 3-digit numbers.
c) In the numerators and the denominators of the two fractions together
you can find every digit once.
How many such pairs of fractions are there?
B.638. E.T. wanted to call home. It happened
back in the days of the impulse telephones. (When you pushed a number, the
telephone sent out that many short electric signals. For example: pressing
1 sent out one short signal, pressing 2 sent out two short signal, ... pressing
9 sent out nine signals, and pressing 0 sent out ten short signals.) E.T.
pressed a (not necessarily different) number with all three of his fingers.
How many different numbers could E.T. dial if the telephone sent out a total
of 10 shirt electric signals?
B.639. We are making 3.5 mm big golden
rings from a 0.5 mm diameter golden wire to be made into a chain. How many
rings do we need for a 50 cm long chain?
B.640. A 1-meter wide concrete stairway
at the terrace has two 60 cm high and 60 cm deep steps. We would like to
have 6 smaller steps instead, so that the steps are still equally high and
equally deep. We do not want anything demolished, we just want to add stairs
made of concrete only. How many cubic meters of concrete do we need the
least?
now: 
new: 