ABACUS International Math Challenge
for
7th and 8th graders
November, 2007
C.601. Find the second smallest positive
whole number that when multiplied by 252 gives you a cube number.
C.602. Find a 4-digit positive whole number
that is 9 times as great as the number you get when you reverse the order
of its digits.
C.603. How many digits of 1 and how many
digits of zero are there in the final result of the following additions:
9 + 99 + 999 + 9999 + ... + 999...999
(There are 100 digits of 9 in the last addend.)
C.604. A solid has 6 octagons, 8 hexagons
and 12 squares on its surface. Every vertex has 3 edges running from it.
How many vertices does this solid have?
C.605. Some whole numbers have the same
value even if you read their digits in a reverse order. How many such numbers
are there between 10,000 and 12,000?
C.606. Newton and Gregory were arguing
about the following problem: If you have a sphere, how many spheres of the
same radius can you place around it so that all of them would touch the
original sphere? The argument was decided only 180 years later. As it turned
out in 1874, Newton was right. What do you think, without proving it, what
is the maximum number of such spheres?
C.607. A company decided that it will raise
the price of a $3.00 item by 3 cents. When I learned about it, I immediately
bought a few of these items for the old price because I figured out that
I could have bought 5 less of these items on the new price for the same
money. How many of these items did I buy?
C.608. In the XVII century Lord de Mere,
a vivid gambler, asked Pascal, one of the great mathematicians of his time:
If a player rolls a die 4 times what should I bet on: a) he rolls at least
one 6; or b) he does not roll a 6 at all? What would you bet on?