ABACUS International Math Challenge
for
7th and 8th graders
November, 1998
C.81. How many six-digit cube numbers are there?
C.82. The sum of two whole numbers (both
greater than 10) is 1000. Prove that the last three digits of the squares
of these two numbers are identical?
C.83. Stick another digit into the six-digit
number 975 312 (even in front or at the end) so that the new number would
be divisible by 468.
C.84. How big could the greatest prime-divisor
of the ababab type 6-digit numbers in base 10 be?
C.85. Find the greatest whole number which
is not the sum of 100 composite numbers.
C.86. A computer responds to the following
six commands:
1) Let the starting value of X be 3, and the starting value of S be
0.
2) Increase the value of X by 2.
3) Increase The value of S by X.
4) If S is at least 10 000, then complete the 5th command, otherwise
go back to the 2nd command.
5) Print the value of X.
6) Stop
What will the computer print out?
C.87. How many 10-digit numbers are there
in which only the digits 2 and 5 appear, and there are no digit 2's next
to each other?
C.88. Is it possible to write the first
six positive whole numbers on the perimeter of a circle so that for any
three a, b, c consecutive numbers on the circle, 
is divisible
by 7?
Please, send your solutions to: