ABACUS International Math Challenge
for
5th and 6th graders
September, 1998
B.65. How much is:
B.66. Break up 45 into the sum of four
numbers so that if you added 2 to one of the numbers, subtracted 2 from
the next, multiplied the third one by 2, and divided the fourth number by
2, you would always get the same number.
B.67. Prove that 1998+2(1+2+3+...+1997)
is a square number.
B.68. Is there such a whole number that
ends with a 3, and if you delete that digit 3 at the end and write it in
front of the number, this new number when divided by 3 will give you the
original number?
B.69. There were 5 competitors on a competition:
A, B, C, D, and E. Somebody guessed the final order of the competitors to
be ABCDE, but as it turned out, none of the competitors finished on these
places, and the person did not even guess the order of any two consecutive
competitors right. Somebody else guessed the order to be DAECB. This guess
turned out to be much better because here exactly two competitors' places
matched the real results and in exactly two cases were the order of two
consecutive competitors right. What could the final order of the competitors
be?
B.70. One of A, B, C, and D broke the window.
When questioned they gave the following answers:
A: "It was C."
B: "It was not me."
C: "It was D."
D: "C is lying."
Who did it if:
a) exactly one person is telling the truth?
b) exactly one person is lying?
B.71. In a group of people the ratio of
the number of women to the number of men is 11:9. The average age of the
women is 22, of the men is 32. What is the average age of the group?
B.72. The first member of a number sequence
is 2, the second is 3, and, from the second member, every member is one
less then the product of its neighbors. What is the sum of the first 1110
members of this sequence?
Please, send your solutions to: