ABACUS International Math Challenge
for
7th and 8th graders
October, 2008
C.649. Is there an "n" positive
whole number such that:
C.650. Out of the numbers divisible by
3, we picked four consecutive numbers. To the sum of these, we added a third
of the sum, and then the half of the third of the sum. We got 225. What
were those four numbers?
C.651. Two students are talking after school:
- What is the average of your grades in Math for September?
- 4.6
- That is impossible since the year just started. You could not possibly
have that many grades.
What was the doubting student thinking of if the possible grades teachers
give out in this school are 1, 2, 3, 4, and 5; 5 being the best grade?
C.652. Sam and his father are talking:
Son, how many pages are there in the book you have to read for
school?
Less than 1000.
Have you started reading it yet?
Yes, on Sunday I already passed page 150.
And how many pages did you read today?
Very many! The sum of the page numbers of the pages I read today
is 2761.
What pages did Sam read on the day of this conversation?
C.653. Is there such an at-least-2-digit
square number, which has all different digits, and if you change the order
of its digits you always get a square number?
C.654. Grandma Olga is baking a cake in
a 24 x 42 cm rectangular baking dish. Up until now grandma always cut her
cakes in the baking dish by 5 straight cuts parallel to the shorter side
of the dish, and 3 cuts parallel to the other side of the dish to get 24
equal pieces. Her handy grandson, however, made an L shape knife for her
with one 6 cm long blade and another 7 cm long blade which is perpendicular
to the shorter one. She just has to push this knife into the cake from the
top to make a cut. At least how many time does grandma have to push this
knife into the cake to make her usual 24 equal pieces?
C.655. Are there eleven consecutive positive
whole numbers whose sum is prime? If yes, find all of them; if no, prove
it!
C.656. In a class 85% of the students received
the highest grade in English Literature, 80% in Math, 75% in History, and
30% in Chemistry. About what percent of the students could you say for sure
that they received the highest grade in all four of these subjects?