ABACUS International Math Challenge
for
7th and 8th graders
December, 2005
C.513. Find the positive whole number below
2005 with the most divisors.
C.514. Kate and Tom open a joint bank account.
Tom: "We should alternate depositing money into this account. Each
time on our turn each of us has to deposit at least one dollar and no more
than $400 but a whole number of dollars. The person who makes the total
on the account be $2005 with his or her deposit can have all the money."
Kate: "OK, but you make the first deposit."
Did Kate make the right decision? How much profit can the winner make
if the bank pays no interest on the account?
by Zsuzsa Bognár, Hungary
C.515. We wrote each of the positive whole
numbers from 1 to 90 on different pieces of paper and put them in a hat.
Now you take them out one-by-one but do not read what numbers are written
on them. After how many pieces of paper taken out can you say for sure that
there are two numbers among the pieces you have taken out with a difference
that is divisible by 11?
C.516. The measuring numbers of the lengths
of the sides of a rectangle in centimeters, and the that of the perimeter
in decimeters are all whole numbers. One side is 1 cm longer than the other
side. Could the measuring number of one of the sides in centimeter be a
square number?
C.517. A mountain climber left the base
for the peak at 7 am and got there at 2 pm on the same day. He spent the
night at the peak, and next morning at 7 am he started to head back to the
base on the same trail and got back at 12 noon already. Is there a point
on the trail where he was at the same time on both days?
C.518. On TV they continually broadcast
the current results of the counting of the votes of a national election.
In each district the candidate with the most votes wins the election. At
one point they announce that in a particular district with three candidates,
after counting 60% of the votes, 80% of the votes counted went to candidate
A, 15% to candidate B, and 5% to candidate C.
a) Can we say for sure that candidate A won the election?
b) Can we say for sure that candidate C cannot win the election?
c) What % of all the votes should have been counted so that we could
say for sure that with the said rations of the votes counted candidate A
surely won the election?
C.519. Two teams are competing. Eva is
a member of one of the teams, Pete is a member of the other team. Eva realized
that the other team has three times as many boys as girls. Pete noticed
that the ration of boys:girls on the other team is 2:3. There are a total
of 49 students competing. How many students could there be on each team?
C.520. Show that the following fraction
is really a whole number. How many zeros does it end with?
