ABACUS International Math Challenge
for
5th and 6th graders
February, 2002
B.297. An old clock chimes incorrectly.
It chimes one more times than it did an hour ago at every full hour, but
if it chimed 11 times an hour ago then it chimes only once now. On Monday
night at 11 pm it chimed 11 times. Which day and at what time will it chime
accurately again next time?
B.298. Anne and Bea are playing a game
with a 3x4 square-sheet of chocolate. They are alternating in breaking one
piece off of the sheet of chocolate along the grid-lines. They agreed that
whoever breaks off a 1x1 piece first wins the whole chocolate. Anne starts
and they both play their best. Does Anne have a chance to win the chocolate?
Would Anne have a chance to win if the rule was that the first person breaking
off a 1x1 piece loses the whole chocolate?
B.299. Peter takes a bus to school every
morning. From one line the buses come to this stop from 6:05am every 15
minutes, from the other line the buses come to this stop from 6:10am every
20 minutes. Peter arrives to the bus stop between 7:00 - 7:30am. What can
we expect, which will be more frequent: that Peter has to wait for more
than 4 minutes, or that Peter has to wait for no more than 4 minutes for
a bus to arrive? (Let's suppose that the buses are always running on schedule.)
B.300. The side of a square-shaped little
garden is 18 meters long. We would like to break it up to 3 sections of
equal areas, so we already built a straight fence from the center of the
garden to one of the corners. We want to build two more straight fences
from the center of the garden. Where should they be?
B.301.
a) The sum of 2001 positive whole numbers is even. Is their product
even or odd?
b) The sum of 2001 positive whole numbers is odd. Is their product even
or odd?
B.302. There are 9 balls of the same size.
All but one has the same weight. The different ball is heavier than the
others. You have a 2-arm balance that can only compare weights. What is
the least number of measurements you need to take to find the heavy ball
even in the worst case scenario?
B.303. It takes a 1 km long freight train
2 minutes to go through a 2 km long tunnel (with its full length) by a steady
speed. How long does this train take to go through a 500 meter long tunnel
by the same speed?
B.304. Your friend, Angie told you about
a small town near your village where she discovered the best icecream place
she had ever seen. You decided to try to find the place. You are biking
down a path in a forest when the path suddenly comes to a fork. You can
choose between two paths to continue your trip. You know that one leads
to where you want to go, but the other leads to a bear cave. Unfortunately,
you do not know which one is the right path for you. However, not all is
lost, because a wise old man is sitting on a stone next to the road. He
tells you that he is willing to answer two simple questions by yes or no
for you. There is a catch though, he will not tell you whether he tells
the truth or not. He also tells you that you may not ask a question that
refers to itself. For example, you may not ask whether he will answer truthfully
to the question you are asking, because that refers to itself. However,
this does not prevent you from referring to the answer of the other question
if you choose to do so. The only thing the wise man promises is that he
will not contradict himself. For example, if you ask whether he will answer
this question truthfully, he must answer yes, because otherwise he would
contradict himself. What two questions would you ask from this old man to
find the way to the icecream place?
by László Miklós Lovász, Sammamish,
Washington, USA
Please, send your solutions to: