ABACUS International Math Challenge
for
7th and 8th graders
January, 2003
C.353. Find the prime numbers a, b, c,
and d, if a + b = c, and a - b = d. (a minus b = d)
C.354. A 182 cm tall man is digging a hole
in the ground. He stops for a moment and says: "I am done with one
quarter of the hole. When I finish the job the top of my head is going to
be three times as far under the ground as far it is above the ground now."
How deep is the hole going to be?
C.355. A man leaves his house in the morning
at the same time every day. He rides his bicycle to work along the train
tracks with a speed of 12 km/h, so he arrives to a railroad crossing every
day at the same time as the train going in the same direction as he does.
The train always goes by the same speed and is always on schedule. One morning
the man did not wake up on time, and left his house 50 minutes late. Therefore,
he met the train 12 km before the railroad crossing. How much longer does
it take the train to get to the railroad crossing?
C.356. There are 8 identical boxes in which
there are 1, 2, 4, 8, ..., 128 pearls, but we do not know which box has
how many pearls in it. Cecilia picks a few of the boxes, and gives the rest
of the boxes to Mary. When they both opened their boxes, it turned out that
Cecilia received 31 more pearls than Mary. How many boxes did Cecilia choose,
and how many pearls were in them each?
C.357. Two wine merchants are going into
town to sell their wine. At the gate they have to pay taxes based on the
value of the wine they want to take into the town. The first merchant has
64 barrels of wine and he has to pay 5 barrels of wine and 40 golden coins.
The other merchant has 20 barrels of wine (the same kind of barrels and
the same wine), and he gives 2 barrels of wine but gets back 40 golden coins.
How many golden coins is the price of a barrel of wine, and how much is
the tax on it?
C.358. You have 12 one-unit-long straight
segments. Using them, create a polygon with an area of 3 units.
C.359. We cut up a rectangle into 9 square-shaped
pieces with sides 1, 4, 7, 8, 9, 10, 14, 15, and 18 units. (Every square
is in one piece, so we cut up the rectangle into 9 pieces.) How long were
the sides of the rectangle if you know that they are whole numbers?
C.360. We extended the BC side of the regular
triangle, ABC, by the half of the side over C to get point D. The straight
segment, starting from point D, going through the midpoint F of side AC,
intersects side AB in point E. What part of the perimeter of ABC is the
length of segment AE?
