ABACUS International Math Challenge
for
7th and 8th graders
October, 2002
C.329. Two pieces of land are separated
by the line ABCD, as shown on the diagram below. AB=30 m, BC=24 m, and CD=10
m. AB, CD, and BC are parallel to the sides of the rectangle. We want to
straighten the border line by the AE straight line, so that the areas of
the two lands does not change. How far is E from D?
C.330. Based on a survey conducted in a
group of 13-year olds, it turned out that they watch TV for an average of
50 minutes every day. The average of the boys in this group was 45 minutes,
the average of the girls in the group was 65 minutes. What is the ratio
of the boys and girls in this group?
C.331. A merchant increased his prices
by 50%. But with the higher prices the traffic in the store decreased, so
he did not make as much money as he did with the old prices. Now he wants
to put everything on sale, but he wants to make 20% more on his items than
the original prices were. What percent sale should he advertise?
C.332. Find all those 2-digit ab
numbers where ab + ba + 5(a +b ) is a square
number.
C.333. There were 5 chess players on a
tournament. Everybody had one game with everybody. You get 1 point for winning,
half a point for a draw, and zero point for losing a game. We know that:
- The winner of the tournament had no draw.
- The second place winner did not lose any game.
- Everybody finished the tournament with a different number of points.
How many points did each player finished with?
C.334. A club of animal lovers has 100
members. 92 of them have dogs, 90 of them have cats, 86 of them have hamsters,
and 80 of them have parrots. At least how many of them have all four kinds
of animals?
C.335. A class organized 3 field trips
during the last school year. 70% of the students went on the first trip,
80% of them went on the second trip, and 90% of them went on the third trip.
This way, 12 students went on all the trips, and everybody else went on
two of the trips. How many students were there in the class?
C.336. How many 2-digit numbers are there
in which the sum of the digits drop to its half when you add 4 to the number?