ABACUS International Math Challenge
for
7th and 8th graders
November, 2000
C.209. Triangles ABC and ABD are isosceles
so that AB=AC=BD. Segments AC and BD intercept each other in point E. What
is the sum of angles ACB and ADB if BD and AC are perpendicular to each
other?
C.210. Using 6 given colors, you color
each side of a cube to a different color, then you write the six numbers
on it so that the numbers 6 and 1; 2 and 5; 3 and 4 are facing each other.
How many different cubes can you make? (Two cubes are considered to be the
same if you can rotate one cube into the position of the other.)
C.211. How many different ways can you
cover a 2x10 rectangle by using 2x1 dominoes?
C.212. In an isosceles triangle T, the
length of one of the segments connecting a vertex and the midpoint of the
opposite side is the same as the length of one of the segments connecting
the midpoints of two sides. How big could the greatest angle of T be?
C.213. A father takes his two twins to
a restaurant for their birthday. The twins' younger brother comes along
also. For the father's dinner they had to pay $4.95, and for the children's
dinner they had to pay $0.45 for each year of the children's ages. How old
is the twins' younger brother if they paid a total of $9.45?
C.214. Find the fraction p/q with the smallest
denominator, so that
99/100 < p/q < 100/101
where p and q are positive whole numbers.
C.215. Steve forgot the combination of
his lock. He remembers only that the first digit is 7, and the fifth digit
is 2. He knows that it is a 6-digit odd number, and that it gives the same
remainder when divided by 3, 4, 7, 9, 11, and 13. What is the number?
C.216. There are 2 American, 1 English,
1 French, 1 Russian, and 3 German swimmers in the final of a swimming competition.
How many different possible final results have at least one American swimmer
on the top three places if every swimmer finished on a different place?
Please, send your solutions to: