ABACUS International Math Challenge
for
7th and 8th graders
December, 2006
C.561. Find the area of the following ABCDEFG
polygon if we know that the angles at vertices B, C, D, E, and F are 90
degrees, and that AB=CD=1, BC=EF=2, DE=4 and FG=8.
C.562. Multiply the numbers from 1 to 2,
then from 1 to 3, then from 1 to 4, and so on until 2006. Add 99 to the
sum of these 2005 numbers. What is the digit on the tens place of this sum?
C.563. Write the numbers from 1 to 25 on
separate pieces of paper, and put them in a box. Then draw numbers one-by-one
from the box until there are two numbers among the picked ones whose product
is a square number. How many numbers do you have to draw the most?
C.564. The numbers AB4, B03, B3C, BA1 are
3-digit numbers in an increasing order, increasing by the same amount every
time. Find these numbers if A, B, and C are different digits.
C.565. We measure the weights of 5 different
men one after the other, and after every measurement we calculate the average
weight of the men already measured. We find that the average increases by
2 kg every time. How much heavier is the man measured last time than the
man measured the first time?
C.566. Eight people are sitting around
a round table. Everybody holds one hand with one other person, even if they
have to reach across the table, to create 4 connecting pairs of hands. How
many different ways can they do this if no two pairs of hands cross each
other?
C.567. In the following multiplication
same letters mean same digits, different letters mean different digits.
What could the value of the product be?
BAG * BAG * BAG = HEAVYBAG
C.568. What is the greatest prime factor
of 87! + 88! ?