ABACUS International Math Challenge
for
7th and 8th graders
October, 2006
C.545. Is there such a triangle in which
every height is less than 1 cm yet its area is greater than 100 square centimeter?
C.546. The remainders when the 5-digit
number abcde is divided by 2, 3, 4, 5, and 6 are a, b, c, d, and
e, in that order. Find this 5-digit number.
C.547. Find the smallest positive whole
number that is not a divisor of the product of the first 80 positive whole
numbers.
C.548. The diagonals of rectangle KLMN
are 12 cm long, and they cross each other at point O. Pick a point A on
side KL, a point B on side LM, and a point C on side MN so that A, O and
C are collienar. What is the minimum of the sum of the lengths of segments
AB and BC?
C.549. There is a 12 cm x 12 cm square
sheet of paper on the table. Its side facing up is white, the other side
facing the table is black. We fold one vertex up so this vertex would be
on the same diagonal that it is was on before. How far is this vertex from
the folding line if the visible black area is the same as the visible white
area?
C.550. By using a straight edge ruler and
compasses, construct a square that has the same area as the shade part of
the following picture:
C.551. How many positive 3-digit numbers
are there in which the sum of the digits is odd?
C.552. Imagine a year in which none of
the months starts with Monday. What day of the week does that year start
with?