ABACUS International Math Challenge
for
7th and 8th graders
March, 2002
C.305. You have a 25-digit whole number
A. Let's say that the abc 3-digit number can be read out of
A if A contains the digits a, b, and c, so
that b is to the right of a, and c is to the right
of b. Prove that there is a 3-digit number (whose first digit is
not zero, and it has 3 different digits), which cannot be read out of A.
C.306. Prove that you can draw a regular
octagon (with 8 equal sides and equal angles) using 8 of the continuing
line shown on the following diagram:
C.307. Prove that there is no such a positive
whole number n for which n+2p is always prime if p is prime.
C.308. Write 2/11 (two elevenths) as the
sum of two different fractions with a numerator of 1 in each.
Kevin Kwok, New York
C.309. You brake up a convex quadrilateral
into 4 triangles with its diagonals. The areas of three of these triangles
are 1, 2, and 3 units. What is the area of the fourth triangle?
C.310. Is there a triangle with heights
of 7cm, 14cm, and 21cm?
C.311. Two points are 13cm apart on a piece
of paper. You have an 8cm-long ruler and compasses. How can you draw the
whole straight segment between these two points if the radius of the greatest
circle you can draw with your compasses is:
a) 13 cm
b) 6 cm?
C.312. a) There are 5 segments with lengths
of 2, 4, 6, 8, and 10 cm. You randomly pick 3 of them. What is the probability
of being able to construct a triangle out of these 3 segments?
b) Draw 5 segments so that if you pick any 3 of them you will not be
able to construct a triangle with them. (Give us the lengths of those segments.)