ABACUS International Math Challenge
for
7th and 8th graders
October, 2001
C.265. An extra terrestrial from Mars came
to visit the Earth. Martians eat only once a day the most, either in the
morning, or at noon, or in the evening, but only if they feel like eating.
They can go without eating for any number of days. While the Martian was
here, it ate 7 times. We also know that it spent 7 mornings, 6 noons, and
7 evenings without eating. How many days did the Martian spend here on Earth?
C.266. There are 3 diagonals running into
the same vertex of a rectangular based column. Prove that you can always
construct a triangle from these three segments.
C.267. Is it true that if the product of
6 positive whole numbers ends in exactly two zeros, then you can find 4
out of these 6 numbers for which the same is true?
C.268. We write all the integers from 1
to 2001 on a circular line in a random order. Then we switch some of the
neighbors using the following procedure. We start at one of the numbers.
We compare this number to its neighbor in the clockwise direction. If the
neighbor is smaller then we switch the two numbers, otherwise we leave them
as they were. In the next step we compare the bigger of these two numbers
to its clockwise neighbor and do a switch if the neighbor is smaller. We
keep repeating these steps. We say that we completed a cycle if we compared
the number in the last position to the number in the position we started
at. How many cycles do we have to complete before the positions of the numbers
are the same as their original positions were?
C.269. Using the grid lines, Tom draws
a rectangle on a paper with unit-square grid lines on it. Staying on the
grid lines, he wants to draw a closed figure inside of this rectangle, so
that he would not go outside of this rectangle, but he would go through
every grid point on the border and the inside of this rectangle exactly
once. How long is the line he has to draw if the dimensions of the rectangle
are
a) 2000 x 2001
b) 2000 x 2002?
C.270. Find such digits x and y for which
the 6-digit number xyxyxy (in base 10) has a 3-digit prime divisor.
C.271. With its 2 diagonals, brake a convex
quadrilateral into 4 triangles. Prove that the products of the areas of
the 2-2 triangles facing each other are the same.
C.272. From the number 12345678901234567890...1234567890,
which has 5000 digits, take out the digits located on an odd number location.
We do the same with the remaining 2500-digit number, and continue until
we have a one-digit number. What number is this last digit?