ABACUS International Math Challenge
for
5th and 6th graders
November, 2000
B.209. Write the digits 0, 1, 2, 3, 4,
5, 6, 7, 8, and 9 into the rectangles so that the number created from every
row is divisible by 97. None of these numbers may start with a zero.
B.210. There is a digit in every field
of the following cross-number puzzle.
Across (Horizontal): 1: Prime number; 3: The square of Down (Vertical)
3
Down (Vertical): 1: Prime number; 2: The square of Down (Vertical) 1;
3: The square root of Across (Horizontal) 3
B.211. The following sequence is made up
of blocks of 2's separated by a 1, so that the nth block of 2's contains
n 2's:
1; 2; 1; 2; 2; 1; 2; 2; 2; 1; 2; 2; 2; 2; 1; 2; 2; 2; 2; 2; 1; 2; ...
What is the sum of the first 1234 elements of this sequence?
B.212. What is the sum of the even digits
of the first 100 positive whole numbers?
B.213. We drew every diagonal of a convex
heptagon (7-sided polygon) H, and found that there is no point inside of
H which belongs to more than 2 diagonals. How many triangles did we create,
all of whose vertices are in the interior of H? (These triangles are empty
triangles, whose sides are pieces of diagonals, and whose interiors are
disjoint of any other diagonal. None of the vertices of these triangles
is on the border of H.)
B.214. Pick 10 points inside of a convex
pentagon. With straight segments, keep connecting these points with each
other and with the vertices of the pentagon, as long as the segments do
not intersect each other. How many triangles did you break the pentagon
into?
B.215. There are 7 men and 4 women in a
group. How many different ways can you pick 6 people out of this group so
that there are at least 2 women among the ones chosen?
B.216. You are given 6 different colors
to paint the sides of a cube. You may use only one color on each side, and
each side has to be a different color. How many different ways can you color
the cube, if two colorings are considered to be the same if you can rotate
one coloring into the other?
Please, send your solutions to: