ABACUS International Math Challenge
for
7th and 8th graders
October, 2003
C.385. There are 32 students in a class.
Everybody studies either French or Spanish. 9 students study both languages.
Prove that a different number of students study French and Spanish.
C.386. Find the smallest positive whole
number that when added to 
, you
get a square number.
C.387. The ratio of the inner angles of
a triangle is 3:5:7. What are the angles between the angle-bisectors of
this triangle?
C.388. We baked a cake in a rectangular
shaped pan. We cut the cake in the pan into rectangular pieces by straight
side-to-side cuts that are parallel to the two sides of the pan. The pieces
that are touching the side of the pan are called outer pieces, and the pieces
that are not touching the side of the pan are called inner pieces. How many
cuts should we use in each direction if we want to portion the cake so that
each portion contains the same number of inner and outer pieces?
C.389. How many different ways can you
go from one vertex of a cube to the vertex farthest away from this, if you
may go on the edges, diagonals of the sides and the diagonals of the cube?
You may go through any vertex only once, and you may change directions only
at a vertex.
C.390. A 4-digit number ends with 22. If
you erase one of the digit 2's, you get a 3-digit number, if you erase both
2's, you get a 2-digit number. If you add these two numbers to the original
number, you get 2022. What is the original number?
C.391. On the AB side of triangle ABC there
is a point D closer to A. On the DB section of side AB there is a point
E. You know that AD=DE=DC=CA, and EC=EB. How big are the inner angles of
triangle ABC?
C.392. Prove that within any eight different
3-digit numbers there are always two such numbers that when written one
after the other, the 6-digit number you got is divisible by 7.