ABACUS International Math Challenge
for
7th and 8th graders
February, 1999
C.105. How many such 5-digit numbers are there
in which the sum of the digits is even?
C.106. Write the numbers 0, 1, 2, ...,
8 in a 3x3 grid so that the 3-digit numbers in every row and every column
(read from top to bottom) is divisible by 6, and by the sum of the digits
of that number.
C.107. Find the smallest such number created
by the digits 3 and 7 only, that is divisible by both 3 and 7.
C.108. How many different ways can 5 people
sit down in a 5-seat balcony in the opera house, so that at least one person
sits where his/her ticket suggests?
C.109. 0, 01, 0110, 01101001, 0110100110010110,
...
In this sequence you can get the next element if you connect the complementer
of the previous element to that previous element. (You get the complementer
of an element by switching the zeros and the 1's in it.) The 12th element
of this sequence has 2048 digits. What digit is on the 1999th place in that
element?
C.110. What is the sum of all those 6-digit
numbers that you can create by a different order of the digits 1, 2, 3,
4, 5, 6?
C.111. The lengths of sides AB, BC, and
CA of triangle ABC are 7, 6, and 7 units respectively. The bisectors of
angle A and B intersect each other in point O. Draw a parallel line with
AB going through O, which intersects the sides of the triangle in points
M and N. What is the perimeter of triangle MNC?
C.112. On Nowhere Island two kinds of people
live: truthful people, who always say the truth, and liars, who always lie.
Once I visited this island. There were three people A, B, and C standing
around on the beach.
"Are you a truthful person?"- I asked A. He answered so quietly
that I could not hear him. So, I asked B: "What did A say?"
"A said no."- answered B. Then C said to me: "Do not
believe B, he is lying."
Is there a truthful person among these three people?
Please, send your solutions to: