ABACUS International Math Challenge
for
7th and 8th graders
December, 2004
C.457. There are 40 pairs of brown, 50
pairs of black, and 30 pairs of white socks in a box. There is no difference
between a right sock and a left sock, and the socks are not in pairs in
the box but all separated. How many socks do you have to take out of the
box in the dark if you want to make sure that there are at least 2 pairs
of white, 5 pairs of brown and 25 pairs of black socks among them?
C.458. Pick as many numbers as you can
out of 1, 2, 3, ..., 50 so that none of the numbers picked has the half
of it among the numbers picked.
C.459. Out of all 6-digit numbers some
can be written as the product of two 3-digit numbers, others are not. Which
kind are there more of?
C.460. There are scheduled buses traveling
between Somewhere and Nowhere. There is only one road between the two villages,
so the buses take this road. From both villages the buses leave for the
other village at every hour and at every half an hour, and travel with the
same speed for the whole trip. If there is no traffic jam, the buses leaving
the villages at the same time meet on the road between the two villages
10 minutes later. One month, however, there was a construction on the road,
and the buses (on the days when the workers were working) had to wait 5
minutes at the border of the village from which they just left. Other than
that, the buses traveled the same way as before. (The workers did not work
every day.) During the 30 days of the road construction the buses met in
an average 14 minutes after leaving. How many days out of the 30 did the
workers work?
C.461. At least how many consecutive integers
do you have to multiply in order for the product to be divisible by 2004
for sure, no matter how you pick that many consecutive numbers?
C.462. The base of the ABC isosceles triangle
is BC. Mark a point D on side BC, and a point E on side AC so that AD=AE.
How big is angle EDC if angle BAD is 30 degrees?
C.463. A math teacher was hit by a car,
which drove away right after the accident. The victim could remember only
that the sum of the digits of the 4-digit number on the plate of the car
was 6. He noticed also that the letters at the beginning of plate number
were his own initials (TD), and that the two middle digits were identical.
In the middle of the night after the accident he also realized that the
sum of the different prime factors of the number on the plate is 172. Next
morning he called the police to let them know the plate number of the car.
What was it?
C.464. In a summer camp Pete played a ping-pong
match with 25 other children, and nobody in the camp played against more
children than him. We know also that if two children played ping-pong with
the same number of children then those two children did not have a mutual
opponent. Prove that one of Pete's opponents played ping-pong against Pete
only.