ABACUS International Math Competition
for
7th and 8th graders
March, 1998
C.49. The sides of the right triangle ABC are
3, 4, and 5. Let P be a point on the hypotenuse. Draw perpendiculars from
P to the legs of triangle ABC, and mark the intersections by Q and R. Where
is P when the segment QR is the shortest possible? How long is the segment
QR then?
C.50. Use parentheses in the following
expression, so that the equality would become true:
1:2:3:4:5:6:7:8:9:10=7
C.51. I wrote down five numbers on a piece
of paper. When you add any two of them you get the following sums: 0, 2,
4, 4, 6, 8, 9, 11, 13, 15. What are the five numbers written on the paper?
C.52. Find all those 4-digit numbers that
end with the digit 9, and divisible by every one of their digits.
C.53. How many such 15-digit numbers are
there that are divisible by 11, and contain only the digits 3 and 8?
C.54. One morning a priest says to his
chaplain: "I met 3 men today. The product of their ages is 2450, and
the sum of their ages is twice your age. How old are these men?" In
the afternoon the chaplain admits that he is not able to answer the question.
So, the priest helps him out: "One of the 3 men is older than me."
How old is the priest?
C.55. Find a 3-digit number such that regardless
whether you increase or decrease it by the sum of its digits, you get a
number with identical digits.
C.56. Is there such a number in which every
non-zero digit is represented once and only once, and a number created from
the first k digits is divisible by k for every k between 1 and 9?
Please, send your solutions to:
Solutions of last year's problems