7th and 8th graders

September, 2014

 
C.969. Sam is writing at-least-2-digit positive whole numbers on a piece of paper. After how many numbers written on the paper can we be sure that there are three of them that start with the same digit and also end with an identical digit. (The first and last digits do not have to be the same.)

C.970.
 Three girls' birth years are 1986, 1989, and 1992. Their last names are Kiss, Hunter, and Goldbach. Their first names are Olga, Maria, and Renata. It happened one year that they realized that everybody is the same age as the number of letters in their own full names (first + last name). What year did this happen and what are their names?

C.971.
 ABC is a 3-digit number in base 10. A, B, and C are positive , and we know that (AxB-C)/(B-C)=0. How many such 3-digit numbers are there?

C.972.
 We are coloring the edges of a cube one-by-one. At the beginning none of the edges is colored, and in every step we may color only an edge that is not in contact with a colored edge. How many edges can you color this way the most?

C.973.
 How many 3-digit numbers are there in which the number of even digits is even?

C.974.
 Take all those 9-digit multiples of 1125 in which every digit is different. How many of them have even digits in the odd places (1st, 3rd, 5th, 7th, and 9th digits)?

C.975.
 One train is 100 meters, another train is 200 meters long. If each of them travels with a constant speed towards each other, they can pass each other completely in 5 seconds. With the same speeds, traveling in the same direction, one can pass the other in 15 seconds. (In this case passing means that it takes the front of the faster train 15 seconds to get from the back of the slower train to the front of the slower train.) What are the speeds of the two trains?

C.976.
 An ant is crawling on the edges of a cube, starting from one vertex. How many edges can it go through the most if it can go on every edge only once?

Please, send your solutions here.


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