GCS 2014-15 December 7th and 8th Grade

7th and 8th graders

December 2014

C.993. 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.994. 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.995. 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.996. 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.997. At least how many consecutive integers do you have to multiply in order for the product to be divisible by 2014 for sure, no matter how you pick that many consecutive numbers?

C.998. 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.999. 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 7. He noticed also that the letters at the beginning of plate number were his own initials (TD). In the middle of the night after the accident he also realized that every digit was different, last digit was the greatest, and that the sum of the prime factors of the number on the plate is 74. Next morning he called the police to let them know the plate number of the car. What was it?

C.1000. 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.

Please, send your solutions here.

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