7th and 8th graders

January 2015

 
C.1001. We filled out an 8x8 grid with natural numbers from 1 to 64 by starting at the top left corner with 1 and going to the right all the way in an order. When we reached the end, we continued on the left side of the second row, and so on ending with 64 in the bottom right corner. Now, we pick 8 numbers from this grid so that we pick exactly one number from each column and each row. Prove that the sum of these 8 numbers is always the same.

C.1002. From a rectangular shape piece of paper we cut out two identical circles. (The two circles do not have any mutual points.) Find a straight cut that divides the paper into two equal areas.

C.1003. The vertices of the gray rhombuses are the midpoints of the sides of the square and the sixths points of the diagonals of the square. How long are the sides of the square (to mm accuracy) if the white area is 3 sq.dm?

C.1004. A princess was kidnapped and the kidnappers told the king that they will ask for a ransom between 1 golden coin and 31 golden coins. Therefore, the king put a total of 31 golden coins in a few bags, so when he meets the kidnappers, he could pay any amount between 1 and 31 coins they ask for without having have to open any of the bags. At least how many bags did the king need?

C.1005. Write down the natural numbers from 1 to 100,000 and count how many digits you had to write down. Then write down the natural numbers from 1 to 1,000,000 and count how many zeros you had to write this time. Prove that these two numbers are the same.

C.1006. You have three cards with one digit written on each. You form all possible 1-, 2-, and 3-digit numbers. The sum of all of these numbers is 5635. What are the digits on the cards?

C.1007. ABC is an equilateral triangle with side 6. Let P be a point with distances x,y,z from the three vertices (where X is less than or equal to Y which is less than or equal to Z). Find the area of the region enclosed by all points such that x+y=z.

C.1008. A fully charged cellular phone can work in a standby position (which means that we do not use it for making phone calls) for 72 hours, or we can talk continually for 3 hours on it, and then we have to recharge it again. This cell phone, after it was fully charged, was in standby position for 27 hours, and the owner conducted a 45-minute conversation, also. How many more minutes could we talk on this phone?

Please, send your solutions here.

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