Problems

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Found: 705

Is it possible to divide the numbers 1, 2, 3, ..., 100 into pairs of one odd and one even number, such that in every pair except one the even number is greater than the odd number

Back in the days when a young mathematician was even younger he could only draw digits “4” and “7”. While looking through the old notes his mother found one piece of paper on which he wrote the numbers with digit sums equal to 18, 22 and 26. Which numbers could be written on this piece of paper?

a) It seems that the young mathematician was making progress quite fast. On the back side of that piece of paper there are numbers with digits adding up to all natural numbers from 18 to 33. And yet all of them consist of only digits “4” and “7”. Make your own list of that kind.

(b) Is it true that any natural number greater than 17 can be equal to the digit sum of some number written with digits “4” and “7”?

(c) Now let’s try the same question for digits “5” and “8”. What values can you get if you consider the sum of the digits of a number written with the help of digits “5” and “8”?

Kate is playing the following game. She has 10 cards with digits “0”, “1”, “2”, ..., “9” written on them and 5 cards with “+” signs. Can she put together 4 cards with “+” signs and several “digit” cards to make an example on addition with the result equal to 2012?

Note that by putting two (three, four, etc.) of the “digit” cards together Kate can obtain 2-digit (3-digit, 4-digit, etc.) numbers.

Jane is playing the same game as Kate was playing in Example 3. Can she put together 5 cards with “+” signs and several “digit” cards to make an example on addition with the result equal to 2012

In the following puzzle an example on addition is encrypted with the letters of Latin alphabet: \[{I}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}+{HE}={US}.\] Different letters correspond to different digits, identical letters correspond to identical digits.

(a) Find one solution to the puzzle.

(b) Find all solutions.

A young mathematician had quite an odd dream last night. In his dream he was a knight on a \(4\times4\) board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either

(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or

(b) he has passed exactly once through all the cells of the board.

For each possibility examine if it could happen or not.

Is it true that if a natural number is divisible by 4 and by 6, then it must be divisible by \(4\times6=24\)?

The number \(A\) is not divisible by 3. Is it possible that the number \(2A\) is divisible by 3?