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2018-11-12 Basic

         

When will it pop?

Six dark-grey symbols (L, H, U, S, T, \(+\)) are drawn on six identical pieces of square paper.

How many of these symbols have the same perimeter as the square paper itself?

You are given two integers \(J\) and \(K.\)

At most, how many of the following statements can be simultaneously true?

  • \(J + K\) is an odd number.
  • \(J - K\) is an odd number.
  • \(J \times K\) is an odd number.
  • \(J \div K\) is an odd number.

I am looking for a positive integer greater than 9 such that the product of all of its digits is equal to the integer itself:

\[\underbrace{{\color{red}a} \times {\color{blue}b} \times {\color{green}c} \times \cdots \times {\color{indigo}x} \times {\color{purple}y} \times {\color{pink}z} = \overline{{\color{red}a}{\color{blue}b}{\color{green}c} \ldots {\color{indigo}x}{\color{purple}y}{\color{pink}z}}}_{\text{The product of the digits is equal to the integer itself.}}.\]

Are there any positive integers that satisfy this condition?

\[ \sqrt1,\ \sqrt{11},\ \sqrt{111},\ \sqrt{1111},\ \sqrt{11111},\ \ldots \]

Is it true that \(\sqrt{1}\) is the only integer in this list of numbers?

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