Problems – Page 9 – We Solve Problems
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#### Cryptography , Puzzles

Try to read the word in the first figure, using the key $($see the second figure$)$.

#### Arithmetic progression

Is the sum of the numbers 1 + 2 + 3 + … + 1999 divisible by 1999?

#### Number theory. Divisibility

Is the number $10^{2002} + 8$ divisible by 9?

#### Decimal number system , Theory of algorithms (other)

In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?

#### Dissections with certain properties

Is it possible to cut a square into four parts so that each part touches each of the other three $($ie has common parts of a border$)$?

#### Pigeonhole principle (other) , Proof by contradiction

Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?

#### Decimal number system , Proof by exhaustion

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

#### Puzzles

What word is encrypted: 22212221265121? Each letter is replaced by its number in the English alphabet.

#### Division with remainder , Proof by exhaustion

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

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