Solution

It is clear that the result is influenced only by the last digits of the numbers written on the cards. Therefore, we can assume that there are 202 cards with the numbers 3, …, 9, 0 and 203 cards with the numbers 1 and 2. The first player can first take a card with the number 2, and then repeat the moves of the second player for as long as possible. At the moment when he can not repeat the move of the second $($ after the second takes the last card with the number 1 $)$, the first takes any of the remaining cards, and then again repeats the moves of the second. And so on. As a result, at the end both cards of each type will be equally divided, except for the cards with the numbers 1 and 2: the first will have one extra card with a deuce and one less card with a one.
Since the sum $100 \times $(0 + 1 + … + 9)$ ends in a zero, the sum of the first players ends in a two, and the sum of the second ends in a one.

Answer

The first player.

Hint

Hint

Try to consider the divisors of the number 1,000,000,000.

Solution

The divisors of the number 1000000000 will be nine deuces, nine fives and any combinations of their works. But as soon as there are a 5 and a 2 at the same time, the number will end in a 0. This means that the only possible factors are $2^9$ and $5^9$ $($since in all other pairs of divisors there are numbers ending in 0$)$. Checking this shows that both these numbers do not contain zeros. So: 1 000 000 000 = $(2^9 \times 5^9)$ = $(512 \times 1 953 125)$.

Answer

$1 000 000 000 = 2^9 \times 5^9= 512 \times 1 953 125$.

Hint

Try to consider the divisors of the number 1,000,000,000.

Solution

If the number 2 in the number 102 is moved upwards, to the place of the exponent, then the original equality takes the form 101 – $10^2$ = 1 and will be true.

Answer

The number 2 in the number 102 should be put in the place of the exponent.

Hint

–

Hint

Try to determine the length of the side of the desired square.

Solution

Such a square cannot be made, because its perimeter should be 50 cm, i.e. the sides are not integers.

Answer

This cannot be done.

Hint

Try to determine the length of the side of the desired square.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $(in our case, from 2 to 9)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1, k, k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Answer

See the table.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

$A= 2011\times 20122012\times 201320132013 = 2011\times 2012 \times 10001 \times 2013 \times 100010001$.

$B= 2013\times 20112011 \times 201220122012 = 2013 \times 2011 \times 10001 \times 2012 \times 100010001 =A$.

Answer

$A=B$.

Hint

Solution

We replace the blots with the letter X: {XX + XX + 1} $\times X $ = XXX. Having reduced by X, we obtain XX + XX + 1 = 111, whence XX = $($111 – 1$)$ / 2 = 55.

Answer

5.

Hint

–

Solution

Nothing to translate.

Hint

–

Hint

Check the example “from right to left”.

Solution

Let’s start by examining the example “from right to left”. In the ranks of the units and tens, everything is in order, but in the hundreds, there is an error. Hence, one of the digits of this category – 1, 8 or 7 – is rearranged. Assuming that Alex moved two cards “inside” the hundreds $($the only option is to swap 7 and 8$)$, then there will still be an error in the tens of thousands. Hence, one of the digits of the hundred digit has been changed with the digit of a digit in a higher rank. To restore equality in the hundreds, the number 1 can only be changed to a 9. There is only one of such a digit in the higher rank digits. But if 1 and 9 are interchanged, then still there will be an error in the tens of thousands.
Digit 8 can be changed only to 6, but not a single digit 6 is in the example there.
So, there is only one possibility – to change the number 7. Instead of it, we have to put the number 9. There is only one $($in the higher rank digits$)$, and if they are interchanged, the correct example is obtained.

Answer

Hint

Check the example “from right to left”.

Hint

Try to form an equation.

Solution

Let us call the original number $x$. Then we can write: $4x+15=15x+4$. Solving this equation gives us that $11=11x$ and so $x=1$.

Hint

Try to form an equation.