7.5. Given the number 1836549, you can take two adjacent non-zero digits and swap their places, after which you subtract 1 from each of them. What is the smallest number that can result from these operations?
Answer: 1010101

Solution: The digits in the number alternate in parity: odd, even, etc. Note that with the described operation, even and odd numbers swap places, and then 1 is subtracted from them, thereby not disrupting the order of parity. Thus, it is impossible to obtain a number less than 1010101 (in each place, the smallest odd or even digit is placed, respectively).

Let's show how this result can be achieved (we perform the operation on the underlined digits, sometimes on two pairs simultaneously): $\underline{\underline{83}} 65 \underline{49}>12 \underline{7658} 3>125 \underline{6743}>12 \underline{56543}>12 \underline{54343}>1 \underline{23} 4343>1 \underline{214343}>$ 1014343

Next, each pair 43 is transformed into 23, then into 21, and then into 01, after which we get 1010101.

## Criteria for Determining Winners and Prize Winners of the All-Siberian Open Mathematics Olympiad for Schoolchildren (2015-2016 academic year)

According to the Regulations, the winners and prize winners of the Olympiad were determined based on the results of the Final Stage of the Olympiad. The total number of winners and prize winners was 380 out of 1578 participants, which is $24.08 \%$. The number of winners was 85, which is $5.38 \%$.

Based on the overall ranking of participants and taking into account the noticeable gaps in the scores of the groups of participants at the top of the ranking, the Olympiad jury developed the following criteria for determining winners and prize winners: The maximum possible score is 35 points.

## 11th Grade:

Winners:

Participants who scored more than $77 \%$ of the maximum possible points, i.e., from 27 to 35 points;

Prize winners:

2nd degree - more than $62 \%$ of the maximum possible points, i.e., from 22 to 26 points

3rd degree - more than $51 \%$ of the maximum possible points, i.e., from 18 to 21 points

10th Grade:

Winners:

Participants who scored more than $85 \%$ of the maximum possible points, i.e., from 30 to 35 points;

Prize winners:

2nd degree - more than $62 \%$ of the maximum possible points, i.e., from 22 to 29 points

3rd degree - more than $51 \%$ of the maximum possible points, i.e., from 18 to 21 points

9th Grade:

Winners:

Participants who scored more than $85 \%$ of the maximum possible points, i.e., from 30 to 35 points;

Prize winners:

2nd degree - more than $65 \%$ of the maximum possible points, i.e., from 24 to 29 points

3rd degree - more than $51 \%$ of the maximum possible points, i.e., from 18 to 23 points

8th Grade:

Winners:

Participants who scored more than $82 \%$ of the maximum possible points, i.e., from 29 to 35 points;

Prize winners:

2nd degree - more than $62 \%$ of the maximum possible points, i.e., from 22 to 28 points

3rd degree - more than $48 \%$ of the maximum possible points, i.e., from 17 to 21 points

7th Grade:

Winners:

Participants who scored more than $85 \%$ of the maximum possible points, i.e., from 30 to 35 points;

Prize winners:

2nd degree - more than $71 \%$ of the maximum possible points, i.e., from 24 to 29 points

3rd degree - more than $42 \%$ of the maximum possible points, i.e., from 15 to 21 points

Co-Chair of the Mathematics Jury

![](https://cdn.mathpix.com/cropped/2024_05_06_16ecf25cad47bb71667ag-3.jpg?height=143&width=314&top_left_y=2367&top_left_x=928)

A.Yu. Avdyushenko