[b]p1.[/b] Ivan Ivanovich came to the store with $20$ rubles. The store sold brooms for $1$ ruble. $17$ kopecks and basins for $1$ rub. $66$ kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: $1$ ruble = $100$ kopecks) [b]p2.[/b] On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another $10$ km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities. [b]p3.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve? [b]p4.[/b] What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from $100$ to $999$? [b]p5.[/b] An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was $13/23$. What fraction was written on the board? [b]p6.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] The square is divided by straight lines into $25$ rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p9.[/b] Petya multiplied all natural numbers from $1$ to his age inclusive. The result is a number $$8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000.$$ How old is Petya? [b]p10.[/b] There are $100$ integers written in a line, and the sum of any three in a row is equal to $10$ or $11$. The first number is equal to one. What could the last number be? List all possibilities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]