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10 problems match “invariants”. Clear

Smallest non-negative signed sum ●●○○○ 20 ptsInsert $+$ or $-$ signs between the numbers $1, 2, 3, \ldots, 1989$ , so that the sum is $\pm1\pm2\pm3\cdots\pm1989$ with the first number taken as $+$ . What is the smallest non-negative value that the sum can take?
Sum after repeated difference insertion ●●○○○ 20 ptsWrite the 4 numbers $1,9,8,8$ in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, and write the result between the two numbers. This completes one operation. Then perform the same operation on the 7 numbers now in a row, and continue this process for a total of 100 operations. Find the sum of the final row of numbers.
Chameleon colour changes in September ●●○○○ 20 ptsOn an island there live red, yellow, green and blue chameleons. On a cloudy day, either one red chameleon changes its color to yellow, or one green chameleon changes its color to blue. On a sunny day, either one red chameleon changes its color to green, or one yellow chameleon changes its color to blue. In September there were 18 sunny days and 12 cloudy days. As a result, the number of yellow chameleons increased by 5. By how much did the number of green chameleons increase?
Erasing triples on a board ●●○○○ 20 ptsOn a board the numbers $1, 2, 3, \ldots, 27$ are written. One step consists of erasing three numbers $a, b, c$ from the board and writing in their place the number $a+b+c+n$ , where $n$ is a fixed positive integer. Determine the positive integer $n$ such that, after 13 steps, the number $n^2$ is written on the board.
Color of the last ball ●●●○○ 50 ptsMilan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p \rfloor$ .
Product after sum-replacement ●●●○○ 50 ptsOn a board there are 2023 distinct real numbers. Each number is simultaneously replaced by the sum of all the other numbers. After this replacement, the 2023 numbers on the board are the same as at the beginning (as a set of numbers). What value can the product of all the numbers on the board take?
Final number from repeated operation ●●●○○ 50 ptsThe 97 numbers $\frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97}$ are written on a blackboard. Repeatedly pick two numbers $a, b$ on the board and replace them by $2ab - a - b + 1$ until only one number remains. Find the final number.
Coin flips moving two numbers together ●●●○○ 50 ptsSteph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips heads, Steph replaces her number $x$ with $2x-1$ , and Jeff replaces his number $y$ with $y+8$ . Every time he flips tails, Steph replaces her number $x$ with $\frac{x+1}{2}$ , and Jeff replaces his number $y$ with $y-3$ . After some (positive) number of coin flips, Steph and Jeff end up with the same number, which is below 2012. How many times was the coin flipped?
Product of the two final numbers ●●●○○ 50 ptsThere are a total of 10040 numbers written on a blackboard: 2006 ones, 2007 twos, 2008 threes, 2009 fours and 2010 fives. Each operation erases four different numbers, one of each of four types, and writes one number of the fifth type (for example, erase one 1, one 2, one 3 and one 4, then write one 5; or erase one 2, one 3, one 4 and one 5, then write one 1). After a finite number of operations exactly two numbers remain on the blackboard. Find the product of these two numbers.
Reachable values on a blackboard ●●●●○ 100 ptsSvitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$ , she can perform one of the following three operations: if $n$ is even, she can replace $n$ with $\frac{n}{2}$ ; if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$ ; and if $n \geq 64$ , she can replace $n$ with $n-64$ . Compute the number of possible values that Svitlana can obtain by doing zero or more operations.