Changing digits and numbers

Inserting a zero into a two-digit number ●○○○○ 10 ptsAnička and Blanka each wrote a two-digit number starting with a seven, and the two girls chose different numbers. Each inserted a zero between the two digits, creating a three-digit number. From this three-digit number, each subtracted their original two-digit number. Both girls got the same result. What was this result?
Sum of squares after two increments ●●○○○ 20 ptsEach of 100 given numbers was increased by $1$ . Then each number was increased by $1$ once more. Given that after the first increase the sum of the squares of the numbers did not change, find by how much the sum of the squares changed after the second increase.
Smallest maximum rounding error ●●○○○ 20 ptsThe sum of the following seven numbers is exactly 19: $a_1 = 2.56,$ $a_2 = 2.61,$ $a_3 = 2.65,$ $a_4 = 2.71,$ $a_5 = 2.79,$ $a_6 = 2.82,$ $a_7 = 2.86.$ Each $a_i$ is approximated by an integer $A_i,$ for $1 \le i \le 7,$ such that the sum of the $A_i$ 's is also $19.$ Let $M$ be the maximum of the seven errors $|A_i - a_i|.$ What is the smallest possible value of $M$ ? Give your answer as a decimal.
Minimum sum of squares after error ●●○○○ 20 ptsAshok multiplied three real numbers. He used 73 instead of 37 for one of the numbers, and as a result the product increased by 720. Find the minimum possible value of the sum of the squares of the other two numbers.
Possible resistances from four resistors ●●●○○ 50 ptsRebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistances $a$ and $b$ ohms, and combines them into one by one of the following methods: (i) connect them in series, producing a resistor of $a+b$ ohms; (ii) connect them in parallel, producing a resistor of $\frac{ab}{a+b}$ ohms; (iii) short-circuit one of the two resistors, producing a resistor of either $a$ or $b$ ohms. Suppose that after three minutes, Rebecca has a single resistor with resistance $R$ ohms. How many possible values are there for $R$ ?
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?
Product preserved after altering two numbers ●●●○○ 50 ptsLucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, and finds that the product of the two original numbers equals the product of the two altered numbers. Compute the minimum possible sum of the original two numbers.

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