# Problem 4. (8 points)

Kolya's parents give him pocket money once a month, calculating it as follows: 100 rubles for each A in math, 50 rubles for a B, 50 rubles are deducted for a C, and 200 rubles are deducted for a D. If the amount turns out to be negative, Kolya simply gets nothing. The math teacher gives a grade for the quarter by calculating the arithmetic mean and rounding according to rounding rules. What is the maximum amount of money Kolya could have received if it is known that his quarterly grade is two points, the quarter lasted exactly 2 months, there were 14 math lessons each month, and Kolya receives no more than one grade per lesson?

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# Solution:

If Kolya received a final grade of 2 for the quarter, then for each 5 he received more than 5 2s, for each 4 - more than 3 2s, and for each 3 - more than 1 2. This means that the number of 2s was greater than the total number of all other grades combined, so Kolya could receive money in at most one of the months.

Thus, the maximum amount is achieved if in one of the months he receives only 2s, and in the other - only 4s and 5s, from which the maximum number of 2s received is 14. Kolya could not receive 3 or more 5s, as in this case he would have 15 or more 2s.

This leaves 3 cases:

a) Kolya has 2 fives, then he can get 1 four, the sum is 250

b) Kolya has 1 five, then he can get 2 fours, the sum is 200

c) Kolya has no fives, in which case he can get 4 fours, the sum is 200.

Thus, the maximum amount of money is 250 rubles.

Criteria:

Correct answer - 2 points

Method of obtaining the correct answer - 2 points

Explanation of why money cannot be received in both months - 2 points

Correct enumeration - 2 points

+1 bonus point for explicitly stating the advantage of receiving all 2s in the second month, as money loses value over time and it is more advantageous to receive it in the first month rather than the second.