Question

ОЛИМПИАДА ОЛИМПИАДА Петя написал на доске 9 различных целых положительных чисел. Оказалось, что среднее арифметическое этих чисел равно 19. Какое наибольшее значение может принимать самое большое из чисел на доске?

Answer 1

Ответ:

135

Пошаговое объяснение:

Чтобы самое большое число было наибольшим, остальные числа должны быть наименьшими. Раз числа различные, то это 1, 2, 3, 4, 5, 6, 7, 8. Их сумма 36. Если среднее арифметическое девяти чисел 19, то сумма их равна 19*9=171. Значит 171-36=135 искомое наибольшее число

Answer 2

Problem Analysis

Petya has written 9 different positive integers on the board, and the average of these numbers is 19. We need to determine the maximum value that the largest number on the board can have.

Solution

To find the maximum value of the largest number on the board, we need to distribute the remaining sum among the other 8 numbers as evenly as possible. This will ensure that the largest number is as small as possible.

Let's assume the largest number on the board is x. Since the average of the 9 numbers is 19, the sum of all the numbers is 9 * 19 = 171. Therefore, the sum of the remaining 8 numbers is 171 - x.

To minimize the largest number, we want to distribute the remaining sum (171 - x) as evenly as possible among the other 8 numbers. This means each of the other 8 numbers should have a value of (171 - x) / 8.

Since all the numbers are positive integers, we need to find the largest value of x that satisfies this condition.

Calculation

To find the maximum value of x, we need to find the largest integer value of (171 - x) / 8.

Let's calculate this value:

(171 - x) / 8 = 21.375 - x/8

Since x must be an integer, the largest value of x that satisfies this condition is when x/8 is the largest possible integer value less than or equal to 21.375.

Therefore, x/8 = 21

Solving for x, we get:

x = 21 * 8 = 168

Answer

Therefore, the largest possible value for the largest number on the board is 168.

Verification

Let's verify our answer using the given information.

If the largest number on the board is 168, then the sum of the remaining 8 numbers is 171 - 168 = 3. Since we distributed the remaining sum as evenly as possible, each of the other 8 numbers should have a value of 3/8 = 0.375.

The average of these 9 numbers is (168 + 0.375 * 8) / 9 = 19.

Therefore, our answer is verified.

Conclusion

The largest possible value for the largest number on the board is 168, given that the average of the 9 numbers is 19.