Question

В городской олимпиаде по математике приняло участие 120 пятиклассников.Каждому из них было предложено для решения 5 задач.После проверки работ выяснилось,что 1/3 всех участников решила ровно по одной задаче,1/4-ровно по две задачи и 1/5- ровно по три задачи.Общее число решенных задач оказалось равным 277 Определите,был ли такой участник олимпиады,который решил все 5 задач,если известно,что каждый участник решил целое число задач

Answer 1

Не было потому что все цифры не сходятся для положительного ответа

Answer 2

Problem Analysis

In the city mathematics Olympiad, 120 fifth-grade students participated, and each of them was given 5 problems to solve. After checking the papers, it was found that 1/3 of all participants solved exactly one problem, 1/4 solved exactly two problems, and 1/5 solved exactly three problems. The total number of problems solved was 277. We need to determine if there was a participant who solved all 5 problems, given that each participant solved a whole number of problems.

Solution

Let's assume that there was a participant who solved all 5 problems. We'll denote the number of participants who solved exactly one problem as x, the number of participants who solved exactly two problems as y, and the number of participants who solved exactly three problems as z.

According to the given information, we can write the following equations:

1. x + y + z = 120 (equation 1) - The total number of participants is 120. 2. (1/3)x + (1/4)y + (1/5)z = 277 (equation 2) - The total number of problems solved is 277.

To solve these equations, we can use the method of substitution.

From equation 1, we can express x in terms of y and z:

x = 120 - y - z

Substituting this value of x into equation 2, we get: