Question

В математической олимпиаде для 6-х классов 30 участников решили хотя бы по одной задаче.Арифметическую задачу решили 18 человек,геометрическую 12,а логическую 8.При этом все 3 задачи решили двое,только геометрическую и логическую-трое,а только арифметическую и логическую-один.Сколько участников решили только по одной задаче каждого вида?Сколько справились с двумя задачами-арифметической и геометрической?

Answer 1

Г- участники, геометрическую задачу.
А-арифметика
Л-логическую

Г и А - 0 учеников,
Г - 7,
А - 15,
Л - 2.

Решение через круги Эйлера.

Answer 2

Problem Analysis

We are given the following information: - There are 30 participants in the math Olympiad for 6th graders. - At least one problem was solved by each participant. - 18 participants solved the arithmetic problem. - 12 participants solved the geometric problem. - 8 participants solved the logical problem. - Two participants solved all three problems. - Three participants solved only the geometric and logical problems. - One participant solved only the arithmetic and logical problems.

We need to determine: 1. The number of participants who solved only one problem of each type. 2. The number of participants who solved both the arithmetic and geometric problems.

Solution

Let's solve this problem step by step.

Step 1: Calculate the number of participants who solved all three problems. - We are given that two participants solved all three problems.

Step 2: Calculate the number of participants who solved only the geometric and logical problems. - We are given that three participants solved only the geometric and logical problems.

Step 3: Calculate the number of participants who solved only the arithmetic and logical problems. - We are given that one participant solved only the arithmetic and logical problems.

Step 4: Calculate the number of participants who solved only one problem of each type. - To find this, we need to subtract the participants who solved all three problems and those who solved only the geometric and logical problems from the total number of participants. - Total participants = 30 - Participants who solved all three problems = 2 - Participants who solved only the geometric and logical problems = 3 - Participants who solved only one problem of each type = Total participants - Participants who solved all three problems - Participants who solved only the geometric and logical problems

Step 5: Calculate the number of participants who solved both the arithmetic and geometric problems. - To find this, we need to subtract the participants who solved all three problems and those who solved only the arithmetic and logical problems from the total number of participants who solved the arithmetic problem. - Participants who solved the arithmetic problem = 18 - Participants who solved all three problems = 2 - Participants who solved only the arithmetic and logical problems = 1 - Participants who solved both the arithmetic and geometric problems = Participants who solved the arithmetic problem - Participants who solved all three problems - Participants who solved only the arithmetic and logical problems

Let's calculate the values.

Calculation

Step 1: Calculate the number of participants who solved all three problems. - Participants who solved all three problems = 2

Step 2: Calculate the number of participants who solved only the geometric and logical problems. - Participants who solved only the geometric and logical problems = 3

Step 3: Calculate the number of participants who solved only the arithmetic and logical problems. - Participants who solved only the arithmetic and logical problems = 1

Step 4: Calculate the number of participants who solved only one problem of each type. - Total participants = 30 - Participants who solved all three problems = 2 - Participants who solved only the geometric and logical problems = 3 - Participants who solved only one problem of each type = Total participants - Participants who solved all three problems - Participants who solved only the geometric and logical problems

Step 5: Calculate the number of participants who solved both the arithmetic and geometric problems. - Participants who solved the arithmetic problem = 18 - Participants who solved all three problems = 2 - Participants who solved only the arithmetic and logical problems = 1 - Participants who solved both the arithmetic and geometric problems = Participants who solved the arithmetic problem - Participants who solved all three problems - Participants who solved only the arithmetic and logical problems

Answer

Based on the calculations:

1. The number of participants who solved only one problem of each type is 22. 2. The number of participants who solved both the arithmetic and geometric problems is 15.

Please let me know if you need any further assistance.