Question

На ста карточках написаны числа от 1 до200. Накаждой карточке по два числа:одно четное и одно не четное, отличающиеся на 1.Вася выбрал 21 карточку. Могла ли сумма 42-х чисел на них оказаться равно 2017

Answer 1

Обратим внимание на два момента 1. числа натуральные от 1 до 200 2. Числа четное и нечетное на карточке, отличаются на 1. 
Есть одно разложение этих чисел на сто карточек
1-2, 3-4, 5-6, ..... 197-198, 199-200 итого сто пар - других разложений нет , иначе бы не выполнялся пункт что разница на каждой карточке равна 1
Сумма на карточках 3 (1*4-1), 7 (2*4-1), 11 (3*4 -1), ....   395 (99*4-1), 399 (4*100-1) то есть можно вывести общую формулу 4*k-1 (k⊂[1 100]) 
Надо теперь определить сумма 21-ой карточки равно 2017 или нет 
сложим 21 карточку 
(4*k₁-1)+(4*k₂-1)+(4*k₃-1)+...+(4*k₂₀-1)+(4*k₂₁-1)=2017
4*(k₁+k₂+k₃+...+k₂₀+k₂₁)-21=2017
4*(k₁+k₂+k₃+...+k₂₀+k₂₁)=2038
k₁+k₂+k₃+...+k₂₀+k₂₁= 2038/4 = 509.5
не может быть , так как слева сумма натуральных чисел и сумма натуральное число, а справа дробь 

Answer 2

Problem Analysis

We are given a set of 200 cards, each containing two numbers: one even and one odd, with a difference of 1. Vasya has chosen the 21st card, and we need to determine if the sum of the 42 numbers on those cards could be equal to 2017.

Solution

To solve this problem, we need to analyze the properties of the numbers on the cards and determine if their sum could be equal to 2017.

Let's consider the even number on the chosen card. Since the difference between the even and odd numbers on each card is 1, the even number on the 21st card must be 20. This is because the even number on the 20th card would be 18, and the even number on the 22nd card would be 22.

Now, let's consider the odd number on the chosen card. Since the difference between the even and odd numbers on each card is 1, the odd number on the 21st card must be 19. This is because the odd number on the 20th card would be 19, and the odd number on the 22nd card would be 21.

Therefore, the sum of the 42 numbers on the cards would be: 20 + 19 + 18 + 17 + ... + 1 + 2 + 3 + ... + 19 + 20

This is an arithmetic series with a common difference of 1. The sum of an arithmetic series can be calculated using the formula:

Sum = (n/2) * (first term + last term)

In this case, the first term is 1, the last term is 20, and the number of terms (n) is 42.

Using the formula, we can calculate the sum of the 42 numbers on the cards:

Sum = (42/2) * (1 + 20) = 21 * 21 = 441

Therefore, the sum of the 42 numbers on the cards is 441, which is not equal to 2017.

So, the sum of the 42 numbers on the chosen cards cannot be equal to 2017.

Conclusion

The sum of the 42 numbers on the chosen cards cannot be equal to 2017. The sum of the numbers on the cards is 441, which is not equal to 2017.