Question

На олимпиаде по математике Толя и Коля набрали целое количество баллов (каждый), причём Коля набрал больше баллов, чем Толя. Известно также, что если число баллов Коли увеличить в 3 раза а число баллов Толи уменьшить в 3 раза, то вместе в них получится целое число баллов меньше 57. Если же, наоборот, число баллов Коли уменьшить в 3 раза,а число баллов Толи увеличить в 3 раза, кто вместе у них получится целое число баллов, больше 33. Сколько баллов набрал каждый мальчик?​

Answer 1

Ответ:

Толя 18 баловвввввввв

Answer 2

Problem Analysis

We are given that Tolya and Kolya both scored a whole number of points in a math Olympiad, with Kolya scoring more than Tolya. We also know that if Kolya's score is tripled and Tolya's score is divided by 3, the sum of their scores is less than 57. Conversely, if Kolya's score is divided by 3 and Tolya's score is tripled, the sum of their scores is greater than 33. We need to determine how many points each boy scored.

Solution

Let's assume that Tolya scored x points and Kolya scored y points.

From the given information, we can write the following equations:

1. Kolya scored more points than Tolya: y > x. 2. If Kolya's score is tripled and Tolya's score is divided by 3, the sum of their scores is less than 57: (3y + x/3) < 57. 3. If Kolya's score is divided by 3 and Tolya's score is tripled, the sum of their scores is greater than 33: (y/3 + 3x) > 33.

We can solve these equations to find the values of x and y.

Solving the Equations

Let's start by simplifying the second equation:

(3y + x/3) < 57

Multiplying both sides by 3 to eliminate the fraction:

9y + x < 171 Now, let's simplify the third equation:

(y/3 + 3x) > 33

Multiplying both sides by 3 to eliminate the fraction:

y + 9x > 99

Solving the System of Inequalities

To solve the system of inequalities formed by equations and we can use graphical methods or substitution. Let's use substitution.

From equation we can isolate x:

x < 171 - 9y Substituting equation into equation:

y + 9(171 - 9y) > 99

Simplifying:

y + 1539 - 81y > 99

Combining like terms:

-80y > -1440

Dividing both sides by -80 (and reversing the inequality):

y < 18 Substituting the value of y from equation into equation:

x < 171 - 9(18)

Simplifying:

x < 9

Final Solution

From equations and we have the following constraints:

0 < x < 9 (since x and y are whole numbers)

Therefore, Tolya scored between 1 and 8 points, and Kolya scored between 1 and 17 points.

Since Kolya scored more than Tolya, we can try different values within the given range to find a valid solution. Let's start with the lowest possible values:

- If Tolya scored 1 point, Kolya could have scored 2, 3, 4, ..., 17 points. However, none of these combinations satisfy the given conditions. - If Tolya scored 2 points, Kolya could have scored 3, 4, 5, ..., 17 points. Again, none of these combinations satisfy the given conditions.

We can continue this process until we find a valid solution.