6.4. When Winnie-the-Pooh came to visit Rabbit, he ate 3 plates of honey, 4 plates of condensed milk, and 2 plates of jam, and after that, he couldn't get out of the hole because he had become too fat from all the food. But it is known that if he had eaten 2 plates of honey, 3 plates of condensed milk, and 4 plates of jam or 4 plates of honey, 2 plates of condensed milk, and 3 plates of jam, he would have been able to leave Rabbit's hospitable burrow without any trouble.

What makes him fatter: jam or condensed milk? Justify your answer.
# Solution:

Method 1. Let one bowl of honey have a caloric value of $a$ units, one bowl of condensed milk $-b$ units, and one bowl of jam $c$ units. (The higher the caloric value of a product, the more it makes one gain weight.) The condition of the problem means that $3 a+4 b+2 c>2 a+3 b+4 c$ and $3 a+4 b+2 c>4 a+2 b+3 c$. The first inequality simplifies to $a>2 c-b$, and the second to $a<2 c-b$, from which $b>c$, meaning the caloric value of condensed milk is higher.

Method 2. Suppose there were 4 Winnie-the-Poohs. The first two each ate 3 bowls of honey, 4 bowls of condensed milk, and 2 bowls of jam (and both couldn't get out), while the third and fourth ate 2, 3, 4 and 4, 2, 3 bowls of honey, condensed milk, and jam, respectively. Then the third and fourth gained less weight than the first and second. Note that the first and second together ate 6 bowls of honey, the same as the third and fourth together, but they ate 8 bowls of condensed milk against 5, which is 3 bowls more, and 4 bowls of jam against 7, which is 3 bowls less. This means that 3 bowls of condensed milk make one gain more weight than 3 bowls of jam.

Method 3. Remove 2 bowls of each product from everywhere. Then it will turn out that, firstly, one bowl of honey and two bowls of condensed milk make one gain more weight than one bowl of condensed milk and two bowls of jam; therefore, one bowl of honey and one bowl of condensed milk give a greater weight gain than two bowls of jam. Secondly, one bowl of honey and two bowls of condensed milk make one gain more weight than one bowl of jam and two bowls of honey; therefore, two bowls of condensed milk give a greater weight gain than one bowl of honey and one bowl of jam. Comparing these two conclusions, we get that one bowl of honey and three bowls of condensed milk are more filling than one bowl of honey and three bowls of jam, and, therefore, condensed milk is more filling than jam.

Answer: Condensed milk makes one gain more weight.

Note: From the condition of the problem, it is impossible to conclude the place of honey in the chain of caloric values: it can be both the most high-caloric and the least high-caloric of the three products, or occupy an intermediate caloric value. Thus, any conclusion about the caloric value of honey and condensed milk (or honey and jam) is incorrect.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| The condition of the problem is correctly written as inequalities, but the problem is not solved to the answer | 3 points |
| Answer without justification | 0 points |