The front tires of the "Gnu Antelope" car wear out after 25,000 km, while the rear tires wear out after 15,000 km. When should O. Bender switch them to maximize the distance the car can travel? What is this distance?

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The first method. On eight pairs of tires (3 pairs of front and 5 pairs of rear), one can travel 75,000 km. Therefore, on two pairs of tires, one can travel a maximum of a quarter of this distance - 18,750 km. To ensure that both pairs of tires are used equally (i.e., in the same position), they should be swapped halfway through the journey.

The second method. To ensure that the tires wear out completely at the same time (which is clearly the most efficient option), they should be swapped halfway through the journey. In this case, each tire will wear out by $3/8$ as a front tire and by $5/8$ as a rear tire (since $25000:15000=5:3$). Therefore, this half of the journey is $5/8 \cdot 15000 = 9375$ km.

## Answer

After 9375 km. 18750 km.

Find all prime numbers that cannot be written as the sum of two composite numbers.

## Solution

We will prove that any prime number $p > 11$ can be represented as the sum of two composite numbers. Since any such number is odd, the number $p-9$ is even and, consequently, composite. Therefore, $p = (p-9) + 9$ is the desired representation.

On the other hand, it can be directly verified that the numbers 2, 3, 5, 7, and 11 cannot be represented as the sum of two composite numbers.