## Task B-3.4.

Determine all real numbers $t$ for which one solution of the equation $x^{2}+2024=t x$ is a prime number, and the other is a composite natural number.
## Solution.

Let's write the given equation in the form $x^{2}-t x+2024=0$.

For the solutions of the equation, we have: $x_{1}+x_{2}=t$ and $x_{1} \cdot x_{2}=2024$.

Since $x_{1}$ and $x_{2}$ are natural numbers, they must be divisors of the number 2024.

Let's factorize the number 2024 into prime factors: $2024=2^{3} \cdot 11 \cdot 23$.

The number 2024 can be written as the product of two natural numbers, with exactly one of them being prime, in three different ways: the prime factor can be the number 2, 11, or 23.

1. possibility: $2024=2 \cdot 1012$

Then $t=2+1012=1014$.

2. possibility: $2024=11 \cdot 184$

Then $t=11+184=195$.

3. possibility: $2024=23 \cdot 88$

Then $t=23+88=111$. 1 point

For the real number $t$, we have $t \in\{111,195,1014\}$.