# Problem 3. (2 points)

What is the maximum number of different reduced quadratic equations that can be written on the board, given that any two of them have a common root, but no four have a root common to all.
Answer: 3

Solution:

Consider three such equations. There are two cases:

1) These equations have a common root $a$. Then they can be represented in the form $(x-a)(x-b)=0, (x-a)(x-c)=0$ and $(x-a)(x-d)=0$, where $b, c$ and $d$ are their remaining roots (they are not equal to each other, otherwise the equations coincide; one of these numbers can coincide with $a$, which is not important).

The fourth equation cannot have the root $a$; then it must have the roots $b, c$ and $d$ simultaneously, which means it is not quadratic.

We get a contradiction.

2) The equations do not have a common root. Then they can be represented as $(x-a)(x-b)=0$, $(x-a)(x-c)=0$ and $(x-b)(x-c)=0$. In this case, the fourth equation can only have one of the roots $a, b$ and $c$, otherwise it coincides with one of the previously described.

If it has a root, for example, $a$, then it cannot have a common root with $(x-$ $b)(x-c)=0$. Again, we get a contradiction.

Therefore, there cannot be more than three different equations.

An example can easily be constructed for the first case by taking arbitrary numbers $a, b, c, d$.

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