In the case of Sándor Mátyás, the leader of the patriots sent an equation to his friends:

$$
2 x^{2}+4 x y+7 y^{2}-12 x-2 y+N=0
$$

The secret message was that the uprising should take place on the $N$-th day of the month. The patriots often said that the uprising was the only solution. This also characterized $N$: the equation was satisfied by only one pair of real numbers $(x, y)$. What is the value of $N$?
I. solution. Consider the given equation as a second-degree parametric equation in $y$. If the equation in two variables has only one solution pair, then the equation

$$
7 y^{2}+(4 x-2) y+2 x^{2}-12 x+N=0
$$

must also have only one solution $y_{0}$, meaning its discriminant, $D_{1}=0$:

$$
D_{1}=(4 x-2)^{2}-28\left(2 x^{2}-12 x+N\right)=-40 x^{2}+320 x-28 N+4=0, \quad \text { rearranging } 10 x^{2}-80 x+(7 N-1)=0
$$

Since the original equation is satisfied by only one $x_{0}$, this equation must also have only one solution, meaning its discriminant, $D_{2}=0$:

$$
D_{2}=80^{2}-40(7 N-1)=0 \Rightarrow 7 N = 6401
$$

Thus, $N=23$.

We have found that if (1) has only one solution $\left(x_{0}, y_{0}\right)$, then the value of $N$ must be 23. However, we need to check if when $N=23$, we indeed get a solution: substituting $x=4$ and $y=-1$ indeed gives the solution to the equation.

Therefore, the uprising took place on (October) 23rd.

Zoltán Szilasi (Debrecen, KLTE Gyak. Gimn., 11th grade)

II. solution. Transform the equation to be equivalent to the original:

$$
2(x+y-3)^{2}+5(y+1)^{2}+N-23=0
$$

This equation has exactly one solution if and only if $N=23$. We can introduce new unknowns $x_{1}=x+y-3$, $y_{1}=y+1$, for which $x=x_{1}-y_{1}+4$ and $y=y_{1}-1$, so the correspondence between $(x, y)$ and $\left(x_{1}, y_{1}\right)$ number pairs is one-to-one. It is clear that the equation $2 x_{1}^{2}+5 y_{1}^{2}=M$ has an even number of solutions if $M \neq 0$, since if, for example, $x_{1} \neq 0$ is part of a solution, then $-x_{1}$ is also.

Remarks. 1. Those who did not check if there is indeed a single solution when $N=23$ received only 3 points.

2. The equation (1) is otherwise an equation of a so-called second-order elliptic curve, which has only one solution if and only if the curve is a point ellipse, that is, if the determinant of its matrix is 0. In his second solution, Zoltán Szilasi calculated the value of $N$ from this.