## Task 3A - 221233A

a) Investigate whether there exists a smallest real number $p$ with the following property:

In every convex quadrilateral, for the side lengths $a, b, c, d$ and the area $F$ of the quadrilateral,

$$
F \leq p \cdot\left(a^{2}+b^{2}+c^{2}+d^{2}\right)
$$

b) Investigate whether there exists a smallest real number $q$ with the following property:

In every triangle, for the side lengths $a, b, c$ and the area $F$ of the triangle,

$$
F \leq q \cdot\left(a^{2}+b^{2}+c^{2}\right)
$$

If such a smallest number $p$ or $q$ exists in a) or b), respectively, determine this number.
a) I. If $A B C D$ is a convex quadrilateral, $F$ its area, and $a=A B, b=B C, c=C D, d=D A$, $\beta=\angle A B C, \delta=\angle C D A$, then

$$
F=\frac{a b}{2} \cdot \sin \beta+\frac{c d}{2} \cdot \sin \delta
$$

because $\sin \beta \leq 1, \sin \delta \leq 1, 0<a b \leq \frac{a^{2}+b^{2}}{2}, 0<c d \leq \frac{c^{2}+d^{2}}{2}$, thus

$$
F \leq \frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right)
$$

II. There are convex quadrilaterals for which, for the side lengths $a, b, c, d$ and the area $F$,

$$
F=\frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right)
$$

holds; for example, there are quadrilaterals with $a=b=c=d$ and $\beta=\delta=90^{\circ}$, which are (squares, thus) convex, and for which $F=a^{2}$, hence (1).

With I. and II., it is proven: There is a smallest real number $p$ with the mentioned property; it is $p=\frac{1}{4}$.

b) I. If $A B C$ is a triangle, $F$ its area, and $a=B C, b=C A, c=A B, s=\frac{1}{2}(a+b+c)$, then by the triangle inequality,

$$
s-a \geq 0 \quad ; \quad s-b \geq 0 \quad ; \quad s-c \geq 0
$$

From this and the inequality between the geometric and arithmetic means of non-negative real numbers, it follows that

$$
\begin{aligned}
F=\sqrt{s(s-a)(s-b)(s-c)} & \leq \sqrt{s\left(\frac{1}{3}(s-a+s-b+s-c)^{3}\right.} \\
& =\frac{1}{3 \sqrt{3}} s^{2}=\frac{1}{12 \sqrt{3}}(a+b+c)^{2} \\
& =\frac{1}{36} \sqrt{3}\left(a^{2}+b^{2}+c^{2}+2 a b+2 a c+2 b c\right) \\
& \leq \frac{1}{36} \sqrt{3}\left(a^{2}+b^{2}+c^{2}+a^{2}+b^{2}+a^{2}+c^{2}+b^{2}+c^{2}\right) \\
& =\frac{1}{12} \sqrt{3}\left(a^{2}+b^{2}+c^{2}\right)
\end{aligned}
$$

II. There are triangles for which, for the side lengths $a, b, c$ and the area $F$,

$$
F=\frac{1}{12} \sqrt{3}\left(a^{2}+b^{2}+c^{2}\right)
$$

holds; for example, for equilateral triangles with side length $a(=b=c)$, $F=\frac{1}{4} \sqrt{3} a^{2}$, hence (2).

With I. and II., it is proven: There is a smallest real number $q$ with the mentioned property; it is $q=\frac{1}{12} \sqrt{3}$.

Adapted from $[5]$