2.15. $C=19 \pi^{2} / 32$. Answer: 0.15. Here $\alpha=\frac{3 \pi}{4}, \beta=\frac{\pi}{8}$. The second triangle with $\beta=\frac{27 \pi}{40}$ does not exist. The exact answer is $\pi^{2} / 64$.

III. Find all pairs of positive numbers $x, y$ that satisfy the equation

$$
\begin{aligned}
& \frac{4 x^{2} y+6 x^{2}+2 x y-4 x}{3 x-y-2}+\sin \left(\frac{3 x^{2}+x y+x-y-2}{3 x-y-2}\right)=2 x y+y^{2}+\frac{x^{2}}{y^{2}}+\frac{2 x}{y}+ \\
& \quad+\frac{2 x y\left(x^{2}+y^{2}\right)}{(3 x-y-2)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \sin \frac{(x+y)^{2}}{x}+y^{2} \sin \frac{(x+y)^{2}}{y^{2}}+2 x y \sin \frac{(x+y)^{2}}{3 x-y-2}\right)
\end{aligned}
$$

In the answer, write the sum of the values $x^{2}+y^{2}$ for all obtained pairs of solutions $(x, y)$, rounding to two decimal places if necessary. If there are no solutions, write -1; if there are infinitely many solutions, write -2.
Solution. The proposed equation can be rewritten as

$$
f\left(\alpha_{1} u_{1}+\alpha_{2} u_{2}+\alpha_{3} u_{3}\right)=\alpha_{1} f\left(u_{1}\right)+\alpha_{2} f\left(u_{2}\right)+\alpha_{3} f\left(u_{3}\right),
$$

where

$$
\begin{gathered}
\alpha_{1}=\frac{x^{2}}{(x+y)^{2}}, \alpha_{2}=\frac{y^{2}}{(x+y)^{2}}, \alpha_{3}=\frac{2 x y}{(x+y)^{2}}, \quad \alpha_{1}+\alpha_{2}+\alpha_{3}=1, \quad \alpha_{1}>0, \alpha_{2}>0, \alpha_{3}>0 \forall x, y>0 ; \\
u_{1}=\frac{(x+y)^{2}}{x}, u_{2}=\frac{(x+y)^{2}}{y^{2}}, u_{3}=\frac{(x+y)^{2}}{3 x-y-2}
\end{gathered}
$$

and the function $f(t)=t^{2}+\sin t-$ is strictly convex on the entire number line. By Jensen's inequality, such equality is possible only if $u_{1}=u_{2}=u_{3}$, from which we have, taking into account $x>0, y>0$

$$
x=y^{2}=3 x-y-2 \Rightarrow 2 y^{2}-y-2=0 \Rightarrow y=\frac{1+\sqrt{17}}{4}, x=\frac{9+\sqrt{17}}{8}
$$

Therefore, $x^{2}+y^{2}=\frac{85+13 \sqrt{17}}{32}=4.331 \ldots$

Answer: 4.33

III. 1 Find all pairs of positive numbers $x, y$ that satisfy the equation

$$
\begin{aligned}
& \frac{4 x^{2} y+6 x^{2}+2 x y-4 x}{3 x-y-2}+\sin \left(\frac{3 x^{2}+x y+x-y-2}{3 x-y-2}\right)=2 x y+y^{2}+\frac{x^{2}}{y^{2}}+\frac{2 x}{y}+ \\
& \quad+\frac{2 x y\left(x^{2}+y^{2}\right)}{(3 x-y-2)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \sin \frac{(x+y)^{2}}{x}+y^{2} \sin \frac{(x+y)^{2}}{y^{2}}+2 x y \sin \frac{(x+y)^{2}}{3 x-y-2}\right)
\end{aligned}
$$

In the answer, write the sum of the values $x^{2}+y^{2}$ for all obtained pairs of solutions $(x, y)$, rounding to two decimal places if necessary. If there are no solutions, write -1; if there are infinitely many solutions, write -2.

Answer: 4.33

III. 2 Find all pairs of positive numbers $x, y$ that satisfy the equation

$$
\begin{aligned}
& \frac{2 x^{2}+14 x y-4 y^{2}+2 x y^{2}-12 y}{2 x-y-3}+\cos \left(\frac{x^{2}+4 x y-2 y^{2}-7 y+2 x-3}{2 x-y-3}\right)=\frac{x^{2}}{y^{2}}+\frac{2 y^{3}}{x}+\frac{2 x}{y}+ \\
& \quad+\frac{x y\left(x y+2 x^{2}\right)}{(2 x-y-3)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \cos \frac{(x+y)^{2}}{2 x-y-3}+2 x y \cos \frac{(x+y)^{2}}{x}+y^{2} \cos \frac{(x+y)^{2}}{y^{2}}\right)
\end{aligned}
$$

In the answer, write the sum of the values $x^{2}+y$ for all obtained pairs of solutions $(x, y)$, rounding to two decimal places if necessary. If there are no solutions, write -1; if there are infinitely many solutions, write -2.

Answer: $16+4 \sqrt{13} \approx 30.42$

Solutions $x=(7+\sqrt{13}) / 2, y=(1+\sqrt{13}) / 2$.

III. 3 Find all pairs of positive numbers $x, y$ that satisfy the equation

$$
\begin{aligned}
& \frac{14 x y+4 y^{2}-2 x^{2} y-8 y}{x+2 y-4}+\cos \left(\frac{2 y^{2}+3 x y+x-2 y-4}{x+2 y-4}\right)=\frac{y^{2}}{x^{2}}+\frac{2 y}{x}+x^{2}+ \\
& \quad+\frac{2 x y\left(x^{2}+y^{2}\right)}{(x+2 y-4)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \cos \frac{(x+y)^{2}}{x^{2}}+y^{2} \cos \frac{(x+y)^{2}}{y}+2 x y \cos \frac{(x+y)^{2}}{x+2 y-4}\right)
\end{aligned}
$$

In the answer, write the sum of the values $x+y^{2}$ for all obtained pairs of solutions $(x, y)$, rounding to two decimal places if necessary. If there are no solutions, write -1; if there are infinitely many solutions, write -2.

Answer: $24-4 \sqrt{17} \approx 7.51$

Solutions $x=(\sqrt{17}-1) / 2, y=(9-\sqrt{17}) / 2$.

III. 4 Find all pairs of positive numbers $x, y$ that satisfy the equation

$$
\begin{aligned}
& \frac{6 x y^{2}-6 x^{2} y+12 x^{2}+2 y^{2}-10 x y+12 x}{3 x-y+3}+\sin \left(\frac{6 x^{2}-2 x y+y^{2}+9 x-y+3}{3 x-y+3}\right)=\frac{y^{2}}{x^{2}}+\frac{2 x^{3}}{y}+\frac{2 y}{x}+ \\
& +\frac{x y\left(x y+2 y^{2}\right)}{(3 x-y+3)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \sin \frac{(x+y)^{2}}{x^{2}}+2 x y \sin \frac{(x+y)^{2}}{y}+y^{2} \sin \frac{(x+y)^{2}}{3 x-y+3}\right)
\end{aligned}
$$

In the answer, write the sum of the values $x+2 y$ for all obtained pairs of solutions $(x, y)$, rounding to two decimal places if necessary. If there are no solutions, write -1; if there are infinitely many solutions, write -2.

Answer: $6+\sqrt{33} \approx 11.74$

Solutions $x=(3+\sqrt{33}) / 4, y=(21+3 \sqrt{33}) / 8$.