Let $x$ and $y$ are reals satisfying $5x^2 + 4xy + 11y^2 = 3. $ Prove that $$xy - 2x + 5y\leq \frac{13}{4}$$

Good.

somebodyyouusedtoknow wrote: It is known that $x$ and $y$ are reals satisfying \[ 5x^2 + 4xy + 11y^2 = 3. \]Without using calculus (differentials/integrals), determine the maximum value of $xy - 2x + 5y$. Are you the author?

No I'm not, I took it from the Indonesian RMO.

sqing wrote: Let $x$ and $y$ are reals satisfying $5x^2 + 4xy + 11y^2 = 3. $ Prove that $$xy - 2x + 5y\leq \frac{13}{4}$$Let $x$ and $y$ are reals satisfying $ x^2 + 2xy + 3y^2 =4. $ Prove that$$xy - 2x + 4y \leq \frac{20}{3}$$ Solution of Zhangyanzong:

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somebodyyouusedtoknow wrote: No I'm not, I took it from the Indonesian RMO. thank you

Let $x$ and $y$ are reals satisfying $ x^2 + 2xy + 3y^2 =4. $ Prove that$$xy - 2x + 4y \leq \frac{20}{3}$$Let $x$ and $y$ are reals satisfying $xy - 2x + 4y = \frac{20}{3}. $ Prove that$$ x^2 + 2xy + 3y^2 \geq 4$$Let $x$ and $y$ are reals satisfying $xy - 2x + 5y=\frac{13}{4}. $ Prove that$$5x^2 + 4xy + 11y^2\geq 3$$