# Problem 6.

B-1

Find the minimum value of the expression $4 x+9 y+\frac{1}{x-4}+\frac{1}{y-5}$ given that $x>4$ and $y>5$.
Answer: 71

Solution. $4 x+9 y+\frac{1}{x-4}+\frac{1}{y-5}=4 x-16+\frac{4}{4 x-16}+9 y-45+\frac{9}{9 y-45}+61 \geq 2 \sqrt{4}+2 \sqrt{9}+61=$ 71 (inequality of means). Equality is achieved when $4 x-16=2,9 y-45=3$, that is, when $x=\frac{9}{2}, y=\frac{16}{3}$.

B-2

Find the minimum value of the expression $9 x+4 y+\frac{1}{x-3}+\frac{1}{y-4}$ given that $x>3$ and $y>4$.

Answer: 53

## B-3

Find the minimum value of the expression $4 x+9 y+\frac{1}{x-1}+\frac{1}{y-2}$ given that $x>1$ and $y>2$.

Answer: 32

## B-4

Find the minimum value of the expression $9 x+4 y+\frac{1}{x-5}+\frac{1}{y-6}$ given that $x>5$ and $y>6$.

Answer: 79

## Lomonosov Olympiad for Schoolchildren in Mathematics

Preliminary Stage 2020/21 Academic Year for 9th Grade

## B-1

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