Q87. If a function $f$ satisfies $f ( \mathrm {~m} + \mathrm { n } ) = f ( \mathrm {~m} ) + f ( \mathrm { n } )$ for all $\mathrm { m } , \mathrm { n } \in \mathbf { N }$ and $f ( 1 ) = 1$, then the largest natural number $\lambda$ such that $\sum _ { k = 1 } ^ { 2022 } f ( \lambda + k ) \leq ( 2022 ) ^ { 2 }$ is equal to $\_\_\_\_$ Q88. Let $f : ( 0 , \pi ) \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { c c } \left( \frac { 8 } { 7 } \right) ^ { \frac { \tan 8 x } { \tan 7 x } } , & 0 < x < \frac { \pi } { 2 } \\ \mathrm { a } - 8 , & x = \frac { \pi } { 2 } \\ ( 1 + | \cot x | ) ^ { \mathrm { b } } | \tan x | , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ where $\mathrm { a } , \mathrm { b } \in \mathbf { Z }$. If $f$ is continuous at $x = \frac { \pi } { 2 }$, then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 }$ is equal to
Q87. If a function $f$ satisfies $f ( \mathrm {~m} + \mathrm { n } ) = f ( \mathrm {~m} ) + f ( \mathrm { n } )$ for all $\mathrm { m } , \mathrm { n } \in \mathbf { N }$ and $f ( 1 ) = 1$, then the largest natural number $\lambda$ such that $\sum _ { k = 1 } ^ { 2022 } f ( \lambda + k ) \leq ( 2022 ) ^ { 2 }$ is equal to $\_\_\_\_$
Q88.\\
Let $f : ( 0 , \pi ) \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { c c } \left( \frac { 8 } { 7 } \right) ^ { \frac { \tan 8 x } { \tan 7 x } } , & 0 < x < \frac { \pi } { 2 } \\ \mathrm { a } - 8 , & x = \frac { \pi } { 2 } \\ ( 1 + | \cot x | ) ^ { \mathrm { b } } | \tan x | , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ where $\mathrm { a } , \mathrm { b } \in \mathbf { Z }$. If $f$ is continuous at $x = \frac { \pi } { 2 }$, then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 }$ is equal to