Some more problems...

Try the followings:

Consider a prism with a triangular base. The total area of the three faces containing a particular vertex $A$ is $K$ . Show that the maximum possible volume of the prism is $\displaystyle \sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.
Consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$ . Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from ach row and one from each column.
If $a_1, \cdots, a_7 \in (1,13)$ are not necessarily distinct reals, show that we can choose three of them such that they are lengths of the sides of a triangle.
Show that there cannot exist a non-constant polynomial $P(x) \in \mathbb{Z}[x]$ such that $P(n)$ is prime for all positive integers $n$ .
(Calculus) Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$ . Suppose that $f$ also satisfies $f(x) \geq 0$ , $f'(x) \leq 0$ and $f''(x) \leq f(x)$ , for all $x \geq 0$ . Show that $f'(0) \geq - \sqrt{2}$ .

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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