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}`	\( \boxed{123} \)

Excel in math and science