Olympiad Proof Problem - Day 7

Find all prime numbers \(p_{1}\), \(p_{2}\), \(p_{3}\) \(\ldots\) \(p_{n}\) such that \[\prod_{i=1}^{n} p_{i} = 10 \sum_{i=1}^{n} p_{i}\]
Given a quadrilateral \(ABCD\) with \(\angle B=\angle D=90^{\circ}\). Point \(M\) is chosen on segment \(AB\) so that \(AD=AM\). Rays \(DM\) and \(CB\) intersect at point \(N\). Points \(H\) and \(K\) are feet of perpendiculars from points \(D\) and \(C\) to lines \(AC\) and \(AN\) respectively. Prove that \(\angle MHN=\angle MCK\).

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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