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 $\angle MHN=\angle MCK$ .

Try more proof problems at Olympiad Proof Problems.

#ProofTechniques #Olympiad #ProofProblems #SuryaPrakash #OlympiadProofProblems

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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}`	$\boxed{123}$

Comments

Sort by:

Top Newest

Just for clarification, in Q1, can two primes be the same, i.e. is it possible to have 5 and 5 as two of the primes?

Sharky Kesa - 4 years, 1 month ago

Yes

Surya Prakash - 4 years, 1 month ago

Q1) A bit of a sleepy proof (I'm gonna sleep soon). Tell me my errors and I'll rectify them.

Firstly, note that the LHS must be a multiple of 10, so 2 of the primes (we'll call them $p_1$ and $p_2$ are 2 and 5. We now have

$p_1 p_2 \ldots p_n = 10(p_1+p_2+\ldots+p_n)$

$p_3 p_4 \ldots p_n = 7+p_3+p_4+\ldots+p_n$

WLOG $p_3 \leq p_4 \leq \ldots \leq p_n$

We will now prove that there is only a maximum of 4 terms in this sequence. Assume that there are $k$ terms for $k \geq 5$ .

$p_3 p_4 \ldots p_k = p_3 + p_4 + \ldots + p_k+7$

We have $p_3+p_4+\ldots+p_k +7 \leq (k-2) p_k + 7$ and $p_3 p_4 \ldots p_k \leq p_k^{k-2}$ . We can prove via induction (or other methods) that $(k-2)p_k+7 < p_k^{k-2}$ for $p_k \geq 3, k \geq 5$ . Thus, no solutions exist when there are 5 or more terms.

We now have 3 cases:

Case 1: 4 terms in the sequence

$p_1 p_2 p_3 p_4 = 10(p_1+p_2+p_3+p_4)$

$p_3 p_4 = 7 + p_3 + p_4$

$(p_3 - 1)(p_4 - 1) = 8$

We must have $p_3 - 1 =2$ and $p_4-1=4$ so $p_3=3$ and $p_4 =5$ . Thus, we have solution $2, 3, 5, 5$ .

Case 2: 3 terms in the sequence

$p_1 p_2 p_3 = 10(p_1+p_2+p_3)$

$p_3 = 7 + p_3$

No solutions here.

Case 3: 2 terms in the sequence

$p_1 p_2 = 10(p_1+p_2)$

$1 = 7$

No solutions here.

Therefore, only $2, 3, 5, 5$ is the solution that satisfies.

Sharky Kesa - 4 years, 1 month ago

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}`	$\boxed{123}$