RMO adventure

1) Solve the equation $y^3 = x^3 + 8x^2 - 6x + 8$ for positive integer x and y.

2) Consider two positive integer a and b which are such that $a^{a}b^{b}$ is divisible by 2000. What is minimum value of ab?

#NumberTheory

Easy Math Editor

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

Solution for $2^{nd}$ question :

$2000|a^{a}b^{b}\Rightarrow 2|a\text{ or }b \text{ and } 5|a \text{ or } b. \\ \text{Therefore, in every case } 10|ab. \\ \text{Thus, the minimum value of }ab \text{ is obviously }\boxed{10}\text{, possible cases }10^{10}1^{1} \text{ and }1^{1}10^{10}.$

Akshat Sharda - 5 years, 8 months ago

Done very well!!!

Dev Sharma - 5 years, 8 months ago

Can you post solution of the first one ?

Akshat Sharda - 5 years, 7 months ago

1)x=0,y=2

naitik sanghavi - 5 years, 7 months ago

x=9,y=11

Can you provide its solution ?

Akshat Sharda - 5 years, 7 months ago

Here is the solution: $y^3=x^3+8x^2-6x+8<x^3+9x^2+27x+27=(x+3)^3 \text{because}\\ x^3=x^3\\ 8x^2<9x^2\\ -6x<27x\\ 8<27$ .Now,let us see when $y^3$ is greater than or equal to $(x+2)^3$ $\Longrightarrow x^3+6x^2+12x+8 \leq x^3+8x^2-6x+8\\ \Longrightarrow 9 \leq x$ .Now,we have that for only one value of $x$ can the given expression be a cube and that happens when, $x=9,y=(x+2)=11$ .And done!

Adarsh Kumar - 5 years, 7 months ago

@Adarsh Kumar – The proof why $y < (x+3)$ .

Consider $(x+3)^3- y^3 = x^2 +33x +19$ . Discriminant of this quadratic $\Delta < 0$ and since the coefficient of $x^2$ is positive, it implies that $(x+3)^3 -y^3$ is always positive. So, $y^3 < (x+3)^3$ . So, $y < x+2$ .

Surya Prakash - 5 years, 7 months ago

@Adarsh Kumar – Actually I have left an integral part,I haven't proved that for $x \leq 8$ ,there are no solutions.

Adarsh Kumar - 5 years, 7 months ago

Sorry ,0 is not an positive integer!!

naitik sanghavi - 5 years, 7 months 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}`	$\boxed{123}$