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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Coloring plane in black
Ryan-asadi   2
N 6 minutes ago by Primeniyazidayi
Source: Iran Team Selection Test - P3
..........
2 replies
Ryan-asadi
3 hours ago
Primeniyazidayi
6 minutes ago
B.Stat & B.Math 2022 - Q8
integrated_JRC   6
N 38 minutes ago by Titeer_Bhar
Source: Indian Statistical Institute (ISI) - B.Stat & B.Math Entrance 2022
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$for real numbers $x$ not multiple of $\frac{\pi}{2}$.
6 replies
integrated_JRC
May 8, 2022
Titeer_Bhar
38 minutes ago
AD=BE implies ABC right
v_Enhance   117
N 42 minutes ago by cj13609517288
Source: European Girl's MO 2013, Problem 1
The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.
117 replies
v_Enhance
Apr 10, 2013
cj13609517288
42 minutes ago
IMO Genre Predictions
ohiorizzler1434   64
N an hour ago by ariopro1387
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
64 replies
ohiorizzler1434
May 3, 2025
ariopro1387
an hour ago
3-var inequality
sqing   2
N an hour ago by pooh123
Source: Own
Let $ a,b\geq  0 ,a^3-ab+b^3=1  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{1}{3}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{1}{3}$$Let $ a,b\geq  0 ,a^3+ab+b^3=3  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{2\sqrt[3]{9}}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{\sqrt[3]{3}}{5}$$
2 replies
sqing
Today at 4:32 AM
pooh123
an hour ago
3-var inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b>0 $ and $\frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \leq \frac{1}{2} . $ Prove that
$$a^2+ab+b^2\geq 3$$$$a^2-ab+b^2 \geq 1 $$Let $ a,b>0 $ and $\frac{1}{a^3+3}+ \frac{1}{b^3+ 3}\leq \frac{1}{2} . $ Prove that
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
1 reply
sqing
an hour ago
sqing
an hour ago
Iranians playing with cards module a prime number.
Ryan-asadi   2
N 2 hours ago by AshAuktober
Source: Iranian Team Selection Test - P2
.........
2 replies
Ryan-asadi
3 hours ago
AshAuktober
2 hours ago
An analytic sequence
Ryan-asadi   1
N 2 hours ago by AshAuktober
Source: Iran Team Selection Test - P1
..........
1 reply
Ryan-asadi
4 hours ago
AshAuktober
2 hours ago
Geometry
gggzul   6
N 2 hours ago by Captainscrubz
In trapezoid $ABCD$ segments $AB$ and $CD$ are parallel. Angle bisectors of $\angle A$ and $\angle C$ meet at $P$. Angle bisectors of $\angle B$ and $\angle D$ meet at $Q$. Prove that $ABPQ$ is cyclic
6 replies
gggzul
Yesterday at 8:22 AM
Captainscrubz
2 hours ago
Need help on this simple looking problem
TheGreatEuler   0
2 hours ago
Show that 1+2+3+4....n divides 1^k+2^k+3^k....n^k when k is odd. Is this possible to prove without using congruence modulo or binomial coefficients?
0 replies
TheGreatEuler
2 hours ago
0 replies
Geometry
Lukariman   5
N 3 hours ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
5 replies
Lukariman
Yesterday at 12:43 PM
Lukariman
3 hours ago
inq , not two of them =0
win14   0
3 hours ago
Let a,b,c be non negative real numbers such that no two of them are simultaneously equal to 0
$$\frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a} \ge \frac{5}{2\sqrt{ab + bc + ca}}.$$
0 replies
win14
3 hours ago
0 replies
Number theory
MathsII-enjoy   5
N 3 hours ago by MathsII-enjoy
Prove that when $x^p+y^p$ | $(p^2-1)^n$ with $x,y$ are positive integers and $p$ is prime ($p>3$), we get: $x=y$
5 replies
MathsII-enjoy
Monday at 3:22 PM
MathsII-enjoy
3 hours ago
Combinatorics
P162008   4
N 4 hours ago by cazanova19921
Let $m,n \in \mathbb{N}.$ Let $[n]$ denote the set of natural numbers less than or equal to $n.$

Let $f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$

Compute the sum of the digits of $f(4,4).$
4 replies
P162008
Today at 5:38 AM
cazanova19921
4 hours ago
IMO ShortList 1998, number theory problem 1
orl   55
N Yesterday at 4:00 PM by reni_wee
Source: IMO ShortList 1998, number theory problem 1
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
55 replies
orl
Oct 22, 2004
reni_wee
Yesterday at 4:00 PM
IMO ShortList 1998, number theory problem 1
G H J
Source: IMO ShortList 1998, number theory problem 1
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orl
3647 posts
#1 • 9 Y
Y by Davi-8191, mathematicsy, samrocksnature, Adventure10, megarnie, Mango247, Tastymooncake2, MAS5236, ItsBesi
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
Attachments:
This post has been edited 2 times. Last edited by orl, Oct 23, 2004, 12:38 PM
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orl
3647 posts
#2 • 5 Y
Y by samrocksnature, Adventure10, megarnie, Mango247, Tastymooncake2
Please post your solutions. This is just a solution template to write up your solutions in a nice way and formatted in LaTeX. But maybe your solution is so well written that this is not required finally. For more information and instructions regarding the ISL/ILL problems please look here: introduction for the IMO ShortList/LongList project and regardingsolutions :)
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orl
3647 posts
#3 • 5 Y
Y by samrocksnature, Adventure10, Mango247, MarioLuigi8972, Tastymooncake2
Unchecked solution by nhat:

find all a,b in positive integer such that:
$\frac {a^{2}*b+a+b}{a*b^2+b+7}$ in positive integer
I want the another solution which is different to the solution of IMO
here my solution
for this case a=b we easy to have the solution
if a<b then $a*b*(a-b)+a-7$<0 (contradition)
hence a>b
let a in $(m*b,(m+1)*b)$ (m i positive integer)
then we easy to prove that : m<$\frac {a^{2}*b+a+b}{a*b^2+b+7}$<(m+1) (contradition)
so a divisible to b
give $a=k*b$ (k in positive integer)
then
$\frac{k^2*b^3+k*b+b}{k*b^3+b+7}$=$\frac{b*{k^2*b^2+k+1}}{k*b^3+b+7}$
if $(a,7)=1$ then $(b,k*b^3+b+7)=1$
so that $\frac {k^2*b^2+k+1}{k*b^3+b+7}$ in positive integer
and we esay to prove that k>b
so easy to prove that \[k\equiv 0 (mod b)\] so $k=q*b$ (b >= 3and q i positive integer)
then $\frac {k^2*b^2+k+1}{k*b^3+b+7}$=${\{q^2*b^4+q*b+1}{q*b^4+b+1}$=$q$+$\frac {1-q}{q*b^4+b+1}$ (contradition)
if b divisible to 7 so that $b=7*u$ (u in positive integer)
and easy to prove that u=k (it's similar for this case two)
so we have the solution $a=7*k^2$,$b=7*k$
my solution hasor hasn't correct :D :D :D
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paladin8
3237 posts
#4 • 5 Y
Y by samrocksnature, Adventure10, Mango247, Tastymooncake2, and 1 other user
The solution given isn't quite correct... there are at least two missing solutions: $(11,1)$ and $(49,1)$.

$11^2+11+1 = 133$, which is divisible by $11+1+7 = 19$.
$49^2+49+1 = 2451$, which is divisible by $49+1+7 = 57$.
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paladin8
3237 posts
#5 • 15 Y
Y by acupofmath, DominicanAOPSer, techniqq, Understandingmathematics, myh2910, samrocksnature, Adventure10, Mango247, Tastymooncake2, MAS5236, and 5 other users
My full solution...

If $(xy^2+y+7)|(x^2y+x+y)$, then there exists a positive integer $k$ such that $k(xy^2+y+7) = x^2y+x+y$ , or

$7k-y = (xy+1)(x-ky)$ .

First we claim that all solutions have both sides nonnegative.

Suppose $7k-y < 0$ . But then we have

$|7k-y| < |y| < |xy+1| < |xy+1||x-ky|$ ,

which is a contradiction. So both sides are nonnegative. Now consider the two cases: (1) both sides are zero, and (2) both sides are positive.

Case 1: Both sides are zero.

Then we have $7k-y = 0 \Rightarrow y = 7k$ . And also $(xy+1)(x-ky) = 0$, but since $xy+1 > 0$ we know $x-ky = 0 \Rightarrow x = ky = 7k^2$ . So we have all solutions of the form $(x,y) = (7k^2, 7k)$ .

Case 2: Both sides are positive.

Then $x > ky$ so the RHS is positive. So we have $7k > 7k-y = (xy+1)(x-ky) > (xy+1) > y^2k+1$. Hence $y < 3$ . So we check $y = 1, 2$ .

For $y = 1$ , we have $(x+8)|(x^2+x+1)$. Since

$\frac{x^2+x+1}{x+8} = x-7+\frac{57}{x+8}$ ,

we have $(x+8)|57 \Rightarrow x = 11, 49$ , giving the solutions $(11,1)$ and $(49,1)$ .

For $y = 2$ , we have $(4x+9)|(2x^2+x+2) \Rightarrow (4x+9)|(4x^2+2x+4)$ . Since

$\frac{4x^2+2x+4}{4x+9} = x-\frac{7x-4}{4x+9}$ ,

so $(4x+9)|(7x-4)$ . But since $2(4x+9) > 7x-4$ , we must have $4x+9 = 7x-4$ , which does not have an integer solution.

Hence our only solutions are $(x,y) = (11,1); (49,1); (7k^2, 7k)$ for positive integers $k$ .
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abdurashidjon
119 posts
#6 • 6 Y
Y by techniqq, samrocksnature, Adventure10, Mango247, Tastymooncake2, and 1 other user
Here my solution I hope there are no mistakes
Let $ab^2+b+7=n$ then $ab^2\cong -b-7 (mod n) \rightarrow a^2b^2\cong -ab-7a (mod n)$
and $a^2b\cong -a-b (mod n) \rightarrow a^2b^2\cong -ab-b^2(mod n)$ from here
$\rightarrow 7a \cong b^2(mod n)$ so $n$ divides both $(ab^2+b+7)$ and $a(b^2-7a)$ so
$(ab^2+b+7)-a(b^2-7a)=b+7+7a^2$ then n divides $b+7+7a^2$ , too.
so $b+7+7a^2\geq ab^2+b+7 \rightarrow 7a \geq b^2$. Lets share it in two parts
1) $7a-b^2=0 \rightarrow (a,b)=(7k^2,7k)$
2) $7a-b^2>0$
Since we have found that $n|(7a-b^2)$ then $7a-b^2\geq ab^2+b+7$ from here we will have
$(7-b^2)(a+1)\geq b+14$. $b+14>0$ then $(7-b^2)(a+1)>0$ also $(a+1)>0$ then we have to have $(7-b^2)>0$ here $b$ is positive integer then $b= 1$ or $2$ otherwise inequality will lead to contradiction
If $b=1$ then the problem will become as $\frac{a^2+a+1}{a+8}$
Let $k=a+8$ then $a^2\cong -a-1 (mod k) (i)$
$a\cong -8(mod k)\rightarrow a^2\cong 64 (mod k) (ii)$
from $(i)$ and $(ii)$ we will have $-a-1\cong 64 (mod k)$ and $a+65$ is divisible by $k$ then

$\frac{a+65}{a+8}=\frac{a+8}{a+8} + \frac{57}{a+8}$ so $\frac{57}{a+8} (iii)$ is an integer, too.
so there are just two solutions for $(iii)$ those are $a=11$ or $49$ ( think didisors of $57$)
If $b=2$ by simliar of previous we will show that there is no solution for $b=2$
So the solutions are $(a,b) = (11,1), (49,1), (7k^2,7k)$
We are done
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dgreenb801
1896 posts
#7 • 3 Y
Y by samrocksnature, Adventure10, Mango247
If $ x < y$ then it is easy to see there are no solutions.
By the division algorithm, we can let $ x = ay + b$, where $ 0 \le b < y$.
Then we have
$ a^2y^3 + 2aby^2 + (b^2 + a + 1)y + b$ is divisible by $ ay^3 + by^2 + y + 7$
Let the ratio between these two numbers be $ a + k$, then
$ a^2y^3 + 2aby^2 + (b^2 + a + 1)y + b = (a + k)(ay^3 + by^2 + y + 7)$
$ aby^2 + b^2y + y + b = aky^3 + kby^2 + ky + 7a + 7k$
If $ k = 0$, then $ aby^2 + b^2y + y + b = 7a$
Case 1:$ b = 0$
Then $ y = 7a$ and we have the solution $ (7a^2,7a)$
Case 2: $ b \ge 0$
Then either $ y = 1$ or $ 2$. If $ y = 1$, then $ b$ can't be more than $ 7$, we try out all values from $ 1$ to $ 7$ for $ b$. $ b = 4$ and $ b = 6$ works, this gives the solutions $ (11,1)$ and $ (49,1)$. If $ y = 2$ then $ b = 1$ and we have no solution.
If $ k > 0$, then since $ b < y$, we must have $ k = 1$ since if $ k > 1$ then the left side is less than the right side (we would have $ aky^3 > aby^2 + b$, $ kby^2 > b^2y$, $ ky > y$). But if $ k = 1$, then $ b = 7a + 7 + (y - b)(ay^2 + by)$. But $ y - b \ge 1$ since $ b < y$, a contradiction.
So $ k < 0$. Then we must have $ y = 1$ or the left side would be greater than the right. Since $ 0 \le b < y$, we must have $ b = 0$. Thus, $ 1 = ak + 7a + 8k$, or $ (a + 8)(k + 7) = 57$. We find $ (a,k) = (11, - 3)$ and $ (49, - 6)$ work, giving the solutions $ (11,1)$ and $ (49,1)$.
So the three solutions are $ (7a^2,7a)$, $ (11,1)$, and $ (49,1)$
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KrazyFK
367 posts
#8 • 6 Y
Y by jam10307, samrocksnature, Adventure10, Mango247, and 2 other users
I thought to myself this morning - "I haven't really tried any IMO problems". I decided to remedy that by solving this one.

First, suppose that $ gcd(y,7)=1$. Then the problem can be restated as follows.

$ xy^2+y+7 \, | \, x^2y+x+y$

$ \Leftrightarrow xy^2+y+7 \, | \, y(x^2y+x+y)-x(xy^2+y+7)$

$ \Leftrightarrow xy^2+y+7 \, | \, y^2-7x$

It is easy to see that since $ x,y$ are positive, then $ xy^2+y+7 > y^2-7x$. Then we must have either $ y^2=7x$ or $ y^2-7x < 0$. The first case cannot arise since $ gcd(7,y)=1$ so we must have $ y^2-7x < 0$.

Then our condition is equivalent to:

$ xy^2+y+7 | 7x - y^2$ and we must have $ 7x - y^2 \ge xy^2 + y + 7$

The second condition implies that $ y < 3$ since otherwise we have:

$ (7x - y^2) - (xy^2+x+y) \le 7x - 9 -9x -x-3 = -3x - 12 < 0$.

Hence $ y=1$ or $ y=2$.

Case 1: $ y=1$.

Our problem restates as $ x+8 | 7x-1$ and so we write:

$ \dfrac{7x-1}{x+8} = k$.

It is clear that $ k$ must be positive. We rewrite this as:

$ x=\dfrac{8k+1}{7-k}$ and since $ x$ is positive, the denominator must be positive and so $ k < 7$. Substituting the values $ k=1,2,3,4,5,6$ into this expression in turn shows that only $ k=4$ and $ k=6$ give integer values for $ x$. These are $ x=11$ and $ x=49$.

So our two solutions are $ (x,y)=(49,1), (11,1)$. Checking in the original problem reveals that they work.

Case 2: $ y=2$.

Starting in the same way as above we get $ x= \dfrac{9k+4}{7-4k}$. In a similar way to above we must have $ k < 2$, but substituting $ k=1$ does not give integer $ x$. Hence, no solutions in this case.


Now we must consider when $ gcd(7,y)=7$. This case is easy. Let $ y=7t$. Any pair $ (x,y)$ which satisfies the problem must also satisfy:

$ xy^2+y+7 | y(x^2y+x+y)-x(xy^2+y+7)$

$ xy^2+y+7 | y^2-7x$

We cannot have $ y=1$ or $ y=2$ as above since $ gcd(y,7)=7$. Thus we must have the other possibility, $ y^2=7x$.

Then we have $ 7x=y^2=49t^2$

So $ x=7t^2$

Then our solution is $ (x,y)=(7t^2,7t)$.

Checking this in the original problem:

$ \dfrac{x^2y+x+y}{xy^2+y+7}$

$ =\dfrac{343t^5+7t^2+7t}{343t^4+7t+7}$

$ =t$ which is an integer.

Hence all solutions are given by:

$ (x,y) = (11,1), (49,1), (7t^2,7t)$
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JSGandora
4216 posts
#9 • 3 Y
Y by samrocksnature, Adventure10, Mango247
KrazyFK wrote:
First, suppose that $ gcd(y,7)=1$. Then the problem can be restated as follows.



$ xy^2+y+7 \, | \, x^2y+x+y$



$ \Leftrightarrow xy^2+y+7 \, | \, y(x^2y+x+y)-x(xy^2+y+7)$



$ \Leftrightarrow xy^2+y+7 \, | \, y^2-7x$
What is the importance of the statement $\gcd(y,7)=1$? Can you not perform the same manipulations even if $\gcd(y,7)=7$?

Also, my solution involved using the division algorithm to get $\frac{a^2b+a+b}{ab^2+b+7}=\frac{a}{b}+\frac{b^2-7a}{ab^3+b^2+7b}$, can this only be done if $a\geq b$? From there I get $b^2-7a=0\implies b^2=7a\implies a=7k,b=7k^2$. Why can't I use the division algorithm to get the same result if $b<a$?
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GlassBead
1583 posts
#10 • 3 Y
Y by samrocksnature, Adventure10, Mango247
The statement of $\gcd(7, y)=1$ is not actually used for the algebraic manipulations, but to set aside a case that forces $y^2-7x$ to be positive (in other words, it's used right after your quote).
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JSGandora
4216 posts
#11 • 3 Y
Y by samrocksnature, Adventure10, Mango247
Oh, so from $ \frac{a^{2}b+a+b}{ab^{2}+b+7}=\frac{a}{b}+\frac{b^{2}-7a}{ab^{3}+b^{2}+7b} $, we must split it into cases. So I forgot the case where $b^2-7a<0$, so $7a-b^2>\frac{ab^{3}+b^{2}+7b}{b}$.
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GlassBead
1583 posts
#12 • 3 Y
Y by samrocksnature, Adventure10, Mango247
Yep.

The $b^2-7a<0$ case leads to the two extra solutions, $(49, 1); (11, 1)$.
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JSGandora
4216 posts
#13 • 3 Y
Y by samrocksnature, Adventure10, Mango247
Thanks, but just from the division algorithm, what is the concrete rational that we must have $ 7a-b^{2}>\frac{ab^{3}+b^{2}+7b}{b} $ (division by $b$ on the RHS)? Is it just that if we factor out the $\frac1b$ on the RHS of $ \frac{a^{2}b+a+b}{ab^{2}+b+7}=\frac{a}{b}+\frac{b^{2}-7a}{ab^{3}+b^{2}+7b} $, we just want $a+\frac{b^2-7a}{ab^2+b+7}$ to be an integer?
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GlassBead
1583 posts
#14 • 4 Y
Y by JSGandora, samrocksnature, Adventure10, Mango247
That's right. From your division algorithm, if we want the LHS to be an integer, then $b$ has to divide $a+\frac{b^2-7a}{ab^2+b+7}$, so it must be an integer, and so $ab^2+b+7$ must divide $7a-b^2$.
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SoyunSerInerte
6 posts
#15 • 2 Y
Y by samrocksnature, Adventure10
Define $S=\left \{(a,b)\in \mathbb{Z}^{+}|\displaystyle \frac{a^2b+a+b}{ab^2+b+7}\in \mathbb{Z}^{+}\right \}$

We note that (i)

$ab^2+b+7|a^2b+a+b\Rightarrow ab^2+b+7|a(b(a^2b+a+b)-a(ab^2+b+7))$

then $ab^2+b+7|ab^2-7a^2\Rightarrow ab^2+b+7|ab^2-7a^2-(ab^2+b+7)$

now we have that

$ab^2+b+7|-7a^2-b-7\Rightarrow ab^2+b+7|7a^2+b+7$

note that $ab^2+b+7>0$ and $7a^2+b+7>0$

therefore $ab^2+b+7\leq 7a^2+b+7\Rightarrow b^2\leq 7a$

if $b^2=7a$ then $a=7t^2$ and $b=7t$ for all $t\in \mathbb{Z}^{+}$

therefore $(7t^2,7t)\in S$

now $b^2<7a$

By (i) we have that

$ab^2+b+7|b^2-7a\Rightarrow ab^2+b+7|7a-b^2$ because $b^2<7a$

now because $b^2<7a$ we have $7a-b^2>0$ and note that $ab^2+b+7>0$

therefore $ab^2+b+7\leq 7a-b^2$ that is impossible for $b\ge 3$ therefore $b<3$

We have two cases:

Case 1: $b=1$

we have $a+8|a^2+a+1\Rightarrow a+8|a^2+a+1-a(a+8)+7(a+8)$

therefore $a+8|57=1\cdot 3\cdot 19$ if we prove all the divisors we have that $(49,1)$ and $(11,1)$ are solutions.

therefore $(11,1),(49,1)\in S$

Case 2: $b=2$

we have $4a+9|2a^2+a+2\Rightarrow 4a+9|4(2(2a^2+a+2)-a(4a+9))+7(4a+9)=79$

therefore $4a+9|79$ no solutions here

Finally we have that $S=\{(11,1),(49,1),(7t^2,7t)\}$

Thanks
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