Number Theory
by AnhQuang_67, Apr 16, 2025, 4:42 PM
Find all pairs of positive integers 
satisfying 
is a perfect square
satisfying 
is a perfect squareSimply equation but hard
by giangtruong13, Apr 16, 2025, 3:29 PM
Find all integer pairs 
satisfy that: 
satisfy that: 
Let \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \) be nonzero real nu
by Jackson0423, Apr 16, 2025, 3:06 PM
Let 
and 
be nonzero real numbers satisfying
![\[
a_1^2 b_1^2 (a_1 + b_1) + a_2^2 b_2^2 (a_2 + b_2) + \cdots + a_n^2 b_n^2 (a_n + b_n) \leq 7,
\]](//latex.artofproblemsolving.com/a/7/e/a7ea7af24ac529090f3b2e305be43e2728b79eb4.png)
![\[
\frac{1}{a_1} + \cdots + \frac{1}{a_n} = \frac{1}{4}, \quad \frac{1}{b_1} + \cdots + \frac{1}{b_n} = \frac{1}{3}.
\]](//latex.artofproblemsolving.com/e/f/3/ef341b850979a9b780f5daaa49f46f1d1324cdbd.png)
Find the maximum value of
![\[
a_1 b_1 + a_2 b_2 + \cdots + a_n b_n.
\]](//latex.artofproblemsolving.com/d/3/3/d338bf5be4f1a218e150bfd84173e988ee92ddad.png)
and 
be nonzero real numbers satisfying![\[
a_1^2 b_1^2 (a_1 + b_1) + a_2^2 b_2^2 (a_2 + b_2) + \cdots + a_n^2 b_n^2 (a_n + b_n) \leq 7,
\]](http://latex.artofproblemsolving.com/a/7/e/a7ea7af24ac529090f3b2e305be43e2728b79eb4.png)
![\[
\frac{1}{a_1} + \cdots + \frac{1}{a_n} = \frac{1}{4}, \quad \frac{1}{b_1} + \cdots + \frac{1}{b_n} = \frac{1}{3}.
\]](http://latex.artofproblemsolving.com/e/f/3/ef341b850979a9b780f5daaa49f46f1d1324cdbd.png)
Find the maximum value of![\[
a_1 b_1 + a_2 b_2 + \cdots + a_n b_n.
\]](http://latex.artofproblemsolving.com/d/3/3/d338bf5be4f1a218e150bfd84173e988ee92ddad.png)
Hard Polynomial Problem
by MinhDucDangCHL2000, Apr 16, 2025, 2:44 PM
Let 
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
. Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
. Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).Centroid Distance Identity in Triangle
by zeta1, Apr 16, 2025, 12:28 PM
Let M be any point inside triangle ABC, and let G be the centroid of triangle ABC. Prove that:
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](//latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](http://latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
Divisibility NT FE
by CHESSR1DER, Apr 14, 2025, 7:07 PM
Find all functions 
such for any 
:
.
such for any 
:
.This post has been edited 3 times. Last edited by CHESSR1DER, Yesterday at 6:45 PM
Number Theory Chain!
by JetFire008, Apr 7, 2025, 7:14 AM
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!
?This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM
Numbers not power of 5
by Kayak, Jul 17, 2019, 12:28 PM
Show that there do not exist natural numbers 
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](//latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
This post has been edited 2 times. Last edited by v_Enhance, Feb 8, 2023, 11:43 PM
Reason: don't \cdots a list
Reason: don't \cdots a list
Silly Sequences
by whatshisbucket, Jun 28, 2018, 7:11 AM
Consider infinite sequences 
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
Proposed by Krit Boonsiriseth
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
Proposed by Krit Boonsiriseth
This post has been edited 1 time. Last edited by whatshisbucket, Jun 29, 2018, 1:05 AM
IMO LongList 1985 CYP2 - System of Simultaneous Equations
by Amir Hossein, Sep 10, 2010, 10:57 PM
Solve the system of simultaneous equations
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](//latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](//latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](//latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](//latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
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