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Integer representation
RL_parkgong_0106 1
N
by Jackson0423
Source: Own
Show that for any positive integer 
, there exists some positive integer 
that makes the following equation have no integer root 
.
, there exists some positive integer 
that makes the following equation have no integer root 
.
1 reply
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
Factor of P(x)
Brut3Forc3 19
N
by xytunghoanh
Source: 1976 USAMO Problem 5
If 
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](//latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
19 replies
2^x+3^x = yx^2
truongphatt2668 1
N
by Jackson0423
Prove that the following equation has infinite integer solutions:
1 reply
FE solution too simple?
Yiyj1 7
N
by ariopro1387
Source: 101 Algebra Problems from the AMSP
Find all functions 
such that the equality 
holds for all pairs of real numbers 
.
I feel like my solution is too simple. Is there something I did wrong or something I missed?
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.I feel like my solution is too simple. Is there something I did wrong or something I missed?
7 replies
1 viewing
A cyclic inequality
KhuongTrang 2
N
by NguyenVanDucThang
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
2 replies
Iran second round 2025-q1
mohsen 3
N
by Parsia--
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
3 replies
and 
: Prove that :
2016 Kmo Final round
Jackson0423 0
Source: 2016 FKMO P4
Let 
with 
.
Find the maximum value of
![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](//latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
with 
.Find the maximum value of
![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
0 replies
Factor sums of integers
Aopamy 1
N
by BR1F1SZ
Let be a positive integer. A positive integer is called a benefactor of if the positive divisors of can be partitioned into two sets 
and 
such that is equal to the sum of elements in minus the sum of the elements in . Note that or could be empty, and that the sum of the elements of the empty set is 
.
For example,
is a benefactor of 
because 
.
Show that every positive integer has at least 
benefactors.
be a positive integer. A positive integer is called a benefactor of if the positive divisors of can be partitioned into two sets 
and 
such that is equal to the sum of elements in minus the sum of the elements in . Note that or could be empty, and that the sum of the elements of the empty set is 
.For example,
is a benefactor of 
because 
.Show that every positive integer
has at least 
benefactors.
1 reply
