Equations
by Jackson0423, Apr 22, 2025, 4:36 PM
Solve the system of equations
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](//latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
2^x+3^x = yx^2
by truongphatt2668, Apr 22, 2025, 3:38 PM
Prove that the following equation has infinite integer solutions:
Integer representation
by RL_parkgong_0106, Apr 22, 2025, 2:23 PM
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 
.
A cyclic inequality
by KhuongTrang, Apr 21, 2025, 4:18 PM
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
FE solution too simple?
by Yiyj1, Apr 9, 2025, 3:26 AM
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?
and 
: Prove that :
Factor sums of integers
by Aopamy, Feb 23, 2023, 3:13 AM
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.Factor of P(x)
by Brut3Forc3, Apr 4, 2010, 2:45 AM
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 
.
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