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)
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
Arrangement of integers in a row with gcd
by egxa, Apr 18, 2025, 5:09 PM
Let 
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:18 PM
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?
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 
.Estonian Math Competitions 2005/2006
by STARS, Jul 30, 2008, 1:17 AM
A 
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?Sum of whose elements is divisible by p
by nntrkien, Aug 8, 2004, 1:29 AM
Let 
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?♪ i just hope you understand / sometimes the clothes do not make the man ♫ // https://beta.vero.site/
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