schur weighted
by Ducksohappi, May 13, 2025, 11:47 PM
Schur-weighted:
let a,b,c be positive. Prove that:
let a,b,c be positive. Prove that:
Something weird with this one FE in integers (probably challenging, maybe not)
by Gaunter_O_Dim_of_math, May 13, 2025, 8:10 PM
This post has been edited 1 time. Last edited by Gaunter_O_Dim_of_math, Yesterday at 8:30 PM
Reason: one mistake
Reason: one mistake
Cyclic inequality with rational functions
by MathMystic33, May 13, 2025, 6:00 PM
Let 
be positive real numbers. Prove the inequality
![\[
\frac{x_1 + 3x_2}{x_2 + x_3}
\;+\;
\frac{x_2 + 3x_3}{x_3 + x_4}
\;+\;
\frac{x_3 + 3x_4}{x_4 + x_1}
\;+\;
\frac{x_4 + 3x_1}{x_1 + x_2}
\;\ge\;8.
\]](//latex.artofproblemsolving.com/5/c/0/5c062a37f6ff120232ae87eb0093a4ce73ee0eb1.png)
be positive real numbers. Prove the inequality![\[
\frac{x_1 + 3x_2}{x_2 + x_3}
\;+\;
\frac{x_2 + 3x_3}{x_3 + x_4}
\;+\;
\frac{x_3 + 3x_4}{x_4 + x_1}
\;+\;
\frac{x_4 + 3x_1}{x_1 + x_2}
\;\ge\;8.
\]](http://latex.artofproblemsolving.com/5/c/0/5c062a37f6ff120232ae87eb0093a4ce73ee0eb1.png)
I got stuck in this combinatorics
by artjustinhere237, May 13, 2025, 4:56 PM
Let 
, where 
is a positive integer.
Prove that there exists a subset of
with exactly 
elements such that the sum of its elements is a prime number.
, where 
is a positive integer.Prove that there exists a subset of
with exactly 
elements such that the sum of its elements is a prime number.forced vertices in graphs
by Davdav1232, May 8, 2025, 8:27 PM
Let 
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.
Prove that for any
-regular graph , there exists a coloring using 
colors such that at least 
of the colors have a forced vertex of that color.
Note: The graph coloring must be valid, this means no
vertices of the same color may be adjacent.
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.Prove that for any
-regular graph , there exists a coloring using 
colors such that at least 
of the colors have a forced vertex of that color.Note: The graph coloring must be valid, this means no
vertices of the same color may be adjacent.Inequality
by nguyentlauv, May 6, 2025, 12:19 PM
Let 
be positive real numbers such that 
and 
, prove that
be positive real numbers such that 
and 
, prove that
Brazilian Locus
by kraDracsO, Sep 9, 2023, 4:45 PM
Let 
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.
Note: The locus is the set of all points of the plane that satisfies the property.
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.Note: The locus is the set of all points of the plane that satisfies the property.
d1-d2 divides n for all divisors d1, d2
by a_507_bc, May 20, 2023, 4:34 PM
Determine all natural numbers 
with at most four natural divisors, which have the property that for any two distinct proper divisors 
and 
of 
, the positive integer 
divides .
with at most four natural divisors, which have the property that for any two distinct proper divisors 
and 
of 
, the positive integer 
divides .P (x^2) = P (x) P (x + 2) for any complex x
by parmenides51, Jul 24, 2021, 8:24 PM
Find all polynomials 
with complex coefficients such that 
for any complex number 
with complex coefficients such that 
for any complex number 
satisfying the equation 
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