Problem about Euler's function
by luutrongphuc, Mar 19, 2025, 4:23 PM
Prove that for every integer 
, we have:
, we have:
Cool Number Theory
by Fermat_Fanatic108, Mar 19, 2025, 1:41 PM
For an integer with 5 digits 
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
Prove that
.
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.Prove that
.classical R+ FE
by jasperE3, Mar 18, 2025, 3:55 PM
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions
for which:
for all 
.
Find all functions
for which:
for all 
.Incenter geometry with parallel lines
by nAalniaOMliO, Apr 16, 2024, 11:50 AM
Let 
be the incircle of triangle 
. Line 
is parallel to side 
and tangent to . Line 
is parallel to side 
and tangent to . It turned out that the intersection point of and lies on circumcircle of
Find all possible values of
be the incircle of triangle 
. Line 
is parallel to side 
and tangent to . Line 
is parallel to side 
and tangent to . It turned out that the intersection point of and lies on circumcircle of Find all possible values of
JBMO Shortlist 2021 N1
by Lukaluce, Jul 2, 2022, 9:14 PM
Find all positive integers 
such that 
, 
, and 
are all equal to
factorials of some positive integers.
Proposed by Nikola Velov, Macedonia
such that 
, 
, and 
are all equal tofactorials of some positive integers.
Proposed by Nikola Velov, Macedonia
This post has been edited 2 times. Last edited by Lukaluce, Jul 2, 2022, 10:14 PM
,
be real numbers, such that 
.
.minimum value of S, ISI 2013
by Sayan, May 12, 2013, 11:04 AM
Let 
be real number greater than 
. Let
![\[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\]](//latex.artofproblemsolving.com/5/9/e/59e1a33017116963165f744b49b8515f126c64ef.png)
Find the minimum possible value of
.
be real number greater than 
. Let![\[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\]](http://latex.artofproblemsolving.com/5/9/e/59e1a33017116963165f744b49b8515f126c64ef.png)
Find the minimum possible value of
.Problem for VASC, SI Book
by hungkhtn, Jun 5, 2007, 4:23 AM
Let be non-negative real numbers such that 
. Prove that
![\[a\sqrt{1+b^{3}}+b\sqrt{1+c^{3}}+c\sqrt{1+a^{3}}\le 5.\]](//latex.artofproblemsolving.com/f/f/a/ffaeec844a6bdf10a96e8e9d298ef97ac839d713.png)
be non-negative real numbers such that 
. Prove that![\[a\sqrt{1+b^{3}}+b\sqrt{1+c^{3}}+c\sqrt{1+a^{3}}\le 5.\]](http://latex.artofproblemsolving.com/f/f/a/ffaeec844a6bdf10a96e8e9d298ef97ac839d713.png)
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