Y by
Let 
be real positive number satisfy that: 
. Prove that: 
be real positive number satisfy that: 
. Prove that: 
This post has been edited 1 time. Last edited by giangtruong13, Yesterday at 3:21 PM
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2^a + 3^b + 1 = 6^c
togrulhamidli2011 3
N
by Tamam
Find all positive integers (a, b, c) such that:
![\[
2^a + 3^b + 1 = 6^c
\]](//latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
3 replies
min A=x+1/x+y+1/y if 2(x+y)=1+xy for x,y>0 , 2020 ISL A3 for juniors
parmenides51 12
N
by mathmax001
Source: 2021 Greece JMO p1 (serves also as JBMO TST) / based on 2020 IMO ISL A3
If positive reals 
are such that 
, find the minimum value of expression 
are such that 
, find the minimum value of expression 
12 replies
1 viewing
3a^2b+16ab^2 is perfect square for primes a,b >0
parmenides51 5
N
by ali123456
Source: 2020 Greek JBMO TST p3
Find all pairs 
of prime positive integers 
such that number 
equals to a square of an integer.
of prime positive integers 
such that number 
equals to a square of an integer.
5 replies
minimum value of S, ISI 2013
Sayan 13
N
by Apple_maths60
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
.
13 replies
classical R+ FE
jasperE3 2
N
by jasperE3
Source: kent2207, based on 2019 Slovenia TST
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 
.
2 replies
Geometry
srnjbr 0
in triangle abc, we know that bac=60. the circumcircle of the center i is tangent to the sides ab and ac at points e and f respectively. the midpoint of side bc is called m. if lines bi and ci intersect line ef at points p and q respectively, show that pmq is equilateral.
0 replies
JBMO Shortlist 2021 N1
Lukaluce 14
N
by ali123456
Source: JBMO Shortlist 2021
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
14 replies
Very easy inequality
pggp 2
N
by ali123456
Source: Polish Junior MO Second Round 2019
Let 
, 
be real numbers, such that 
. Prove that 
.
, 
be real numbers, such that 
. Prove that 
.
2 replies
Problem 5
blug 1
N
by WallyWalrus
Source: Polish Junior Math Olympiad Finals 2025
Each square on a 5×5 board contains an arrow pointing up, down, left, or right. Show that it is possible to remove exactly 20 arrows from this board so that no two of the remaining five arrows point to the same square.
1 reply
Cool Number Theory
Fermat_Fanatic108 6
N
by epl1
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
.
6 replies
The return of an inequality
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Maybe:
,
and 
be positive numbers such that 
.
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.
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