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For positive integers \( a, b, c \), find all possible positive integer values o
Jackson0423 11
N
by zoinkers
For positive integers 
, find all possible positive integer values of
![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](//latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
, find all possible positive integer values of![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
11 replies
Set summed with itself
Math-Problem-Solving 1
N
by pi_quadrat_sechstel
Source: Awesomemath Sample Problems
Let 
be the set of the first 
perfect squares of nonzero integers. Suppose that 
for some 
. Here 
stands for the set 
. Prove that 
holds for every 
.
be the set of the first 
perfect squares of nonzero integers. Suppose that 
for some 
. Here 
stands for the set 
. Prove that 
holds for every 
.
1 reply
+1 w
(x+y) f(2yf(x)+f(y))=x^3 f(yf(x)) for all x,y\in R^+
parmenides51 12
N
by MuradSafarli
Source: Balkan BMO Shortlist 2015 A4
Find all functions 
such that 
(Albania)
such that 
(Albania)
12 replies
Advanced topics in Inequalities
va2010 9
N
by Strangett
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
9 replies
,
satisfy that: 
Hard Polynomial Problem
MinhDucDangCHL2000 1
N
by Tung-CHL
Source: IDK
Let 
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
. Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
. Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).1 reply
IMO LongList 1985 CYP2 - System of Simultaneous Equations
Amir Hossein 13
N
by cubres
Solve the system of simultaneous equations
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](//latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](//latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](//latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](//latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
13 replies
Centroid Distance Identity in Triangle
zeta1 5
N
by DottedCaculator
Let M be any point inside triangle ABC, and let G be the centroid of triangle ABC. Prove that:
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](//latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](http://latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
5 replies
Numbers not power of 5
Kayak 34
N
by cursed_tangent1434
Source: Indian TST D1 P2
Show that there do not exist natural numbers 
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](//latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
34 replies
