G
Topic
First Poster
Last Poster
Distinct Integers with Divisibility Condition
tastymath75025 15
N
by cursed_tangent1434
Source: 2017 ELMO Shortlist N3
For each integer 
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.
Proposed by Daniel Liu
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.Proposed by Daniel Liu
15 replies
+1 w
and 
.
An easy FE
oVlad 1
N
by pco
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](//latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
1 reply
Fractions and reciprocals
adihaya 34
N
by de-Kirschbaum
Source: 2013 BAMO-8 #4
For a positive integer 
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
, consider the 
fractions 
The product of these fractions equals 
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of for which this is possible and prove that you have found them all.
34 replies
GCD Functional Equation
pinetree1 60
N
by cursed_tangent1434
Source: USA TSTST 2019 Problem 7
Let 
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .
Ankan Bhattacharya
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and such that 
for all integers .Ankan Bhattacharya
60 replies
1 viewing
such that: 
.
Easy geo
oVlad 3
N
by Primeniyazidayi
Source: Romania EGMO TST 2019 Day 1 P1
A line through the vertex 
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
3 replies
abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla 28
N
by mihaig
Let 
be positive real numbers such that 
Prove that we have
![\[(a+b)(b+c)(c+a)\geq 8.\]](//latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
Also determine the case of equality.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
Also determine the case of equality.
28 replies
NT with repeating decimal digits
oVlad 1
N
by kokcio
Source: Romania EGMO TST 2019 Day 1 P2
Determine the digits 
such that for any positive integer 
there exists a positive integer such that the last digits of 
are equal to 
such that for any positive integer 
there exists a positive integer such that the last digits of 
are equal to 
1 reply
Inequalities make a comeback
MS_Kekas 2
N
by ZeroHero
Source: Kyiv City MO 2025 Round 1, Problem 11.5
Determine the largest possible constant 
such that for any positive real numbers 
, which are the sides of a triangle, the following inequality holds:
![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](//latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
Proposed by Vadym Solomka
such that for any positive real numbers 
, which are the sides of a triangle, the following inequality holds:![\[
\frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C.
\]](http://latex.artofproblemsolving.com/9/1/c/91c8a4af2bd2f3107c8921d2e06f8a09dedd0164.png)
Proposed by Vadym Solomka
2 replies
