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Hojoo Lee problem 73
Leon 25
N
by sqing
Source: Belarus 1998
Let 
, 
, 
be real positive numbers. Show that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]](//latex.artofproblemsolving.com/0/9/5/0959464bbb9b821a11df292f9e6addded7326371.png)
, 
, 
be real positive numbers. Show that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]](http://latex.artofproblemsolving.com/0/9/5/0959464bbb9b821a11df292f9e6addded7326371.png)
25 replies
Almost Squarefree Integers
oVlad 3
N
by Primeniyazidayi
Source: Romania Junior TST 2025 Day 1 P1
A positive integer 
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
3 replies
+1 w
Math camp combi
ErTeeEs06 3
N
by genius_007
Source: BxMO 2025 P2
Let 
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
3 replies
Benelux fe
ErTeeEs06 10
N
by genius_007
Source: BxMO 2025 P1
Does there exist a function 
such that 
for all 
?
such that 
for all 
?
10 replies
IMO Shortlist Problems
ABCD1728 0
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
0 replies
Geometric inequality in quadrilateral
BBNoDollar 0
Source: Romanian Mathematical Gazette 2025
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
0 replies
A coincidence about triangles with common incenter
flower417477 2
N
by flower417477
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
2 replies
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
Consecutive sum of integers sum up to 2020
NicoN9 2
N
by NicoN9
Source: Japan Junior MO Preliminary 2020 P2
Let and be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.
All among those pairs
, find the pair such that achieves the minimum.
and be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.All among those pairs
, find the pair such that achieves the minimum.
2 replies
Range of a^3+b^3-3c
Kunihiko_Chikaya 1
N
by Mathzeus1024
Let 
be real numbers such that 
and
Find the range of
Proposed by Kunihiko Chikaya/September 23, 2020
be real numbers such that 
and
Find the range of
Proposed by Kunihiko Chikaya/September 23, 2020
1 reply
