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Topic
First Poster
Last Poster
JBMO Shortlist 2022 N1
Lukaluce 8
N
by godchunguus
Source: JBMO Shortlist 2022
Determine all pairs 
of positive integers that satisfy
of positive integers that satisfy
8 replies
P(x) | P(x^2-2)
GreenTea2593 4
N
by GreenTea2593
Source: Valentio Iverson
Let 
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](//latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of .
Proposed by Valentio Iverson, Indonesia
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of .Proposed by Valentio Iverson, Indonesia
4 replies
USEMO P6 (Idk what to say here)
franzliszt 16
N
by MathLuis
Source: USEMO 2020/6
Prove that for every odd integer 
, there exist integers 
such that, if we let 
, then the following conditions hold:
we have 
;
the number 
is divisible by 
; and
the numbers 
each have a prime factor not dividing .
, there exist integers 
such that, if we let 
, then the following conditions hold:
we have 
; the number 
is divisible by 
; and the numbers 
each have a prime factor not dividing .
16 replies
Prove that the fraction (21n + 4)/(14n + 3) is irreducible
DPopov 110
N
by Shenhax
Source: IMO 1959 #1
Prove that the fraction 
is irreducible for every natural number 
.
is irreducible for every natural number 
.
110 replies
Let \( a, b, c \) be positive real numbers satisfying \[ a^2 + c^2 = b(a + c). \
Jackson0423 3
N
by Mathzeus1024
Let 
be positive real numbers satisfying
![\[
a^2 + c^2 = b(a + c).
\]](//latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let
![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](//latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.3 replies
real+ FE
pomodor_ap 3
N
by MathLuis
Source: Own, PDC001-P7
Let 
be a function such that
for all 
. Determine all such functions 
.
be a function such that
for all 
. Determine all such functions 
.
3 replies
Inspired by hlminh
sqing 1
N
by sqing
Source: Own
Let 
be real numbers such that 
Prove that 
Where 
be real numbers such that 
Prove that 
Where 
1 reply
Is this FE solvable?
ItzsleepyXD 3
N
by jasperE3
Source: Original
Let 
. Find all 
such that for all 
. Find all 
such that for all 
3 replies
PQ bisects AC if <BCD=90^o, A, B,C,D concyclic
parmenides51 2
N
by venhancefan777
Source: Mathematics Regional Olympiad of Mexico Northeast 2020 P2
Let 
, 
, 
and 
be points on the same circumference with 
. Let 
and 
be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
, 
, 
and 
be points on the same circumference with 
. Let 
and 
be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
2 replies
Inequality with three conditions
oVlad 3
N
by sqing
Source: Romania EGMO TST 2019 Day 1 P3
Let 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](//latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
3 replies
