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Regional Olympiad - FBH 2018 Grade 9 Problem 3
gobathegreat 5
N
by justaguy_69
Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018
Let 
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
5 replies
R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3 3
N
by jasperE3
Source: p24734470
Find all functions 
such that for all positive real numbers 
and 
:
such that for all positive real numbers 
and 
:
3 replies
Nice functional equation
ICE_CNME_4 2
N
by Pi-Oneer
Determine all functions 
that satisfy the equation
![\[
f(x) + 3f(-x) + f\left( \frac{1}{x} \right) = x, \quad \text{for all } x \in \mathbb{R}^*.
\]](//latex.artofproblemsolving.com/6/a/2/6a2e35c42644036a62451b5e1794b36ffa728591.png)
that satisfy the equation![\[
f(x) + 3f(-x) + f\left( \frac{1}{x} \right) = x, \quad \text{for all } x \in \mathbb{R}^*.
\]](http://latex.artofproblemsolving.com/6/a/2/6a2e35c42644036a62451b5e1794b36ffa728591.png)
2 replies
Balkan Mathematical Olympiad
ABCD1728 0
Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!
0 replies
1 viewing
Prove that the fraction (21n + 4)/(14n + 3) is irreducible
DPopov 111
N
by reni_wee
Source: IMO 1959 #1
Prove that the fraction 
is irreducible for every natural number 
.
is irreducible for every natural number 
.
111 replies
Concentric Circles
MithsApprentice 63
N
by QueenArwen
Source: USAMO 1998
Let 
and 
be concentric circles, with in the interior of . From a point 
on one draws the tangent 
to (
). Let 
be the second point of intersection of and , and let 
be the midpoint of . A line passing through intersects at 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
and 
be concentric circles, with in the interior of . From a point 
on one draws the tangent 
to (
). Let 
be the second point of intersection of and , and let 
be the midpoint of . A line passing through intersects at 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
63 replies
D is incenter
Layaliya 7
N
by rong2020
Source: From my friend in Indonesia
Given an acute triangle 
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
7 replies
IMO Genre Predictions
ohiorizzler1434 73
N
by CrazyInMath
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
73 replies
Divisibility
emregirgin35 13
N
by Andyexists
Source: Turkey TST 2014 Day 2 Problem 4
Find all pairs 
of positive odd integers, such that 
and 
.
of positive odd integers, such that 
and 
.
13 replies
Self-evident inequality trick
Lukaluce 13
N
by ytChen
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
13 replies
1 viewing
