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Prove that
Self-evident inequality trick
Lukaluce 19
N
by pooh123
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?
19 replies
Nice concurrency
navi_09220114 4
N
by quacksaysduck
Source: TASIMO 2025 Day 1 Problem 2
Four points 
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments 
and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments 
and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
4 replies
Numbers on a circle
navi_09220114 3
N
by quacksaysduck
Source: TASIMO 2025 Day 1 Problem 1
For a given positive integer 
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
3 replies
D1033 : A problem of probability for dominoes 3*1
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Let 
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
1 reply
f(n)=f(n-1)+1
Seventh 9
N
by cursed_tangent1434
Source: Problem 3, Brazilian MO 2015
Given a natural 
and its prime fatorization 
, its false derived is defined by 
Prove that there exist infinitely many naturals such that 
.
and its prime fatorization 
, its false derived is defined by 
Prove that there exist infinitely many naturals such that 
.
9 replies
Functional equation meets inequality condition
Lukaluce 1
N
by sarjinius
Source: 2025 Macedonian Balkan Math Olympiad TST Problem 3
Find all functions 
that satisfy
![\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]](//latex.artofproblemsolving.com/4/7/f/47f094ce57fe35d339ec3b93152ec3459a22ec48.png)
for every 
, and 
for every 
and 
, such that 
.
that satisfy![\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]](http://latex.artofproblemsolving.com/4/7/f/47f094ce57fe35d339ec3b93152ec3459a22ec48.png)
for every 
, and 
for every 
and 
, such that 
.
1 reply
Moving points in a plane
IAmTheHazard 2
N
by shanelin-sigma
Source: ELMO Shortlist 2024/C5
Let 
be a set of 
points in a plane that lie within a disk of radius 
billion. Define a 
as picking a point 
and reflecting it across 's centroid. Does there always exist a sequence of at most 
moves after which all points of are contained in a disk of radius ?
Advaith Avadhanam
be a set of 
points in a plane that lie within a disk of radius 
billion. Define a 
as picking a point 
and reflecting it across 's centroid. Does there always exist a sequence of at most 
moves after which all points of are contained in a disk of radius ?Advaith Avadhanam
2 replies
thank you !
Nakumi 0
Given two non-constant polynomials 
such that for every real number 
, 
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
such that for every real number 
, 
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
0 replies
Same divisor
sam-n 16
N
by AbdulWaheed
Source: IMO Shortlist 1997, Q14, China TST 2005
Let 
be positive integers such that 
and 
Prove that if 
and 
have the same prime divisors, then 
is a power of 2.
be positive integers such that 
and 
Prove that if 
and 
have the same prime divisors, then 
is a power of 2.
16 replies
for the contest high achievers, can you share your math path?
HCM2001 20
N
by Yrock
Hi all
Just wondering if any orz or high scorers on contests at young age (which are a lot of u guys lol) can share what your math path has been like?
- school math: you probably finish calculus in 5th grade or something lol then what do you do for the rest of the school? concurrent enrollment? college class? none (focus on math competitions)?
- what grade did you get honor roll or higher on AMC 8, AMC 10, AIME qual, USAJMO qual, etc?
- besides aops do you use another program to study? (like Mr Math, Alphastar, etc)?
You're all great inspirations and i appreciate the answers.. you all give me a lot of motivation for this math journey. Thanks
Just wondering if any orz or high scorers on contests at young age (which are a lot of u guys lol) can share what your math path has been like?
- school math: you probably finish calculus in 5th grade or something lol then what do you do for the rest of the school? concurrent enrollment? college class? none (focus on math competitions)?
- what grade did you get honor roll or higher on AMC 8, AMC 10, AIME qual, USAJMO qual, etc?
- besides aops do you use another program to study? (like Mr Math, Alphastar, etc)?
You're all great inspirations and i appreciate the answers.. you all give me a lot of motivation for this math journey. Thanks
20 replies
