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cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)
Amir Hossein 12
N
by AylyGayypow009
Source: Greece JBMO TST 2017, Problem 1
Positive real numbers 
satisfy 
. Prove that
Also, find the values of for which the equality happens.
satisfy 
. Prove that
Also, find the values of for which the equality happens.
12 replies
R to R, with x+f(xy)=f(1+f(y))x
NicoN9 2
N
by dangerousliri
Source: Own.
Find all functions 
such that![\[
x+f(xy)=f(1+f(y))x
\]](//latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
2 replies
Brilliant guessing game on triples
Assassino9931 1
N
by Sardor_lil
Source: Al-Khwarizmi Junior International Olympiad 2025 P8
There are 
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
Shubin Yakov, Russia
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?Shubin Yakov, Russia
1 reply
in n^2-9 has 6 positive divisors than GCD (n-3, n+3)=1
parmenides51 7
N
by AylyGayypow009
Source: Greece JBMO TST 2016 p3
Positive integer 
is such that number 
has exactly 
positive divisors. Prove that GCD 
is such that number 
has exactly 
positive divisors. Prove that GCD 
7 replies
a deep thinking topic. either useless or extraordinary , not yet disovered
jainam_luniya 5
N
by jainam_luniya
Source: 1.99999999999....................................................................1. it this possible or not we can debate
it can be a new discovery in world or NT
5 replies
Divisibilty...
Sadigly 4
N
by jainam_luniya
Source: Azerbaijan Junior NMO 2025 P2
Find all 
consecutive even numbers, such that the sum of their squares divides the square of their product.
consecutive even numbers, such that the sum of their squares divides the square of their product.
4 replies
ioqm to imo journey
jainam_luniya 2
N
by jainam_luniya
only imginative ones are alloud .all country and classes or even colleges
2 replies
Inequality
Sadigly 5
N
by jainam_luniya
Source: Azerbaijan Junior MO 2025 P5
For positive real numbers 
satisfying 
, find the biggest value the following equation could acquire:
satisfying 
, find the biggest value the following equation could acquire:
5 replies
D'B, E'C and l are congruence.
cronus119 7
N
by Tkn
Source: 2022 Iran second round mathematical Olympiad P1
Let 
and 
on 
and 
respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of 
and to respectively prove that 
and are congruence.
and 
on 
and 
respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of 
and to respectively prove that 
and are congruence.
7 replies
a set of $9$ distinct integers
N.T.TUAN 17
N
by hlminh
Source: APMO 2007
Let 
be a set of 
distinct integers all of whose prime factors are at most 
Prove that contains distinct integers such that their product is a perfect cube.
be a set of 
distinct integers all of whose prime factors are at most 
Prove that contains distinct integers such that their product is a perfect cube.
17 replies
