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Topic
First Poster
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3 var inquality
sqing 5
N
by sqing
Source: Own
Let 
be reals such that 
and 
Prove that
Let be reals such that 
and Prove that
![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](//latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](//latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
be reals such that 
and 
Prove that
Let be reals such that 
and Prove that![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](http://latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](http://latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
5 replies
such that: 
.
such that 
then find minimal value of 
functional equation on natural numbers ! CMO 2015 P1
aditya21 18
N
by NicoN9
Source: Canadian mathematical olympiad 2015
Let 
be the set of positive integers. Find all functions 
, defined on 
and taking values in , such that 
for every positive integer 
.
be the set of positive integers. Find all functions 
, defined on 
and taking values in , such that 
for every positive integer 
.
18 replies
Prove that 
Where 
Number of Polynomial Q such that P(x) | P(Q(x))
IndoMathXdZ 16
N
by Ilikeminecraft
Source: IZHO 2021 P6
Let 
be a nonconstant polynomial of degree with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials 
of degree less than with rational coefficients such that divides 
a) is finite
b) does not exceed.
be a nonconstant polynomial of degree with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials 
of degree less than with rational coefficients such that divides 
a) is finite
b) does not exceed
.
16 replies
Quick Oly question
Alpaca31415 0
What is China second round? Just asking because I did a few questions and I'm wondering about the difficulty. Also, are there mohs ratings for non-IMO ISL questions?
0 replies
Rectangular line segments in russia
egxa 1
N
by Quantum-Phantom
Source: All Russian 2025 9.1
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
1 reply
old and easy imo inequality
Valentin Vornicu 212
N
by Sleepy_Head
Source: IMO 2000, Problem 2, IMO Shortlist 2000, A1
Let 
be positive real numbers so that 
. Prove that
![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](//latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
be positive real numbers so that 
. Prove that![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
212 replies
Killer NT that nobody solved (also my hardest NT ever created)
mshtand1 1
N
by YaoAOPS
Source: Ukraine IMO 2025 TST P8
A positive integer number 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where
![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](//latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
1 reply
