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hard problem
Cobedangiu 1
N
by lbh_qys
Let a,b,c>0, a^3+b^3+c^3=3. Find min D(and prove)
D= 5(a^2+b^2+c^2)+4(1/a+1/b+1/c)
D= 5(a^2+b^2+c^2)+4(1/a+1/b+1/c)
1 reply
and 
Prove that
Numbers Theory
Amin12 9
N
by Ihatecombin
Source: Iran 3rd round 2017 Numbers theory final exam-P3
Let 
be a positive integer. Prove that there exists a poisitve integer 
such that
be a positive integer. Prove that there exists a poisitve integer 
such that
9 replies
cyclic ineq not tight
RainbowNeos 1
N
by lbh_qys
Source: own
Given 
and 
with sum 
. Show that
![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](//latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
and 
with sum 
. Show that![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
1 reply
Something nice
KhuongTrang 22
N
by KhuongTrang
Source: own
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
22 replies
and 
Prove that
![\[ a^2+b^2+c^2+abc=4\]](http://latex.artofproblemsolving.com/e/3/0/e30d8ebeabd9ea1c2892d3de685e518a75f62da3.png)
![\[ab+bc+ca-\frac{a+b+c-1}{2}\cdot abc\le 2\]](http://latex.artofproblemsolving.com/8/4/6/8464472df19ce2853344510353965107392c7243.png)
Send me your problem.
B1t 0
Does anyone have a non-famous hard problem (Category C or N is great) with a Mohs rating of 35M( harder than average imo p2) or higher? If you do, please send me the problem via private message along with the solution. I will propose it for the Olympiad. Please do not post it here.
0 replies
and 
Prove that
integral points
jhz 3
N
by flower417477
Source: 2025 CTST P17
Prove: there exist integer 
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](//latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，and any rectangular strip of width 1 covers at most two points of S.
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，and any rectangular strip of width 1 covers at most two points of S.
3 replies
