[/quote]
,
and 
Prove that
![\[
a + b + c = 0 \quad \text{and} \quad abc = -16.
\]](http://latex.artofproblemsolving.com/6/b/1/6b1532d4de660e7ab4300ee1f7ec1742cbde9dc2.png)
Find the minimum value of the expression![\[
W = \frac{a^2 + b^2} {c} + \frac{b^2 + c^2} {a} + \frac{c^2 + a^2} {b}.
\]](http://latex.artofproblemsolving.com/2/2/3/223e5abc7ae4f5a92dde90a74b0988adb18bdf4e.png)
such that there exists a 
-
of ![\([6000]\)](http://latex.artofproblemsolving.com/4/0/6/4064bfdf860714008484748e944421eca61e0746.png)
satisfying the following condition: if we define ![$T = \{(i, j, k) | i, j, k \in [2000], i + j + k \in A\}$](http://latex.artofproblemsolving.com/6/8/2/6826f34515ccdd5b0703452933f38f8d89ff1402.png)
,
,![\[\sum_{(i, j, k) \in T} a_i a_j a_k \leq c \left( \sum_{i=1}^{2000} a_i \right)^3.\]](http://latex.artofproblemsolving.com/4/6/a/46a61cfde609848e0758ec3dc93e6e53981142f3.png)
Proposed by Tianqin Li, High School Affiliated to Renmin University of China
of positive real numbers,
.
,![\[\frac{1}{a^2+b+1}+\frac{1}{b^2+c+1}+\frac{1}{c^2+a+1}\le1\]](http://latex.artofproblemsolving.com/d/4/3/d431a035ed0b9f699bd232b7c1d0129e659e3309.png)
of real numbers such that the following conditions hold for all 
:
be a trapezium, 
,
be points on the side
,
meets 
again at 
,
meets 
again at 
.
are concurrent.
be a triangle with incentre 
.
tangent to 
at 
.
tangent to 
again at 
.
is tangent to the incircle of 
,
and 
be circles with centres 
and 
and 
intersects 
,
lie on 
.
meets 
and meets 
.
be the orthocentre of triangle 
.
.
such that for all 
both not 
,
?
Y by
A 9-digit number 
is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation?
is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation?
This post has been edited 1 time. Last edited by Sadigly, May 8, 2025, 5:29 PM
.
numbers should be 
We also have 
,
as 
.
.
,
.
