,
integers and 
is a positive integer. Suppose that the integers
satisfy: 
are divisible by k. Prove that:
is divisible by k
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
denote the increasing arithmetic sequence of integers whose first term is 
and whose common difference is 
is the sequence 
.
?
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in 
Y by
1. Real numbers a,b,c are satisfy a+1/b = b+1/c = c+1/a =x. If a,b,c are distinct, what is the value of x?
2. If x^2+y^2=1, then what is the value of : root(x^2-2x+1) + root(xy-2x+y-2) ?
3. Find the value of the sequence 2^2 + (3^2+1) + (4^2+2) + … + (97^2+95) + (98^2+96).
4. If x^2+x-1=0, then evaluate (1-x^2-x^3-x^4-…-x^2022-x^2023)/x^2022 .
5. If triangle XYZ has 3 sides that are all whole numbers, and the perimeter of XYZ is 24, what is the probability XYZ is a right triangle?
Note: If someone can latex-ify this it would help.
2. If x^2+y^2=1, then what is the value of : root(x^2-2x+1) + root(xy-2x+y-2) ?
3. Find the value of the sequence 2^2 + (3^2+1) + (4^2+2) + … + (97^2+95) + (98^2+96).
4. If x^2+x-1=0, then evaluate (1-x^2-x^3-x^4-…-x^2022-x^2023)/x^2022 .
5. If triangle XYZ has 3 sides that are all whole numbers, and the perimeter of XYZ is 24, what is the probability XYZ is a right triangle?
Note: If someone can latex-ify this it would help.
This post has been edited 1 time. Last edited by hashbrown2009, Apr 19, 2025, 11:02 PM
,
,
satisfy 
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has 
is 
,
