[/quote]
,
be a natural number. Prove that,![\[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \]](http://latex.artofproblemsolving.com/5/6/0/560a9568b2c09512cac1d317a011931a439e6a02.png)
and the sets 
If 
has exactly three elements, prove that 
cannot be an integer.
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
such that
holds for all 
.
such that:
with all 
be a triangle, and let 
,
,
be collinear points such that 
,
,
.
,
,
are the reflections of 
,
,
,
is a nondegenerate triangle, then its circumcenter lies on the circumcircle of 
be three positive real numbers satisfying 
Prove that
There is a very elegant proof :-D Could anyone think of it?
and has exactly 
then
Y by Mogmog8, bestzack66, eagles2018, akliu, Cokevending56, Mango247
In some countries, automobile fuel efficiency is measured in liters per 
kilometers while other countries use miles per gallon. Suppose that 
kilometer equals 
miles, and gallon equals 
liters. Which of the following gives the fuel efficiency in liters per kilometers for a car that gets 
miles per gallon?
kilometers while other countries use miles per gallon. Suppose that 
kilometer equals 
miles, and gallon equals 
liters. Which of the following gives the fuel efficiency in liters per kilometers for a car that gets 
miles per gallon?
This post has been edited 1 time. Last edited by programmeruser, Nov 11, 2022, 4:15 PM
So, E.
.
.
