[/quote]
,
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
such that
is an integer.
be a triangle with 
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
be the point of intersection of 
and 
.
.
be a point on segment 
.
be a fixed circle passing through 
be a variable point on 
be the intersection of the tangent to the circumcircle of 
at 
and the tangent to the circumcircle of 
at 
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be the sum of the remainders when 
.
.
.
be the set of all positive integers. Find all functions 
such that 
and 
have the same number of 
'
.
,
,
,
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
that are relatively prime be called intertwined if among any two divisors of 
and among any two divisors of 
is intertwined.
for which there exists an intertwined pair 
is equal to the product of the first 
distinct prime numbers.
and 
have a common tangent that meets 
and 
.
where 
than 
meets circle 
.
bisects angle 
.
be real numbers satisfying 
and 
Prove that 
Let 
and 
Prove that 
be a positive integer. Call a rearrangement 
of 
,
is not divisible by 
are non-negative real numbers and 
Prove that
the equality hold for 
or 
or any
Y by Adventure10, Mango247
Dear Mathlinkers,
1. ABC a triangle
2. D the foot of the A-inner bissector
3. E the midpointAC
4. P the point of intersection of BE and AD
5. F the point of intersection of CP and AB
6. Pb the parallel to CF through B
7. M the point of intersection of Pb and DF
8. X the symmetric of B wrt AD.
Prove : MX is parallel to AB.
Sincerely
Jean-Louis
1. ABC a triangle
2. D the foot of the A-inner bissector
3. E the midpointAC
4. P the point of intersection of BE and AD
5. F the point of intersection of CP and AB
6. Pb the parallel to CF through B
7. M the point of intersection of Pb and DF
8. X the symmetric of B wrt AD.
Prove : MX is parallel to AB.
Sincerely
Jean-Louis
This post has been edited 1 time. Last edited by jayme, Aug 1, 2018, 11:34 AM
Reason: typo
Reason: typo
.
,
is a parallelogramm and we are done.
.
hence 
.
is a parallelogramm, and thus 
,
