[/quote]
,
term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
is the largest value of the constant 
such that ![\[{ab+ac+ad+bc+bd+cd-6}\leqslant{k{\left(a+b+c+d-1\right)}{\left(a+b+c+d-4\right)}}\]](http://latex.artofproblemsolving.com/b/e/7/be787de190655249cbfa6db6705a0d614231ad18.png)
holds for any nonnegative real numbers 
satisfying 
.
be an acute triangle with orthocenter 
.
,
,
is congruent to triangle 
such that 
,
for all reals 
.
,
and 
,
.
.
is ______.
be the circumcircle of triangle 
is tangent to the line segment 
and is tangent to 
at a point lying on the same side of the line 
.
intersects 
and 
.
.
,
Prove that:
When does the equality hold ?
.
be the midpoint of 
and let 
be the midpoint of 
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
and 
be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose 
is odd and the sets 
and 
are the same. Prove that there exists an integer 
such that 
.
Y by
What happens when you rotate a line of slope 
by 
degrees? All I've got is that if 
, then the new line should still have slope since 
. :/
On a second thought, is it even possible to rotate such a line?
I remember reading about this because "five points determine a conic" and "three points are sufficient to determine a circle". So there are two special points on all circles (smth like that).
by 
degrees? All I've got is that if 
, then the new line should still have slope since 
. :/On a second thought, is it even possible to rotate such a line?
I remember reading about this because "five points determine a conic" and "three points are sufficient to determine a circle". So there are two special points on all circles (smth like that).
This post has been edited 3 times. Last edited by eduD_looC, Dec 19, 2024, 2:39 PM
