[/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 
.
Y by
P1. Find 
where 
is a monic polynomial of degree 
satisfying 
P2. Find
where is a polynomial of smallest degree possible satisfying 
P3. Find
and 
, if polynomial satisfies 
P4. Find where is a polynomial of smallest degree satisfying 
P5. Find
and 
if polynomial satisfies 
P6. If
is a polynomial of degree 
such that 
Evaluate 
where 
is a monic polynomial of degree 
satisfying 
P2. Find
where is a polynomial of smallest degree possible satisfying 
P3. Find
and 
, if polynomial satisfies 
P4. Find
where is a polynomial of smallest degree satisfying 
P5. Find
and 
if polynomial satisfies 
P6. If
is a polynomial of degree 
such that 
Evaluate 
This post has been edited 1 time. Last edited by P162008, Jun 6, 2025, 12:31 AM
Reason: Typo
Reason: Typo
