[/quote]
,
be a right angled triangle with 
.
is the midpoint of side 
And 
be an arbitrary point on 
.
over 
intersects lines 
and 
and 
,
and 
.
lies on 
.
be a trapezoid with parallel sides 
.
and 
lie on the line segments 
and 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
is greater than 0 if 
and 
.
and since 
?
be a positive integer. A board with a format 
is divided in 
(or 
)
positive integers 
such that 
for any 
,
such that 
?
![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
.
,
is the smallest number different from 
which doesn't divide 
.
.
and 
meet the sides 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
.
and 
lie on the circles 
and 
,
and 
.
and 
are concyclic.
and the recursive relation
for all natural number 
.
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This post has been edited 1 time. Last edited by Jaxman8, Apr 11, 2025, 3:08 AM
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