AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
be different points on a circle 
such that 
is not a diameter. Let 
be the tangent line to at . Point 
is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let 
be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to .
be a right angled triangle with 
. Point 
is the midpoint of side 
And 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
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.
be a positive integer. A board with a format 
is divided in equal squares.Determine all integers ≥3 such that the board can be covered in 
(or 
) pieces so that there is exactly one empty square in each row and each column.
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 
.
. For 
, 
is the smallest number different from 
which doesn't divide 
. Proof that every number large enough appears in the sequence.
be a triangle for which there exists an interior point such that 
. Let the lines 
and 
meet the sides and at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
, the point is the foot of the altitude from , and is a point on the segment 
. The lines through parallel to and meet at and , respectively. Points 
and 
lie on the circles 
and 
, respectively, such that 
and 
.
and 
are concyclic.
and the recursive relation
for all natural number . Find the general form of .
.
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point is chosen on 
such taht 
and point is the foot of the perpendicular from to 
. Ray mets again at . Point is chosen on line 
such that 
and 
lie on a line in that order. Finally, let be a point satisfying 
and 
. Prove that lies on .
satisfying
Prove
and 
Prove that
with area 
and a point 
in the same plane, determine the area of polygon 
where 
is the image of 
around 
by 