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.
is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.
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 .
+1 w
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 .
.
, where, of course, a = BC, b = CA, c = AB.
. Thus, we get , and Lemma 1 is proven.
and 
.
.
and thus
.
and 
, so that
.
,
.
;
.
.
, and we are done.
and 
Prove that
satisfying
Prove
and 
be its projections on the sides 
and 
respectively. Denote by 
and 
the inradii of the triangles 
and 
Prove that![\[
r_{1}+r_{2}+r_{3}=r_{1}^{\prime }+r_{2}^{\prime }+r_{3}^{\prime }.
\]](http://latex.artofproblemsolving.com/e/9/1/e9195cfc1c992ca3619a8d2bf598aa479508d49d.png)