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.
satisfies the following conditions:![\begin{align*}
&f(4xy)=2y[f(x+y)+f(x-y)]\text{ for all }x,y\in\mathbb R\\
&f(5)=3
\end{align*}](http://latex.artofproblemsolving.com/6/2/5/625b8046a00d07680dd2a8d2aa3231d5daa4fcf2.png)
.
be real numbers with 
that satisfy the equations
} can be written in the form 
, where 
, and 
are positive integers. What is 
?
is the diameter of a semicircle centered at 
, which intersects 
and
at 
and 
respectively. Suppose that 
, 
, and 
. Find the
.
, how many positive integers 
are there such that there are exactly four powers of 2 with exactly 
digits? (Note that 1 is considered a power of 2)
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.
Prove that
Where 
and 
Prove that
satisfying
Prove
