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.
such that 
is also a prime number.
be a cyclic quadrilateral. Let 
, 
, 
be the feet of the perpendiculars from 
to the lines 
, 
, 
, respectively. Show that 
if and only if the bisectors of 
and 
are concurrent with 
.
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
such that for all real numbers 
and 
,
be a quadrilateral inscribed in a circle of center 
. Let M and N be the midpoints of diagonals 
and 
, respectively and let 
be the intersection point of the diagonals and of the given quadrilateral .It is known that the points 
are distinct. Prove that the points 
are concyclic if and only if the points 
are concyclic.
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
such that has exactly 2000 prime divisors and divides 
?
be a polynomial with nonnegative integer coefficients. If 
and 
, what is the remainder when 
is divided by 
has side length 
. Let the midpoint of be 
. What is the area of the overlapping region between the circle centered at with radius 
and the circle centered at with radius ? (You may express your answer using inverse trigonometry functions of noncommon values.)
occurs when 
for the function 
. room configurations:
Configuration 
: a one-room double, which is a square with side length of , Configuration 
: a two-room double, which is two connected rooms, each of them squares with a side length of .) to be exactly m
larger than the total area of a two-room double. Find the number of possible pairs of side lengths 
, where 
, 
, such that 
.
islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands and if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?
and 
be the roots of the polynomial 
. Compute 
.
, 
for prime and 
for 
. Find the smallest 
such that 
is a perfect power of 
.
be a triangle whose -excircle, -excircle, and 
-excircle have radii 
, 
, and 
, respectively. If 
and the perimeter of is 
, what is the area of ?
of functions 
satisfying:
,
,
.
can be written as 
where 
are distinct primes, compute 
.
and that the first (leftmost) two digits of 
are 10. Compute the number of integers with 
such that 
starts with either the digit 
or 
(in base 
).
be the circumcenter of . Let 
be the midpoint of , and let and 
be the feet of the altitudes from and , respectively, onto the opposite sides. 
intersects at . The line passing through and perpendicular to intersects the circumcircle of at 
(on the major arc ) and 
, and intersects at . Point lies on the line 
such that 
is perpendicular to 
. Given that 
and 
, compute 
.
be the isosceles triangle with side lengths 
. Arpit and Katherine simultaneously choose points and 
within this triangle, and compute 
, the squared distance between the two points. Suppose that Arpit chooses a random point within . Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of . Compute this value. with sides, let 
denote the regular polygon with 
sides such that the vertices of are the midpoints of every other side of . Let 
denote the polygon that results after applying f a total of k times. The area of 
where is a pentagon of side length , can be expressed as 
for some positive integers 
where 
is not divisible by the square of any prime and does not share any positive divisors with 
and 
. Find 
.
. This function can be expressed in the form 
for sequences of integers 
, 
. Determine 
., let 
be the centroid and let the circumcenters of 
, 
, and 
be 
, and , respectively. The line passing through 
and the midpoint of intersects 
at 
. If the radius of circle is 
, the radius of circle 
is , and 
, what is the length of 
?
and 
not is a perfect square for all 
.
such that 
.
Prove that
Let 
Prove that
of a triangle 
are positive integers. Let
for any positive integer 
and 
,