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.
be a polynomial with nonnegative integer coefficients. If 
and 
, what is the remainder when 
is divided by 
be a positive integer and consider the set 
.
of numbers not previously chosen such that for every distinct 
, neither 
divides 
nor divides .
, all multiples of every element in , including those in itself, are removed from 
. until no moves are possible., which player has a winning strategy if any
be a triangle and 
an interior point of . Show that at least one of the angles 
is less than or equal to 
.
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
be a triangle with incenter and let 
be an arbitrary point on the side 
. Let the line through perpendicular to 
intersect 
at 
. Let the line through perpendicular to intersect at 
. Prove that the reflection of across the line 
lies on the line .
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 
.
be an acute triangle. Let 
be the reflection of point over the line . Let and 
be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let and be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.![$p(x)\in\mathbb{R}[x]$](http://latex.artofproblemsolving.com/9/a/e/9aecb18edc17f7391a44db6e4f9d1502ab1d92f7.png)
such that for all 
:
such that:
be a triangle. Circle 
passes through , meets segments 
and 
again at points and respectively, and intersects segment at and such that lies between and . The tangent to circle 
at and the tangent to circle 
at meet at point . Suppose that points and are distinct. Prove that line 
is parallel to . with circumcircle and incenter , point bisects arc 
and line 
meets at 
. The excircle opposite to touches 
at point . Point 
on the circumcircle of 
is such that 
. Prove that the lines 
and 
meet on .
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Prove that
Let 
Prove that
?