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.
for some base 
, 
. The points 
, 
, and 
lie on the graph of 
. If 
, find the value of 
.
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an integer. The numbers 
are split into 
groups of . Prove that there are at least 
quadruples 
such that they are all in different groups, 
and 
.
, it holds that
be positive real numbers so that 
. Prove that![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
and 
not is a perfect square for all 
.
be an acute triangle with orthocenter 
. Prove that there is a line 
which is parallel to 
and tangent to the incircles of 
and 
.
are three positive real numbers, prove that 
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a positive integer. A positive integer is called a -good number if it satisfies has exactly 
digits in decimal representation (it cannot have leading zeros). digits and the last digits of are considered as two separate -digit.
is a -good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
, determine 
representation of 
?
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 be real numbers sach that 
and 
if 
where 
and are relatively prime positive integers, what is 
and 
defined, and with another symmetric equation in 
would be to attempt to write the third equation in terms of the symmetric sums of 
then substitute the first and third sums so we can find the second, its then pretty trivial to find 
