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 natural number and 
be a prime number. How many different pairs 
satisfy the equation:
such that 
is a perfect cube.
digits are distinct?
Solve for .
, 
, 
, and 
. Let 
be the point on 
such that 
, and let 
be the midpoint of 
. If 
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of . Points 
, 
, 
, and 
are chosen on 
, , 
, and respectively, such that 
, 
, 
, and 
are length 
. Compute the area of quadrilateral 
.
satisfies for all integers , 
If 
and 
, compute 
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class? such that the set 
can be partitioned into n Akash subsets.
. We call a distribution the act of distributing the integers from 
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
such that the following equations are both true:
be reals such that 
Prove that
of the triangle 
touch its side at . Let the line intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
and 
are concyclic.
be a triangle with 
. is the midpoint of 
. Let the angle bisector of 
cut 
at and 
be the centroid of . 
meets at 
and is the projection of onto . Prove that 
lie on a common circle.
be 
. Points 
and 
are two moving points on the circle 
: 
, satisfying 
. Find the maximum area of 
.
. What is the least possible positive integer value of 
such that there exists a non-negative integer 
for which the set 
is fragrant?
is a friend of 
, then is a friend of .) designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a -best friend. More generally, if 
is a positive integer, then a user is an -best friend provided that they have been designated the best friend of someone who is an 
-best friend. Someone who is a -best friend for every positive integer is called popular.
such that the equation![\[f(xf(x)+y) = f(y) + x^2\]](http://latex.artofproblemsolving.com/2/0/a/20a245460634f0455fc703d7ce41c475c1ae4970.png)
holds for all rational numbers and 
.
denotes the set of rational numbers.
we have 
. Let be a point in segment such that 
. Show that 
.
and 
Show that
Let 
Show that
:
and 
Show that
Where 
Where 
and 
and join 
is an equilateral triangle. Now, 
.
and as 
.
also 
.
.