AoPS Academy
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)
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 positive integers. Maximize 
if 
.
such that ![\[133^5 + 110^5 + 84^5 + 27^5 = n^5.\]](http://latex.artofproblemsolving.com/8/4/f/84f861cc58e03fafd8008e312693ba46e25ce95e.png)
Find the value of .
such that 
divides 
.
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
such that:
be a natural number and 
be a prime number. How many different pairs 
satisfy the equation:
such that 
and 
are both perfect cubes.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
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.
be 
, and 
. What is the sum 
? through , shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
.
and 
are elementary symmetric polynomials give: