AoPS Academy
and 
not is a perfect square for all 
.
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 
?
be a polynomial with nonnegative integer coefficients. If 
and 
, what is the remainder when 
is divided by 
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?
on ![$[1,n]$](http://latex.artofproblemsolving.com/2/2/4/2240bb122bdadbe086c0a6291dcd5a3eb82ad5cf.png)
as a word with 
in some order, hence 
is a permutation on 5 symbols. We write 
for the wellness of namely the amount of pairs 
that appear in the same order in the word. Here 
.
for the number of 
with wellness . Hence we want to determine 
. hence 
. The symbol k adds 
to the wellness hence now we count the permutations on ![$[1,7]\setminus \{k\}$](http://latex.artofproblemsolving.com/2/a/3/2a3fb0361b0937dbfba193cae8ae422eb3328c31.png)
with wellness . Since we do not consider the symbol anymore, we may as well count the number of permutations on ![$[1,6]$](http://latex.artofproblemsolving.com/a/b/0/ab036e7da17bee704b3b7ee0ca5ea79f424465e5.png)
with wellness . This equals 
.
.. If the first symbol is 
then it contributes 
to the wellness hence 
. We find 
. The lower bound may be 
, in that case we start counting at 
. This reduces the problem to finding the numbers 
. Etc.
and then work your way upwards.
of 
by a unique 
-tuple of integers 
such that 
for all valid , where each 
represents the number of pairs of integers 
satisfying 
. Thus, we seek the number of -tuples such that for all valid and 
.
in the expansion of 
. But notice that 
is a root of 
. Hence, the sum of the coefficients of the terms of with degree divisible by is 
. Observe that the coefficients of 
and 
are trivially each , and the coefficients of 
and 
are equal due to symmetry. Thus, the coefficient of is simply 
.
satisfying 
and 
.