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for which there exists positive integers 
and 
satisfying ![\[ \gcd(a,b) + \operatorname{lcm}(b,c) = \operatorname{lcm}(c,a)^3. \]](http://latex.artofproblemsolving.com/0/6/0/0602a5b960fc784175f1208cf3db1d244c067377.png)
for which the roots of the equation 
are integers.
for some base 
, 
. The points 
, 
, and 
lie on the graph of 
. If 
, find the value of 
.
and 
not is a perfect square for all 
.
be a scalene triangle with circumcircle 
and incenter 
. Ray 
meets 
at 
and meets again at 
; the circle with diameter 
cuts again at 
. Lines 
and 
meet at 
, and 
is the midpoint of 
. The circumcircles of 
and 
intersect at points 
and 
. Prove that passes through the midpoint of either 
or 
.
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 
?-by- board of unit squares for some odd positive integer . We say that a collection of identical dominoes is a maximal grid-aligned configuration on the board if consists of 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let 
be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
be a polynomial with nonnegative integer coefficients. If 
and 
, what is the remainder when 
is divided by 
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
Prove that
Let 
Prove that
. Prove that
