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the product of all 
's positive divisors, where 
. such that ![$\sqrt[2024]{p(n)} \in \mathbb{Z}$](http://latex.artofproblemsolving.com/7/0/4/70442d4ec6ae8b7a3cf9717719e862bf3b5d4a28.png)
and is not a perfect power of another integer.
of the triangle 
is extended beyond 
to 
so that 
. The side 
is extended beyond 
to 
so that 
. Prove that, if 
, then the triangle is right-angled.
, we construct equilateral triangle 
such that 
is on the same side of 
as and . It is given that 
intersects 
at 
. Prove that there exists a point 
on 
such that 
is equilateral.
+1 w
where 
are non-negative integers that are not necessarily distinct. Find the smallest positive integer such that no multiple of is a fancy number. , on the hypotenuse 
, the point chosen 
is such that 
. Inside the 
, a point 
lies on the bisector of 
, such that 
. Prove that:
is isosceles.
, on the hypotenuse , the point chosen is such that . Inside the , a point lies on the bisector of , such that . Prove that: is isosceles.
as givens.
and thus, 
. Thus, since , we have 
.
are cyclic
.
. Thus, 
, and 
. Now, if we let 
, we have![\[\angle AXB = 180-\angle XAB-\angle XBA = 180-60-30=90\]](http://latex.artofproblemsolving.com/3/5/9/359bc6779cc2cb5e20bf1385865b15eba53125cc.png)
Thus, part (a) is done, 
.
, then trig ceva's on F gives![\[\sin 15 \cdot \sin 30 \cdot \sin (45-x) = \sin 75 \cdot \sin 15 \cdot \sin x \]](http://latex.artofproblemsolving.com/d/c/e/dce66decec8740321a34bb95f5622f48255c503d.png)
Solving gives the solution 
since 
.
. Thus, 
. Thus, 
and 
is isosceles and we are done with (b).
