AoPS Academy
to the system of equations![\[\begin{aligned}
\begin{cases}
x^2+y^2-xy-3x+3=0,\\
x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/4/a/b/4ab8e94e2e39be219c80264aa5ea23456b74b3e8.png)
such that 
and 
, evaluate 
.
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
such that 
.
Is there an elegant way to prove this?
Product of all even divisors of 
If 
dice are rolled, Then number of ways of getting sum at least 
is
, 
, 
, 
such that ![\[\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.\]](http://latex.artofproblemsolving.com/3/6/2/362fccfa6353ffcbc4d29f24ccbfce73269080bf.png)
Compute 
.
be an isosceles triangle such that 
. Let 
be a dot on the 
side.
at point intersects the circumcircle of 
at 
. Prove that CD
AB
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of . Let 
intersect 
at , 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
be the intersection point of medians in the acute-angled triangle 
,
and 
.Prove that is equilateral triangle.
be the point of symmetry of with respect to the midpoint of 
,then quadrilateral 
is a parallelogram,which follows that 
,and hence 
are concyclic.Therefore 
,and 
.So 
,then 
is a diameter of the circumcircle of 
,and 
.Combining 
,we get that is equilateral,as desired.
Prove that
and 
be the orthocentre and circumcentre of 
and 
,
lies in the same circle. But we also know that 
collinear and 
,
are same, and thus