[/quote]
,
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be a point on side 
be a point on side 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
be the intersection point of 
lies on the arc 
.
.
Y by Adventure10, Mango247
Let 
be a trapezoid of bases 
and 
. Let 
be the intersection point of the diagonals 
and 
. If the area of the triangle 
is 
and the area of the triangle 
is 
, calculate the area of the triangle 
.
be a trapezoid of bases 
and 
. Let 
be the intersection point of the diagonals 
and 
. If the area of the triangle 
is 
and the area of the triangle 
is 
, calculate the area of the triangle 
.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:36 AM
![$[ABC]+[ABD]-[ACD]-[BCD]=150+150-120-120=60$](http://latex.artofproblemsolving.com/0/e/6/0e655ed90d88ed3bbbca42da0c2a7e3db52d9ee4.png)
.![$[AOB]-[CDO]=\frac{60}{2}=30$](http://latex.artofproblemsolving.com/d/7/d/d7dfa9e014a8b61b6c888ca41b2b11485a031d07.png)
.![$\sqrt{270}=\sqrt{[AOB]}+\sqrt{[CDO]}$](http://latex.artofproblemsolving.com/f/9/2/f92033c0f45e3665eb095a7d35d6b25e16ae88f1.png)
.![$[CDO]=[AOB]-30$](http://latex.artofproblemsolving.com/3/b/8/3b8b4ffcaeedf29ab5c152404ef47b3aad0f985e.png)
.![$$\sqrt{270}=\sqrt{[AOB]}+\sqrt{[CDO]}$$](http://latex.artofproblemsolving.com/4/6/8/4682147fa983dedc89a83a35b2202b44a6d1f664.png)
![$$\left(\sqrt{270} \right)^2=\left(\sqrt{[AOB]}+\sqrt{[AOB]-30} \right)^2$$](http://latex.artofproblemsolving.com/d/c/8/dc84dfa6c9e558569bd16412575b0273591e8d02.png)
![$$270=[AOB]+[AOB]-30+2\cdot \sqrt{[AOB]} \cdot \sqrt{[AOB]-30}$$](http://latex.artofproblemsolving.com/f/3/6/f36f833e70f0c5c5651fcba283ebde242f03f4b1.png)
![$$300=2[AOB]+2\sqrt{[AOB]\left([AOB]-30\right)}$$](http://latex.artofproblemsolving.com/3/1/4/31495e0f1549b0849f4730ac8bc93a2b70fee912.png)
![$$150=[AOB]+\sqrt{[AOB]^2-30[AOB]}$$](http://latex.artofproblemsolving.com/8/2/4/8241e329e249043ff727cd7ff58a5d9dfbe9303b.png)
![$$150-[AOB]=\sqrt{[AOB]^2-30[AOB]}$$](http://latex.artofproblemsolving.com/a/7/4/a74c2a999fc88591342b627d1291548f02ecb6ff.png)
![$$(150-[AOB])^2=\left(\sqrt{[AOB]^2-30[AOB]}\right)^2$$](http://latex.artofproblemsolving.com/2/4/7/247214da129a0e07d706c1f1002c403d8da94e5b.png)
![$$150^2-300[AOB]+[AOB]^2=[AOB]^2-30[AOB]$$](http://latex.artofproblemsolving.com/1/e/d/1edf94c253d9f964793af213ac8f0b38ca134b5b.png)
![$$22500=[AOB]^2-[AOB]^2-30[AOB]+300[AOB]$$](http://latex.artofproblemsolving.com/1/c/6/1c6f07969aa4ebfbb7c483d2c9eb8a75d0d422f2.png)
![$$22500=270[AOB]$$](http://latex.artofproblemsolving.com/d/2/a/d2a8e5b281281192e2a23f41ea612f230cd27f31.png)
![$$\frac{22500}{270}=[AOB]$$](http://latex.artofproblemsolving.com/5/1/1/511b406fab710059d94f22cb642c7d765cdc0192.png)
![$$\frac{250}{3}=[AOB]$$](http://latex.artofproblemsolving.com/e/9/e/e9efa077b70fa137a23cc84923b4dc5d6f0e4008.png)
Then ![$[BCO]=[ABC]-[AOB]=150-\frac{250}{3}=\boxed{\frac{200}{3}}$](http://latex.artofproblemsolving.com/4/1/d/41d6fd7d547da32d9d895019e3cac886040ef8e2.png)
![$[BOC]=[AOD]$](http://latex.artofproblemsolving.com/1/5/0/1507e2424c0a891bc8b6b7b69d2d1141ca6eee16.png)
and ![$[BOC]^2=[ODC][OAB]$](http://latex.artofproblemsolving.com/2/c/5/2c544efd7d1a70f033298d6aa945cebdb7786509.png)
![${[BOC]^2} = \left( {120 - [AOD]} \right)\left( {150 - [BOC]} \right) \Leftrightarrow {[BOC]^2} = 120 \cdot 150 - 270[BOC] + {[BOC]^2} \Leftrightarrow $](http://latex.artofproblemsolving.com/1/4/3/143b1f78562e2c59c08e9223ce96d209fe812966.png)
![$\boxed{[BOC] = \frac{{200}}{3}}$](http://latex.artofproblemsolving.com/1/f/8/1f8fc68f2f3d0d8dd0cafda1bdd6e5efe2fb84fa.png)
