AoPS Academy
![$\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$](http://latex.artofproblemsolving.com/a/f/3/af33799c6924b61378db323645ea3e1b9c4e8c9d.png)
if ![$\color[rgb]{0.35,0.35,0.35}\sin x=1/3$](http://latex.artofproblemsolving.com/c/e/a/cea5c46445caa512d94d655350170fb19518a2c8.png)
and ![$\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$](http://latex.artofproblemsolving.com/9/8/b/98b7d75c944002b8b9c11dbe8cca0d402e6fbb67.png)
.
+1 w![$\color[rgb]{0.35,0.35,0.35}v_1=2$](http://latex.artofproblemsolving.com/2/0/4/204b13717fb1dec258e4917232ca9c96344031d2.png)
, ![$\color[rgb]{0.35,0.35,0.35}v_2=4$](http://latex.artofproblemsolving.com/b/c/2/bc24e9589aaf75cb848e9fddeef43fb7a6fea2c3.png)
and ![$\color[rgb]{0.35,0.35,0.35} v_{n+1}=3v_n-v_{n-1}$](http://latex.artofproblemsolving.com/8/7/3/8734d09c683e906771abf049d89b273196175585.png)
, prove that ![$\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$](http://latex.artofproblemsolving.com/b/1/e/b1efbac9eef93c5649dab70be54ae2286ca5b2f0.png)
, where the terms ![$\color[rgb]{0.35,0.35,0.35}F_n$](http://latex.artofproblemsolving.com/0/2/5/025c95f57f5317da3d9739c1620448d8f7cf4e47.png)
are the Fibonacci numbers.
, and square 
. Point 
and 
are on the circle, and 
is tangent to the circle at point 
. Let 
represent the midpoint of 
and 
represent the intersection between and circle. Prove that 
.
be a prime number. Define the set![\[
M_q = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2q^{2025} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/9/8/b/98b266a869cf5686cea86be51152ef234486993e.png)
.
be positive real number satisfy that: 
.Find the minimum: 
, holds 
with 
, holds 
and 
are points on the segments 
and 
, respectively, such that 
and 
. If the area of triangle 
is 
, then what is the area of triangle 
?
![$\frac{[AXY]}{[BXY]}=\frac{AX}{BX}=\frac12$](http://latex.artofproblemsolving.com/2/a/9/2a915730c9421688ecd5dd77f0f60dfe0d3cda51.png)
(Collinear Bases with common vertex (since their altitude is the same, the ratio of area is equal to the ratio of bases))![$[BXY]=20,[BYA]=30$](http://latex.artofproblemsolving.com/c/1/1/c11a0d75f8c6afb22d9554224f6486925cd1e0bd.png)
![$\frac{[BYA]}{[BYC]}=\frac{AY}{YC}=\frac21$](http://latex.artofproblemsolving.com/7/9/5/7953177214258046e77563174589ae9bf1d29dcc.png)
![$[BYC]=15$](http://latex.artofproblemsolving.com/c/0/a/c0a16d4aee2e2d6e983ffc7cd18a78f83d9a3f89.png)
![$[ABC]=10+20+15 = 45$](http://latex.artofproblemsolving.com/0/0/8/008536cc0eaa4126052ed67d163c57fc2ff24788.png)
(Collinear Bases with common vertex (since their altitude is the same, the ratio of area is equal to the ratio of bases))
.
be real numbers so that 
and 
Show that
![\[ \frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}. \]](http://latex.artofproblemsolving.com/c/f/0/cf007b7ce24aa95493617d1f4011e451e7c49126.png)
.
?
![$$[AXY]=\frac12 AY \cdot AX \cdot \sin\angle XAY$$](http://latex.artofproblemsolving.com/c/1/6/c164ab44d74478c0782b3ba13e3ec2b2f33d05da.png)
and ![$$[ABC]=\frac12 AC\cdot AB \cdot \sin\angle BAC=\frac92 AC\cdot AB \cdot \sin\angle XAY.$$](http://latex.artofproblemsolving.com/e/8/9/e89845bd6394a47149e5a69ea81809fead89b7b9.png)
From this, we conclude ![$[ABC]=9\cdot [AXY]=\boxed{90}$](http://latex.artofproblemsolving.com/0/7/a/07ad2b699249fc61c924e6dd687410f28213d9cf.png)
as the answer