MathDB

15

Part of 2022 AIME Problems

Problems(2)

Circumradii???

Source: 2022 AIME I #15

2/9/2022
Let xx, yy, and zz be positive real numbers satisfying the system of equations \begin{align*} \sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\ \sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\ \sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}. \end{align*}Then [(1−x)(1−y)(1−z)]2\big[ (1-x)(1-y)(1-z) \big] ^2 can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME I
Another Hexagon Problem?

Source: 2022 AIME II Problem 15

2/17/2022
Two externally tangent circles ω1\omega_1 and ω2\omega_2 have centers O1O_1 and O2O_2, respectively. A third circle Ω\Omega passing through O1O_1 and O2O_2 intersects ω1\omega_1 at BB and CC and ω2\omega_2 at AA and DD, as shown. Suppose that AB=2AB = 2, O1O2=15O_1O_2 = 15, CD=16CD = 16, and ABO1CDO2ABO_1CDO_2 is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("ω1\omega_1",8*dir(110),SW); label("ω2\omega_2",5*dir(70)+(15,0),SE); label("O1O_1",O1,W); label("O2O_2",O2,E); label("BB",B,N+1/2*E); label("AA",A,N+1/2*W); label("CC",C,S+1/4*W); label("DD",D,S+1/4*E); label("1515",midpoint(O1--O2),N); label("1616",midpoint(C--D),N); label("22",midpoint(A--B),S); label("Ω\Omega",o.C+(o.r-1)*dir(270)); [/asy]
AMCAIMEAIME II