[/quote]
,
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
be positive real numbers such that 
.![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
with integer coefficients such that for all positive number 
and prime 
satisfying 
,
.
be reals such that 
P
,
,
is __
satisfying 
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
and 
Prove that
with bases 
and 
,
is a point on side 
.
and 
intersect again at 
.
and 
intersect line 
and 
respectively. Let 
and 
meet at 
.
is tangent to the circumcircle of 
.
be the circle passing through 
and tangent to 
,
be the circle passing through 
and tangent to 
,
be the circle passing through 
be the circle passing through 
be the centres of 
respectively. Prove that 
is cyclic.
,
,
and 
respectively. Let 
be an intersection point of 
be the common external tangent to 
respectively such that 
at 
intersect 
,
,
are concurrent.
and 
.
of 
meet 
hit 
at 
.
and 
,
intersects 
at 
.
is the midpoint of 
always passes through one of the intersection points of 
with the circumcircle of 
.
,
and 
.
intersects 
and 
,
intersects 
,
passes through 
,
,
are concurrent.
Y by snail2, sjaelee, xliu, waver123, dinoboy, pi37, NewAlbionAcademy, fermat007, AWCMABMV1, thecmd999, fractals, blasterboy, vanu1996, Super, parmenides51, Adventure10, Mango247, and 1 other user
Hey,
cwein3 and I wrote some geometry problems over the past week and put compiled them into a mock geometry USAMO. There's 6 questions organized over 2 "days", and I think they're probably around the difficulty of a real USAMO (1/4, 2/5, 3/6). The questions can be solved casually or under olympiad conditions, whichever preferred. PM me & cwein3 the solutions, and we'll look at them and release good ones in a week or two.
Thanks to Zhero for taking the time to proofread the questions.
Day 1:
Day 2:
cwein3 and I wrote some geometry problems over the past week and put compiled them into a mock geometry USAMO. There's 6 questions organized over 2 "days", and I think they're probably around the difficulty of a real USAMO (1/4, 2/5, 3/6). The questions can be solved casually or under olympiad conditions, whichever preferred. PM me & cwein3 the solutions, and we'll look at them and release good ones in a week or two.
Thanks to Zhero for taking the time to proofread the questions.
Day 1:
Day 2:
![[asy]
import graph; size(15cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.859553719008266, xmax = 20.44044628099176, ymin = -6.311520661157035, ymax = 6.602479338842981; /* image dimensions */
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draw((3.462000000000004,0.1440000000000002)--(8.682000000000010,-0.5160000000000006));
draw((1.982000000000002,-1.916000000000002)--(9.602702210663205,-2.879537061118338));
draw((9.602702210663205,-2.879537061118338)--(8.682000000000010,-0.5160000000000006));
draw((3.462000000000004,0.1440000000000002)--(1.982000000000002,-1.916000000000002));
draw(shift((4.174027878556640,-1.929204495273702)) * scale(2.192067649290265, 2.192067649290265)*unitcircle);
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draw((5.379658183295740,-0.09846252892244933)--(6.680058435438281,4.623189443920845));
draw((-0.7076865493527664,0.6712017476193160)--(6.680058435438281,4.623189443920845));
draw((1.982000000000002,-1.916000000000002)--(5.379658183295740,-0.09846252892244933));
draw((9.602702210663205,-2.879537061118338)--(5.379658183295740,-0.09846252892244933));
draw((-0.7076865493527664,0.6712017476193160)--(1.982000000000002,-1.916000000000002));
draw((1.982000000000002,-1.916000000000002)--(4.272335015139070,-4.119066656680650));
draw((4.272335015139070,-4.119066656680650)--(9.602702210663205,-2.879537061118338));
draw((9.602702210663205,-2.879537061118338)--(15.86249252699034,-1.423878365481537));
draw((15.86249252699034,-1.423878365481537)--(8.682000000000010,-0.5160000000000006));
draw((3.462000000000004,0.1440000000000002)--(-0.7076865493527664,0.6712017476193160));
draw((6.680058435438281,4.623189443920845)--(15.86249252699034,-1.423878365481537));
draw((3.462000000000004,0.1440000000000002)--(4.272335015139070,-4.119066656680650));
draw((4.272335015139070,-4.119066656680650)--(8.682000000000010,-0.5160000000000006));
/* dots and labels */
dot((3.462000000000004,0.1440000000000002),dotstyle);
label("$A$", (3.060446280991744,0.7284793388429741), NE * labelscalefactor);
dot((8.682000000000010,-0.5160000000000006),dotstyle);
label("$B$", (9.550446280991753,0.02447933884297325), NE * labelscalefactor);
dot((1.982000000000002,-1.916000000000002),dotstyle);
label("$D$", (1.476446280991742,-2.505520661157030), NE * labelscalefactor);
dot((9.602702210663205,-2.879537061118338),dotstyle);
label("$C$", (9.990446280991753,-3.715520661157031), NE * labelscalefactor);
dot((5.379658183295740,-0.09846252892244933),dotstyle);
label("$E$", (5.766446280991747,0.3544793388429737), NE * labelscalefactor);
dot((6.680058435438281,4.623189443920845),dotstyle);
label("$J$", (6.712446280991748,5.172479338842980), NE * labelscalefactor);
dot((4.272335015139070,-4.119066656680650),dotstyle);
label("$F$", (3.764446280991745,-4.661520661157033), NE * labelscalefactor);
dot((-0.7076865493527664,0.6712017476193160),dotstyle);
label("$H$", (-1.339553719008262,0.3764793388429737), NE * labelscalefactor);
dot((15.86249252699034,-1.423878365481537),dotstyle);
label("$G$", (16.30444628099176,-1.911520661157029), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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currentpicture = yscale(0.9999999999999444) * currentpicture;
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/7/8/2/782c037bfd0d4a8c234526bb926bd08e89a40092.png)
.
,
,
and consequently lies on their radical axis 
.
,
is cyclic; similarly quadrilateral 
is too. It follows that ![\[\angle FJH=\angle FBH=\angle ABC-\angle FBC=\angle DAB-\angle FEC=\angle DEC-\angle FEC=\angle DEF=\angle DAF=\angle JGF\]](http://latex.artofproblemsolving.com/6/b/4/6b45c686575d503d3c999bd6c89a729a2dba744d.png)
as desired. 
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
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dot((5.700000000000006,4.640000000000005),dotstyle);
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dot((14.56000000000002,-1.540000000000002),dotstyle);
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dot((1.882569703431851,0.1145542321812508),dotstyle);
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label("$O_3$", (15.76000000000002,4.900000000000006), NE * labelscalefactor);
dot((6.276763644492712,-0.3778437070865145),dotstyle);
label("$O_5$", (5.880000000000006,-1.200000000000001), NE * labelscalefactor);
dot((-0.2221408068317213,2.813893150047443),dotstyle);
label("$O_6$", (-0.9600000000000011,2.940000000000003), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/8/b/5/8b5e59d8bfeccc59da690e4ebbf7b900c0bc710e.png)
,
be the reflections of 
,
,![\[\angle OAO_1=90-\angle OO_1A-\angle O_1AD=90-\angle ABD-(\angle BAD-90)=180-\angle ABD-\angle BAD=\angle ADB\]](http://latex.artofproblemsolving.com/3/5/e/35ea9229f6a1f3b7e00ac4b6e5bb9e4119fdf074.png)
and ![\[OBO_2=90-\angle O_2OB-\angle O_2BC=90-\angle BAC-(\angle ABC-90)=180-\angle BAC-\angle ABC=\angle ACB\]](http://latex.artofproblemsolving.com/c/a/1/ca1928ff40bb65db923a0272b227b5925b2fab05.png)
it follows that 
,
;
.
,
is cyclic. But this follows from 
,
as desired. ![[asy]
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dot((4.273306006905960,1.849809175737665),dotstyle);
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/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/6/b/8/6b83d248544dc3ef736fd9a65abe12578ce30113.png)
,
.
.
and consequently quadrilateral 
is cyclic. We prove a lemma.
.
.
.
passes through the midpoint of the arc intercepted by 
in 
does too, whence 
.
,
has the same power with respect to 
.
,
is cyclic and consquently 
.
,
.
is cyclic, whence 
by 
.
and consequently ![[asy]
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dot((6.854204497792615,4.630248519334468),dotstyle);
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dot((6.777611414820138,-2.790737466733881),dotstyle);
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dot((13.37303405584031,4.320490746967888),dotstyle);
label("$G$", (14.00000000000002,4.460000000000005), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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[/asy]](http://latex.artofproblemsolving.com/8/b/0/8b026321b81c0a00359479bd210291f1c13e2f63.png)
.
,
,
.
by Inscribed Angles, it follows that 
,
.
,
is cyclic and consequently ![[asy]
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pen dotstyle = black; /* point style */
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draw(circle((20.33788971058545,-2.520679281190164), 7.508418971312913));
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draw((9.621819796694226,10.70940187107439)--(-1.973852203305790,-5.846640928925633));
draw((-1.973852203305790,-5.846640928925633)--(26.48790176628613,-6.828080720980520));
draw((9.621819796694226,10.70940187107439)--(13.90577639669423,-6.394214328925631));
draw((-1.973852203305790,-5.846640928925633)--(12.83931209990264,-2.136375670306571));
draw((2.994905439290894,1.247640816337410)--(13.90577639669423,-6.394214328925631));
draw((2.994905439290894,1.247640816337410)--(26.48790176628613,-6.828080720980520));
draw((26.48790176628613,-6.828080720980520)--(6.931869783045715,6.868751018254012));
draw((26.48790176628613,-6.828080720980520)--(9.621819796694226,10.70940187107439));
draw((5.965962096694221,-6.120427628925633)--(16.27371695899529,3.792707709483206));
/* dots and labels */
dot((-1.973852203305790,-5.846640928925633),dotstyle);
label("$A$", (-2.779107203305792,-6.877367328925635), NE * labelscalefactor);
dot((13.90577639669423,-6.394214328925631),dotstyle);
label("$B$", (13.22936219669423,-7.424940728925635), NE * labelscalefactor);
dot((9.621819796694226,10.70940187107439),dotstyle);
label("$C$", (9.492978996694225,11.51465687107439), NE * labelscalefactor);
dot((12.83931209990264,-2.136375670306571),dotstyle);
label("$A'$", (12.26305619669423,-3.205404528925630), NE * labelscalefactor);
dot((2.994905439290894,1.247640816337410),dotstyle);
label("$B'$", (1.698110596694214,1.078552071074374), NE * labelscalefactor);
dot((9.146971483396705,-3.061199214553821),dotstyle);
label("$H$", (9.009825996694225,-4.171710528925631), NE * labelscalefactor);
dot((26.48790176628613,-6.828080720980520),dotstyle);
label("$P$", (27.36963999669425,-7.585991728925635), NE * labelscalefactor);
dot((5.965962096694221,-6.120427628925633),dotstyle);
label("$M$", (5.756595796694220,-7.392730528925635), NE * labelscalefactor);
dot((14.18787765488478,1.786722158600196),dotstyle);
label("$X$", (14.06682739669423,2.656851871074376), NE * labelscalefactor);
dot((16.27371695899529,3.792707709483206),dotstyle);
label("$Y$", (15.06534359669423,3.687578271074377), NE * labelscalefactor);
dot((6.931869783045715,6.868751018254012),dotstyle);
label("$P'$", (5.821016196694220,7.391751271074382), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/0/9/d/09d1fd0ea7cc81650ebca457ed84e81648a1954d.png)
.
,
.
,
,
.
,
is cyclic, so 
by Power of a Point and consequently quadrilateral 
is cyclic. It follows that 
as desired. ![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.341487603305790, xmax = 23.33851239669423, ymin = -8.538016528925631, ymax = 5.821983471074389; /* image dimensions */
/* draw figures */
draw(circle((5.838512396694219,-3.198016528925624), 4.933153149862676));
draw(circle((5.194533122133551,-0.3496239360040415), 2.012870685874950));
draw(circle((12.25853954014948,-3.667774612593097), 7.127678227422344));
draw((5.194533122133551,-0.3496239360040415)--(7.202036938385524,-0.4965144591444284));
draw((5.194533122133551,-0.3496239360040415)--(6.015521775548982,-0.7352640238803320));
draw((6.015521775548982,-0.7352640238803320)--(7.202036938385524,-0.4965144591444284));
draw((7.400274667539871,2.212734505964979)--(5.194533122133551,-0.3496239360040415));
draw((7.598512396694217,4.921983471074386)--(0.9185123966942146,-2.838016528925624));
draw((0.9185123966942146,-2.838016528925624)--(10.75851239669422,-3.558016528925624));
draw((10.75851239669422,-3.558016528925624)--(7.598512396694217,4.921983471074386));
draw((7.598512396694217,4.921983471074386)--(6.648300073278480,-8.064251615607350));
draw((6.779495663839028,0.8911589376064901)--(5.251547887258935,-2.361686985367154));
draw((9.323130937282285,0.2938932102557798)--(0.9185123966942146,-2.838016528925624));
draw((4.750655853390131,1.613695031547480)--(10.75851239669422,-3.558016528925624));
/* dots and labels */
dot((7.598512396694217,4.921983471074386),dotstyle);
label("$V$", (7.558512396694221,5.401983471074389), NE * labelscalefactor);
dot((0.9185123966942146,-2.838016528925624),dotstyle);
label("$P$", (0.2985123966942140,-3.258016528925624), NE * labelscalefactor);
dot((10.75851239669422,-3.558016528925624),dotstyle);
label("$Q$", (11.11851239669422,-3.698016528925625), NE * labelscalefactor);
dot((4.750655853390131,1.613695031547480),dotstyle);
label("$N$", (4.358512396694217,2.141983471074384), NE * labelscalefactor);
dot((9.323130937282285,0.2938932102557798),dotstyle);
label("$R$", (9.818512396694222,0.3619834710743811), NE * labelscalefactor);
dot((7.348014660938355,1.498514415744268),dotstyle);
label("$B$", (7.798512396694221,1.481983471074383), NE * labelscalefactor);
dot((6.998157367108419,-3.282868599931541),dotstyle);
label("$U$", (7.078512396694220,-3.158016528925624), NE * labelscalefactor);
dot((6.648300073278480,-8.064251615607350),dotstyle);
label("$C$", (6.718512396694220,-7.938016528925632), NE * labelscalefactor);
dot((7.202036938385524,-0.4965144591444284),dotstyle);
label("$A$", (7.598512396694221,-0.1380165289256197), NE * labelscalefactor);
dot((7.400274667539871,2.212734505964979),dotstyle);
label("$W$", (7.718512396694221,2.421983471074384), NE * labelscalefactor);
dot((5.194533122133551,-0.3496239360040415),dotstyle);
label("$O$", (4.458512396694219,-0.4380165289256202), NE * labelscalefactor);
dot((6.779495663839028,0.8911589376064901),dotstyle);
label("$S$", (6.738512396694220,0.3019834710743810), NE * labelscalefactor);
dot((5.251547887258935,-2.361686985367154),dotstyle);
label("$T$", (5.538512396694219,-2.878016528925623), NE * labelscalefactor);
dot((5.899710010557591,-0.9818155313067233),dotstyle);
label("$M$", (6.258512396694220,-1.178016528925621), NE * labelscalefactor);
dot((6.015521775548982,-0.7352640238803320),dotstyle);
label("$O'$", (5.998512396694219,-1.558016528925622), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/f/5/5/f55ee5ddced75787e1303b6220d5781e013f8f23.png)
,
,
,
.
be the center of 
be the radical center of 
.
and consequently 
.
,
,
,
is cyclic.![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.300000000000004, xmax = 23.38000000000002, ymin = -8.060000000000010, ymax = 6.300000000000007; /* image dimensions */
/* draw figures */
draw(circle((8.748760599019485,-2.840996216591040), 4.982724865980863));
draw((5.074655876575544,5.435841981376004)--(3.838006744831395,-3.684823019697467));
draw((5.074655876575544,5.435841981376004)--(13.66000000000002,-2.000000000000000));
draw((8.880000000000011,2.140000000000002)--(3.838006744831395,-3.684823019697467));
draw((4.240000000000005,-0.7200000000000008)--(13.66000000000002,-2.000000000000000));
draw((8.880000000000011,2.140000000000002)--(-2.269854448419569,-4.732539595362065));
draw((-2.269854448419569,-4.732539595362065)--(13.66000000000002,-2.000000000000000));
draw((10.34800117536609,0.8685512832603393)--(-2.269854448419569,-4.732539595362065));
/* dots and labels */
dot((4.240000000000005,-0.7200000000000008),dotstyle);
label("$A$", (3.500000000000004,-0.6400000000000007), NE * labelscalefactor);
dot((8.880000000000011,2.140000000000002),dotstyle);
label("$B$", (9.300000000000011,2.540000000000003), NE * labelscalefactor);
dot((13.66000000000002,-2.000000000000000),dotstyle);
label("$C$", (14.24000000000002,-2.460000000000003), NE * labelscalefactor);
dot((3.838006744831395,-3.684823019697467),dotstyle);
label("$D$", (3.420000000000004,-4.460000000000005), NE * labelscalefactor);
dot((6.176590583262808,-0.9831460665155412),dotstyle);
label("$E$", (6.240000000000007,-1.720000000000002), NE * labelscalefactor);
dot((-2.269854448419569,-4.732539595362065),dotstyle);
label("$F$", (-2.960000000000003,-5.300000000000006), NE * labelscalefactor);
dot((10.34800117536609,0.8685512832603393),dotstyle);
label("$G$", (10.80000000000001,1.140000000000002), NE * labelscalefactor);
dot((11.27000000000001,0.07000000000000015),dotstyle);
label("$M$", (11.70000000000001,0.3200000000000003), NE * labelscalefactor);
dot((5.074655876575544,5.435841981376004),dotstyle);
label("$H$", (4.780000000000006,5.900000000000007), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/d/5/5/d559fefd043ca574b3f236613cc058581ae58449.png)
.
.
,
is the midpoint of 
,
and consequently the midpoint of 
.
,
is cyclic, or equivalently, that 
.
,
by Ceva/Menelaus.
,
,
is tangent to 
.
be the midpoint of 
,
is cyclic, whence 
and consequently 
