translated_imo_2000_p1

a b = segment a b; g1 = on_tline g1 a a b; g2 = on_tline g2 b b a; m = on_circle m

g1 a, on_circle m g2 b; n = on_circle n g1 a, on_circle n g2 b; c = on_pline c m a
b, on_circle c g1 a; d = on_pline d m a b, on_circle d g2 b; e = on_line e a c,
on_line e b d; p = on_line p a n, on_line p c d; q = on_line q b n, on_line q c d ?
cong e p e q
translated_imo_2000_p6
a b c = triangle a b c; h = orthocenter h a b c; t1 t2 t3 i = incenter2 t1 t2 t3 i
a b c; h1 = foot h1 a b c; h2 = foot h2 b c a; h3 = foot h3 c a b; x1 = reflect x1
h1 t1 t2; x2 = reflect x2 h2 t1 t2; y2 = reflect y2 h2 t2 t3; y3 = reflect y3 h3 t2
t3; z = on_line z x1 x2, on_line z y2 y3 ? cong i z i t1
translated_imo_2002_p2a
b c = segment b c; o = midpoint o b c; a = on_circle a o b; d = on_circle d o b,
on_bline d a b; e = on_bline e o a, on_circle e o b; f = on_bline f o a, on_circle
f o b; j = on_pline j o a d, on_line j a c ? eqangle e c e j e j e f
translated_imo_2002_p2b
b c = segment b c; o = midpoint o b c; a = on_circle a o b; d = on_circle d o b,
on_bline d a b; e = on_bline e o a, on_circle e o b; f = on_bline f o a, on_circle
f o b; j = on_pline j o a d, on_line j a c ? eqangle c e c j c j c f
translated_imo_2003_p4
a b c = triangle a b c; o = circle o a b c; b1 = on_circle b1 o a, on_bline b1 c a;
d1 = on_circle d1 o a, on_bline d1 c a; x = on_line x b b1, on_line x a c; d =
on_line d d1 x, on_circle d o a; p = foot p d b c; q = foot q d c a; r = foot r d a
b ? cong p q q r
translated_imo_2004_p1
a b c = triangle a b c; o = midpoint o b c; m = on_circle m o b, on_line m a b; n =
on_circle n o b, on_line n a c; r = angle_bisector r b a c, angle_bisector r m o n;
o1 = circle o1 b m r; o2 = circle o2 c n r; p = on_circle p o1 r, on_circle p o2
r ? coll p b c
translated_imo_2004_p5
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; p = on_aline p b c
a b d, on_aline p d c a d b ? cong a p c p
translated_imo_2005_p5
a b c = triangle a b c; d = eqdistance d a b c; e = on_line e b c; f = on_line f a
d, eqdistance f d e b; p = on_line p a c, on_line p b d; q = on_line q e f, on_line
q b d; r = on_line r e f, on_line r a c; o1 = circle o1 a p d; o2 = circle o2 b p
c; m = on_circle m o1 p, on_circle m o2 p ? cyclic p q r m
translated_imo_2007_p4
a b c = triangle a b c; o = circle o a b c; r = on_circle r o a, on_bline r a b; l
= midpoint l c a; k = midpoint k c b; p = on_line p o k, on_line p c r; q = on_line
q o l, on_line q c r; l1 = foot l1 l c r; k1 = foot k1 k c r ? eqratio k k1 l l1 r
q r p
translated_imo_2008_p1a
a b c = triangle a b c; h = orthocenter h a b c; d = midpoint d b c; e = midpoint e
a c; f = midpoint f a b; a1 = on_circle a1 d h, on_line a1 b c; a2 = on_circle a2 d
h, on_line a2 b c; b1 = on_circle b1 e h, on_line b1 c a; b2 = on_circle b2 e h,
on_line b2 c a; c1 = on_circle c1 f h, on_line c1 a b; c2 = on_circle c2 f h,
on_line c2 a b ? cyclic c1 c2 b1 b2
translated_imo_2008_p1b
a b c = triangle a b c; h = orthocenter h a b c; d = midpoint d b c; e = midpoint e
a c; f = midpoint f a b; a1 = on_circle a1 d h, on_line a1 b c; a2 = on_circle a2 d
h, on_line a2 b c; b1 = on_circle b1 e h, on_line b1 c a; b2 = on_circle b2 e h,
on_line b2 c a; c1 = on_circle c1 f h, on_line c1 a b; c2 = on_circle c2 f h,
on_line c2 a b ? cyclic c1 c2 b1 a1
translated_imo_2008_p6
x@4.96_-0.13 y@-1.0068968328888160_-1.2534881080682770 z@-2.8402847238575120_-
4.9117762734006830 = triangle x y z; o = circle o x y z; w@6.9090049230038776_-
1.3884003936987552 = on_circle w o x; a = on_tline a z o z, on_tline a x o x; b =
on_tline b z o z, on_tline b w o w; c = on_tline c y o y, on_tline c w o w; d =
on_tline d x o x, on_tline d y o y; i1 = incenter i1 a b c; i2 = incenter i2 a c d;
f1 = foot f1 i1 a c; f2 = foot f2 i2 a c; q t p s = cc_tangent q t p s i1 f1 i2 f2;
k = on_line k q t, on_line k p s ? cong o k o x
translated_imo_2009_p2
m l k = triangle m l k; w = circle w m l k; q = on_tline q m w m; p = mirror p q m;
b = mirror b p k; c = mirror c q l; a = on_line a b q, on_line a c p; o = circle o
a b c ? cong o p o q
translated_imo_2010_p2
a b c = triangle a b c; o = circle o a b c; i = incenter i a b c; d = on_line d a
i, on_circle d o a; f = on_line f b c; e = on_aline e a c b a f, on_circle e o a; g
= midpoint g i f; k = on_line k d g, on_line k e i ? cong o a o k
translated_imo_2010_p4
s c p = iso_triangle s c p; o = on_tline o c s c; a = on_circle a o c; b =
on_circle b o c, on_line b s a; m = on_line m c p, on_circle m o c; l = on_line l b
p, on_circle l o c; k = on_line k a p, on_circle k o c ? cong m k m l
translated_imo_2011_p6
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; q = on_tline q p o
p; pa = reflect pa p b c; pb = reflect pb p c a; pc = reflect pc p a b; qa =
reflect qa q b c; qb = reflect qb q c a; qc = reflect qc q a b; a1 = on_line a1 pb
qb, on_line a1 pc qc; b1 = on_line b1 pa qa, on_line b1 pc qc; c1 = on_line c1 pa
qa, on_line c1 pb qb; o1 = circle o1 a1 b1 c1; x = on_circle x o a, on_circle x o1
a1 ? coll x o o1
translated_imo_2012_p1
a b c = triangle a b c; m l k j = excenter2 m l k j a b c; f = on_line f m l,
on_line f b j; g = on_line g m k, on_line g c j; s = on_line s f a, on_line s b c;
t = on_line t g a, on_line t c b ? cong m s m t
translated_imo_2012_p5
c a b = r_triangle c a b; d = foot d c a b; x = on_line x c d; k = on_line k a x,
on_circle k b c; l = on_line l b x, on_circle l a c; m = on_line m a l, on_line m b
k ? cong m k m l
translated_imo_2013_p4
a b c = triangle a b c; h = orthocenter h a b c; m = on_line m h b, on_line m a c;
n = on_line n h c, on_line n a b; w = on_line w b c; o1 = circle o1 b n w; o2 =
circle o2 c m w; x = on_line x o1 w, on_circle x o1 w; y = on_line y o2 w,
on_circle y o2 w ? coll x h y
translated_imo_2014_p4
a b c = triangle a b c; p = on_line p b c, on_aline p a b b c a; q = on_line q b c,
on_aline q a c c b a; m = mirror m a p; n = mirror n a q; x = on_line x b m,
on_line x c n; o = circle o a b c ? cong o x o a
translated_imo_2015_p3
a b c = triangle a b c; h = orthocenter h a b c; f = on_line f h a, on_line f b c;
m = midpoint m b c; o = circle o a b c; q = on_dia q a h, on_circle q o a; k =
on_dia k h q, on_circle k o a; o1 = circle o1 k q h; o2 = circle o2 f k m ? coll o1
o2 k
translated_imo_2015_p4
a b c = triangle a b c; o = circle o a b c; d = on_line d b c; e = on_line e b c,
on_circle e a d; f = on_circle f o a, on_circle f a d; g = on_circle g o a,
on_circle g a d; o1 = circle o1 f b d; o2 = circle o2 g c e; k = on_circle k o1 b,
on_line k a b; l = on_circle l o2 c, on_line l a c; x = on_line x f k, on_line x l
g ? coll x o a
translated_imo_2016_p1
a b z = triangle a b z; f = angle_bisector f b a z, on_bline f a b; c = on_tline c
b f b, on_line c a f; d = on_line d a z, on_bline d a c; e = angle_mirror e c a d,
on_bline e a d; m = midpoint m c f; x = parallelogram e a m x; y = on_line y f x,
on_line y e m ? coll y b d
translated_imo_2017_p4
r s = segment r s; t = mirror t r s; o = on_bline o r s; j = on_circle j o s; o1 =
circle o1 j s t; a = on_tline a r o r, on_circle a o1 s; b = on_tline b r o r,
on_circle b o1 s; k = on_line k j a, on_circle k o s ? perp k t o1 t
translated_imo_2018_p1
a b c = triangle a b c; o = circle o a b c; d = on_line d a b; e = on_line e a c,
on_circle e a d; f = on_bline f b d, on_circle f o a; g = on_bline g e c, on_circle
g o a ? para d e f g
translated_imo_2019_p2
a b c = triangle; a1 = on_line b c; b1 = on_line a c; p = on_line a a1; q = on_line
b b1, on_pline p a b; p1 = on_line p b1, eqangle3 p c a b c; q1 = on_line q a1,
eqangle3 c q b c a ? cyclic p q p1 q1
translated_imo_2019_p6
a b c = triangle a b c; d e f i = incenter2 d e f i a b c; r = on_tline r d e f,
on_circle r i d; p = on_line p r a, on_circle p i d; o1 = circle o1 p c e; o2 =
circle o2 p b f; q = on_circle q o1 p, on_circle q o2 p; t = on_line t p q, on_line
t i d ? perp a t a i
translated_imo_2020_p1
p a b = triangle p a b; x = angle_bisector p b a; y = angle_bisector p a b; z =
on_aline z a p a b x; t = on_aline t p a p a z; d = on_aline d p t p b a, on_line a
z; u = on_aline u b p b a y; v = on_aline v p b p b u; c = on_aline c p v p a b,
on_line b u; o = angle_bisector a d p, angle_bisector p c b ? cong o a o b
translated_imo_2021_p3
a b c = triangle; d = angle_bisector b a c; e = on_aline d a d c b, on_line a c; f
= on_aline d a d b c, on_line a b; x = on_bline b c, on_line a c; o1 = circle a d
c; o2 = circle e x d; y = on_line e f, on_line b c ? coll o1 o2 y
translated_imo_2022_p4
b c = segment; d = free; e = eqdistance d b c; t = on_bline b d, on_bline c e; a =
eqangle2 b t e; p = on_line a b, on_line c d; q = on_line a b, on_line c t; r =
on_line a e, on_line c d; s = on_line a e, on_line d t ? cyclic p q r s