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This post has been edited 5 times. Last edited by B1t, Apr 30, 2025, 6:56 AM
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Balkan MO 2022/1 is reborn
Assassino9931 7
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by Rayvhs
Source: Bulgaria EGMO TST 2023 Day 1, Problem 1
Let 
be a triangle with circumcircle 
. The tangents at 
and 
intersect at 
. The circumcircle of triangle 
intersects the line 
at 
and 
is the midpoint of 
. Prove that the lines 
and 
intersect on .
be a triangle with circumcircle 
. The tangents at 
and 
intersect at 
. The circumcircle of triangle 
intersects the line 
at 
and 
is the midpoint of 
. Prove that the lines 
and 
intersect on .
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weird Condition
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Source: Mongolian TST 2025 P4
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Y by
Let 
midpoint of arc 
on 
then let 
notice that 
so 
is cyclic and thus by the perpendiculars we get 
colinear, now let 
the A-excenter of 
, let 
the E,F excenters of 
respectively then finally let 
the D'-queue point of 
, notice how we have 
from the condition but also from radax 
colinear which forces that 
and 
and thus we are done by I-E Lemma.
midpoint of arc 
on 
then let 
notice that 
so 
is cyclic and thus by the perpendiculars we get 
colinear, now let 
the A-excenter of 
, let 
the E,F excenters of 
respectively then finally let 
the D'-queue point of 
, notice how we have 
from the condition but also from radax 
colinear which forces that 
and 
and thus we are done by I-E Lemma.
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#8
Apr 29, 2025, 11:59 PM
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Sadigly wrote:
Probably from IMO SL, since this problem is also on Azerbaijan IMO TST, so please remove this post
No, it's not from IMO SL, but it's very curious where is it from then, given the coincidence.
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Assassino9931 wrote:
Sadigly wrote:
Probably from IMO SL, since this problem is also on Azerbaijan IMO TST, so please remove this post
No, it's not from IMO SL, but it's very curious where is it from then, given the coincidence.
What should we do with this problem — should we keep it or remove it? Problem 6 was similarly a duplicate and was removed before anyone wrote a solution. This one, however, has already been discussed, but I can still remove the statement if necessary — should I?
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