[/quote]
,
be an acute triangle and 
it's incenter. Let 
,
and 
be the projections of 
,
and 
respectively. 
and 
is the projection of 
.
be the middle of the arc 
and 
the second intersection of 
and the circumcircle of 
.
is the midpoint of 
,
,
and 
,
is cyclic.
denote the midpoint of side 
and 
intersect in point 
.
be the second point of intersection of the circumcircles of triangles 
and 
.
and 
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
and 
are 
and 
,
intersects line 
and 
intersects line 
and 
.
.
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
such that points 
are fixed, while 
intersects 
is tangent to a fixed circle whose radius is half the radius of 
and 
be the circumcenter, centroid and symmedian point of 
respectively. 
Show that the 
falls on 
such that if 
,
by 
also belongs to 
.
,
,
is at most 
denotes the image of 
be positive integers such that 
is a square. Prove that 
is not prime.
with parallel sides 
and 
,
on line 
outside segment 
inside segment 
such that 
.
the point of intersection of 
,
the point of intersection of 
be the midpoint of segment 
if and only if 
.
such that 
is the square of an integer.
prove that
satisfy 
.
Y by
The numbers 
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:24 PM
