[/quote]
,
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
and 
Prove that
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.
is recursively defined such that 
is not a power of 
,
:
is even, then 
is the largest odd factor of 
where 
is the smallest prime factor of 
for all 
.
such that 
is the square of an integer.
denote the midpoint of side 
and 
intersect in point 
.
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 
.
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 
.
Y by
A non-degenerate triangle 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?
Note: The function is called injective if for any 
, 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?Note: The function
is called injective if for any 
, 
This post has been edited 1 time. Last edited by 827681, Apr 20, 2025, 6:24 PM
