be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
.
be the second intersection of the circumcircles of triangles 
and 
.
,
,
-
are concurrent.
that satisfy the condition 
for the natural number 
Here 
and 
are natural numbers. Is there a natural number 
such that each of the numbers 
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
we have 
.
.
such that
and 
Prove that
be a function defined in any real numbers 
with 
Prove that on the 
plane, the line 
passes through the fixed point which isn't on the 
axis in regardless of the value of 
and 
.
foot radius. Bruno completes 
laps every hour, while Brutus completes 
laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
Prove that
Is there an elegant way to prove this?
Product of all even divisors of 
If 
dice are rolled, Then number of ways of getting sum at least 
is
Y by Adventure10
Let 
be an acute triangle and let 
, 
, 
be points on the sides 
and 
, respectively. Show that the triangles and 
are similar (
) if and only if the orthocentre of the triangle and the circumcentre of the triangle coincide.
be an acute triangle and let 
, 
, 
be points on the sides 
and 
, respectively. Show that the triangles and 
are similar (
) if and only if the orthocentre of the triangle and the circumcentre of the triangle coincide.This post has been edited 2 times. Last edited by parmenides51, Dec 18, 2022, 7:09 PM
