be a triangle, and let 
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, Mango247
and 
are perpendicular diameters of a circle. 
is the tangent to the circle at 
. 
is a variable point on the minor arc 
. The ray 
meet the line at 
and 
respectively. Show that 
.Show how to construct the point
which gives
parallel to 
.If the lines
and 
meet at 
find the locus of .The lines
and meet the tangent at 
at 
and 
respectively. Find the relation between and '.The lines
P and 
meet the line 
at 
and 
respectively. Find the relation between and .
.
,
.
.
.
.
cuts the line 
.
,
with 
.
