AoPS Academy
be a triangle and 
, 
, 
be the midpoints of arcs 
, 
, 
on the circumcircle. Line 
passes through the feet of the perpendiculars from 
to 
and 
. Line 
passes through the feet of the perpendiculars from to and 
. Let 
denote the intersection of lines and . Define points 
and 
similarly. Prove that triangle 
and 
are similar to each other.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
which satisfies all the following properties:
. Determine the smallest Norwegian number.
.)
m above the sea. The plane will aim an enemy submarine in the distance of 
m. To be aimed on the target, a missile shot by the plane plunges into the sea with a depth of 
m. Suppose the surface of the sea to be 
-axis and the plane's motion is represented by a function 
for 
. Find the value of 
.
is a function satisfying 
and 
, 
.
then minimum possible number of values of 
satisfying 
, for ![$x\in[0,170]$](http://latex.artofproblemsolving.com/3/d/2/3d2c6ead3059f96f853bba6cfc7440e394227ab9.png)
is?
$ is symmetrical about which line?
is inscribed in the circle. Construct a point 
such that the points of intersection of lines 
and 
with this circle are the vertices of an equilateral triangle.
is inscribed in the circle. Construct a point such that the points of intersection of lines and with this circle are the vertices of an equilateral triangle.
and 
such that for any point of the circle which lies on the same side of the plane as 
with respect to 
. Consider an analogous circle passing through and . Let be the point of their intersection. Then ; 
; 
. It's not hard to see that this point suffices the conditions
then
aand 
Prove that
prove that 
and 
Prove that
Prove that
