AoPS Academy
?
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
. Each step the pig has probability 
of moving 
unit in the positive direction and probability 
of moving unit in the negative direction. Find the expected number of steps until the yellow pig visits 
.
, the sum of the products of the elements taken 2 at a time is 
; and the sum of the products of the elements taken 3 at a time is 
. Let 
be the sum of the products of the first 2000 positive integers, taken n at a time. For example, 
= the sum of the first 2000 positive integers, and 
= the sum of the products of the 2000 positive integers, taken two at a time, etc. What is the value of ++
+..........+
+
?
be positive integers such that 
. Find 
of 
be a cyclic quadrilateral. Points 
and 
lie on the sides 
and 
in such a way that 
and 
. Let 
be the midpoint of 
. Prove that 
.
be the reflection of 
with respect to . The quadrilateral 
will be a parallelogram, then the triangle 
will be congruent to the triangle 
, from here we get that 
. Therefore, 
is an isoscele triangle, and since 
is the median, it will also be the height.
intersects 
at 
.
and 
are isoscele triangles. Let 
and 
be the midpoints of 
and 
.
is similar to 
and take 
such that these triangles are similar to 
that sends 
to 
. Thus there exists a spiral similarity with center that sends 
to 
hence 
. Similarly 
is rhomboid 
,
be the point such that 
is a parallelogram. A bit of angle chasing (computing angles at combined with the fact that 
and 
are isosceles) yields 
. This means is the circumcenter of 
and the perpendicular bisectors of 
and 
are perpendicular. But these bisectors are precisely 
and , respectively. The end.
in natural numbers m and n.
is an integer. Prove that 
are integers.
(mod 
)
