AoPS Academy
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment if and only if the following condition holds: distinct from 
and 
, there exists 
such[/center]
is directly similar to either 
or 
.[/center]
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to , 
to , and 
to 
.
and lie in a straight line in that order. Suppose that 
is a point not on the line and that 
and 
Find the area of 
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
and be positive integers with 
. Let 
be a polynomial of degree with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers 
such that the polynomial 
divides , the product 
is zero. Prove that has a nonreal root.
people around in a circle, who at 
each have one penny. After each second every person gives deals penny they have to the left or right, or keeps that penny with equal probability. What is the expected value of seconds such that one person has all pennies? (Express answer in terms of ).
but I’m not sure. Purely guessing.
the answer is 
because from any arrangement there is exactly 
way to get to any of the other arrangements (consider the problem from the coin’s perspective as they switch humans. Label the humans 
. Therefore the initial arrangement is . Every time we add or subtract or leave alone each of the numbers. We can only obtain, say, 
(where every coin is in the second person’s hand), in one way: that is 
. This is true for every possible arrangement of coins from any possible arrangement of coins. Hence the probability of landing on 
or or 
is 
and the answer is .
?
grid if the centres of no four colored squares form a parallelogram.
