[/quote]
,
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
be circumcenter of a non-isosceles triangle 
.
be foots of perpendicular lines from 
.
is cyclic and 
is the circumcenter of 
.
bisects 
such that 
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
.
is at least rational, so that 
must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs 
for which this happens are 
and 
,
,
The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?
interesting if 
of 
and 
is also a divisor of 
such that 
holds for all real numbers 
and 
.
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
be a triangle with 
.
be the midpoint of 
,
be a point on segment 
.
be the point of intersection, different from 
,
and line 
.
and 
as the points of intersection of line 
with 
and 
,
.
with 
on side 
be the reflection of 
such that lines 
and 
and 
meet again at 
.
.
Y by
So someone pointed out to me that the last five problems on my previous practice AMC 10 test were rather low quality. Here are some problems that are (hopefully) better.
21.
A partition of a positive integer n is writing n as the sum of positive integer(s), where order does not matter. Find the number of partitions of 6.
A - 10, B - 11, C - 12, D - 13, E - 14
22.
Let n be the smallest positive integer that satisfies the following conditions:
- n is even
- The last digit of n is not 2 or 8
- n^2 + 1 is composite
Find the sum of the digits of n.
A - 3, B - 5, C - 8, D - 9, E - 10
23.
Find the sum of the coordinates of the reflection of the point (6, 9) over the line x + 2y + 3 = 0.
A - (-17.7), B - (-17.6), C - (-17.5), D - (-17.4), E - (-17.3)
24.
Find the number of ordered pairs of integers (a, b), where both a and b have absolute value less than 69, such that a^2 + 42b^2 = 13ab.
A - 21, B - 40, C - 41, D - 42, E - 69
25.
Let f(n) be the sum of the positive integer factors of n, where n is an integer. Find the sum of all positive integers n less than 1000 such that f(f(n) - n) = f(n).
A - 420, B - 530, C - 690, D - 911, E - 1034
21.
A partition of a positive integer n is writing n as the sum of positive integer(s), where order does not matter. Find the number of partitions of 6.
A - 10, B - 11, C - 12, D - 13, E - 14
22.
Let n be the smallest positive integer that satisfies the following conditions:
- n is even
- The last digit of n is not 2 or 8
- n^2 + 1 is composite
Find the sum of the digits of n.
A - 3, B - 5, C - 8, D - 9, E - 10
23.
Find the sum of the coordinates of the reflection of the point (6, 9) over the line x + 2y + 3 = 0.
A - (-17.7), B - (-17.6), C - (-17.5), D - (-17.4), E - (-17.3)
24.
Find the number of ordered pairs of integers (a, b), where both a and b have absolute value less than 69, such that a^2 + 42b^2 = 13ab.
A - 21, B - 40, C - 41, D - 42, E - 69
25.
Let f(n) be the sum of the positive integer factors of n, where n is an integer. Find the sum of all positive integers n less than 1000 such that f(f(n) - n) = f(n).
A - 420, B - 530, C - 690, D - 911, E - 1034
