AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+1 w
denote the sum of the 
th powers of the roots of the polynomial 
. In particular, 
, 
, and 
. Let 
, 
, and 
be real numbers such that 
for 
, 
, 
What is 
?
+1 w
by 
different methods. The same inequality can be used twice but in different ways, dm me if you have any question about the rules.
+1 w
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at 
. Prove that 
.
let 
be the foot of the altitude from 
to the side 
and 
, 
, 
be the incenter, -excenter, and 
-excenter, respectively. Denote by 
and 
the other intersection points of the circle 
with the lines 
and 
, respectively. Prove that 
.
be a set that intersects all non-finite integer arithmetic progressions, 
be the set of prime divisors of 
and 
be the set of prime divisors of 
. Suppose 
. Prove that 
, 
be a triangle, and let 
be centered at 
, 
and tangent to line at 
, respectively. Let line 
intersect 
again at 
and let line 
intersect 
again at 
. If 
is the other intersection of the circumcircles of triangles and 
, then prove that lines 
, , and 
either concur or are all parallel., , 
and 
are the points of tangency of incircle with the center of to , 
and respectively. Let 
be the foot of the perpendicular from to 
. is on 
such that 
. If 
is the orthocenter of 
, prove that 
bisects 
.
, where 
are reals. Let be a circle with diameter 
and let be any other point on . Line 
meets the x-axis again at . Prove that angle 
.
, let and 
denote the midpoints of and 
. Let be the projection of onto and let be the reflection of over the midpoint of 
. Assume lies in the interior of quadrilateral . Prove that 
.
and be given as: 
, 
(where 
can be 0), and let be the count of numbers 
such that 
and 
(
can also be 0). (Here, 
is considered the same as 
).
is 
, then is called a perfect number. For example, the sum of the positive divisors of 6 is 
, hence 6 is a perfect number.
, where 
are positive integers, and 
are odd primes.
be positive integers, and let 
all be 
-th roots of unity, satisfying:![\[
\alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k).
\]](http://latex.artofproblemsolving.com/3/3/b/33be8eef29cd506bf1132a86def9992ec8d21cb4.png)
Prove that among , each -th root of unity appears the same number of times.
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
Solving with Cramer's Rule gives 
or .
people. Then portia's has 
. Thus 
. Thus 
, and 
. Then Portia's high school has students.
From the above result, the number of students in Lara's school is 
. Hence the answer is (C) 
.
--> 
--> 
so (C)
to get 
.
Then, 
,
.
and 
students, respectively. We have the following systems of equations: 
Plugging the first equation into the second, we have 
Finally, we have 
students and Portia's school have 
students. So 
.
.
so 
so the answer is 
.