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.
such that there exists a sequence of positive real numbers 
, 
, with the property that for every it holds that:![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](http://latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
and 
such that![\[
\frac{1}{\gcd(x, y)} + \frac{3}{xy} + \frac{y}{\operatorname{lcm}(x, y)} = y,
\]](http://latex.artofproblemsolving.com/c/8/2/c828f866a97dbb38fe5b6d7b8d190b3684365938.png)
where 
denotes the greatest common divisor of and , and 
denotes their least common multiple.
that satisfies: for all different 
, we have 
?
be a triangle and let 
and 
be points on the sides 
and 
, respectively , such that 
is parallel to 
. Let 
be any point interior to triangle 
, and let 
and 
be the intersections of with the lines 
and 
, respectively. Let 
be the second intersection points of the circumcircles of triangles 
and 
. Prove that the points 
are collinear .
indicates my favorites.
is defined over the positive integers as follows: 
, 
for 
prime, and for all relatively prime positive integers 
and 
, 
. If 
is the smallest positive integer such that 
, find the units digit of .
If convex quadrilateral 
satisfies 
, 
, 
, 
, and 
, what is the value of 
? Express your answer in simplest radical form.
and 
be the greatest odd divisor of . Let 
for even 
denote the product of every odd positive integer less than . If 
for positive integers 
and 
where 
is minimized, find the number of divisors of 
. There exists exactly one positive real number such that the graph of the equation 
consists of a line and a point not on the line. The distance from the point to the line can be expressed as 
, where and are positive integers and is not divisible by any square greater than 
. Find 
.
. If 
are positive real numbers such that 
, 
, and 
, find . Express your answer as a common fraction. Let 
be a convex pentagon satisfying 
, 
, 
. Let 
be the intersection of lines 
and 
. If 
has a perimeter of 
and an area of 
, find the area of .
Let 
be an acute scalene triangle with longest side 
. Let 
be the circumcenter of . Points and 
are chosen on such that 
and 
. If 
, 
, and 
, the area of the circumcircle of can be expressed as 
. Find .
such that 
is divisible by .
Alex has a 
by grid of squares. Let be the number of ways that Alex can fill out each square with one of the letters 
, 
, 
, or 
such that in every row and column, the number of 's and 's are the same, and the number of 's and 's are the same. (For example, a row with squares labeled 
or 
is valid, while a row with squares labeled 
or 
is not valid.) Find the remainder when is divided by 
.
by grid of squares along the gridlines into two or more pieces such that if three pieces meet at a point 
, then there are actually four pieces with a vertex at ? An example is shown below.
be real numbers satisfying the system of equations
What is the value of 
? In , the incircle is tangent to 
at , and 
is the reflection of across the midpoint of . Suppose that the inradii of 
and 
are and respectively, and the distance between their incenters is 
. What is the inradius of ? Express your answer as a common fraction. be a positive integer and let be the set of all -tuples of 
's and 's. Two elements of are said to be neighboring if and only if they differ in only one coordinate. Bob colors the elements of red and blue such that each blue -tuple is neighboring to exactly two red -tuples and no two red -tuples neighbor each other. If 
, find the least possible value of .
rectangle with 
dominoes?
, 
, 
, 
be circles having a common point . Denote by 
the intersection point, other than , of 
and 
(
). Let 
be the common point of the 4 circles 
, 
, 
, 
.
be the Simson line of with respect to 
. Define 
, 
, 
cyclically.
be the Simson line of with respect to 
. Define 
, 
, 
cyclically., , , are concurrent, then, , , , are also concurrent.
, is the point of tangency of the incircle with the side 
. Points 
and 
are the intersections of the angle bisectors of 
and 
with the circumcircle of , respectively. Let 
be the antipodal point of in the circumcircle of 
, and let 
be the antipodal point of in the circumcircle of 
.
and 
are similar.
Then there are two cases either 
or 
.When we are admitting the case we cannot divide by . So, in the case we apply divison by 
then is a solid prerequisite to do so.Thus, 
from 
we must take the assumption in hand that . For example take the extension of the same problem in 
then the same problem reads 
,implying only one solution with optimistic repetation of root , two times that is the multiplicity of in 
. Thankfully, we are lucky enough that we are in the field of 
.The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of .
Then there are two cases either or .When we are admitting the case we cannot divide by . So, in the case we apply divison by then is a solid prerequisite to do so.Thus, from we must take the assumption in hand that . For example take the extension of the same problem in then the same problem reads ,implying only one solution with optimistic repetation of root , two times that is the multiplicity of in . Thankfully, we are lucky enough that we are in the field of .The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of ..
Then there are two cases either or .When we are admitting the case we cannot divide by . So, in the case we apply divison by then is a solid prerequisite to do so.Thus, from we must take the assumption in hand that . For example take the extension of the same problem in then the same problem reads ,implying only one solution with optimistic repetation of root , two times that is the multiplicity of in . Thankfully, we are lucky enough that we are in the field of .The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of ...The assumption whenever we get 
else we get the case .I can explain you more but the fact is I cannot use argument of field theory to explain it in total details.The reason why it's actually the case lies in field theory logics.
Which means that the roots of this equation have to be real so we can use an method that is lost in the darkness called factoring. We can factor out the from each of the terms on the left hand side and get which with more logic we can find that the possible outcomes is that if the Parantheses are 0 or the x=0 so first we can subsitute a value of x into that to make the value 0 so we get that x=5 and we finally get the solutions of and and to wrap up our proof we can prove that factoring is the best way to go because with quadratics you would only find 2 possibiliteis or 1 depending on the plus minus. And the other thing is that if you divide by x and just solve it with algebra then you will only get the solution of 5. Thus, proving that factoring is the best method out of all of them. We can use the remainder theorem to prove this which can be done easily. 
is lost? right. Well forget about field theory and all that, Let me explain it in layman's term what is actually happening. In second solution , the solution is not actually lost. The reason we are getting but not is beacuse when we are dividing by we making an assumption that and since we are making this assumption the solution is lost. For example when we divide by 
the solution is lost why 
and we are assuming 
which is equivalent to 
. Now divison by zero is not possible which is not at all very easy to explain. Now and is not possible at the same time. Thus either or . and 
can be same when we are in a field of 
. If you are avoid knowing what is field that's perfectly fine to learn later, but just in layman's term note that 
is possible in finite fields of charecteristic .
