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.
, but skips any number containing digits 
, 
or 
. If the last number he writes is 
, how many numbers has Penrick written in total?
of the set 
, define its colorfulness as the greatest natural number 
such that:
, 
, if 
, then 
.
? Determine how many such permutations have maximal colorfulness.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
. The segment 
measures 40 units. Let 
be the orthocenter of triangle and 
its circumcenter. Let 
be the foot of the altitude from 
and 
the foot of the altitude from 
. Additionally, point divides the segment such that 
. If the perpendicular bisector of segment 
passes through point , calculate the area of quadrilateral 
.
be real number such that 
Prove that
Let 
be real number . Prove that
be a triangle. The points 
and 
lie on the segments 
and 
respectively, such that the lines 
and 
intersect in a common point. Prove that it is possible to choose two of the triangles 
and 
whose inradii sum up to at least the inradius of the triangle .
and 
be two positive integer numbers such that 
.Prove that 
divides 
be a point interior to triangle (with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively . The tangent line at 
to meets the line 
at 
. Show that from 
follows 
.
integers is 
. After adding 
to each of the first four and subtracting 
from each of the last seven, what is the new mean?
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 .
and 
Prove that
Prove that
Let 
Prove that
