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.
are the roots of the polynomial 
. Let 
be the unique monic polynomial whose roots are 
. Find 
.
is inscribed in a circle. Through 
and 
, tangents to the circle are drawn, meeting at point 
. If 
and 
is the radian measure of 
, then 
be the real roots of![\[x^3 + 7x^2 -6x - 13 = 0\]](http://latex.artofproblemsolving.com/6/6/3/6639be4c9e92b5ad4466359681db0d2a461ec8dd.png)
Find 
is written on the board. A move consists of adding either , 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
, then in a move a player can't choose the number , but he can choose either or ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be a triangle. Let 
be a circle passing through intersecting 
at and 
at 
. Let 
be the intersection of 
and . Further, let 
and 
be the intersections of 
and 
with the tangent to 
at 
. Now, let 
be the second intersection of 
and . Then, prove that , , , 
and are concyclic.
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers such that 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.
:
Square again and subtract to get 
Thus, 
Roots are 
Negatives are bad because lhs and rhs would have differing signs, and 
fails. Thus, all are bogus except 
. Now observe that for 
, 
, so we must have 
. So, 
, giving 
.
for can be proven by casework.)
:
.
such that 
coprime, 
and 
.
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
because the LHS is always positive, so the RHS is also always positive. Now we split it into cases:
This gives 
solving gives 
This gives 
so 
again.
