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.
denote 
. For any real numbers 
prove that![\[ \prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\ldots,x_k) \geq \prod_{i=1}^k F_i(x_i,x_i,\ldots,x_i)\]](http://latex.artofproblemsolving.com/a/4/d/a4d515f1102b31cff69d6823df7a58482aec1171.png)
. Yael and Ziad play a game as follows, involving a monic polynomial of degree 
. With Yael going first, they take turns to choose a strictly positive real number as the value of one of the coecients of the polynomial. Once a coefficient is assigned a value, it cannot be chosen again later in the game. So the game rounds, until Ziad assigns the final coefficient. Yael wins if 
for some real
. Otherwise, Ziad wins. Decide who has the winning strategy.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.
be an isosceles triangle with 
and circumcircle 
. The line through 
perpendicular to 
is denoted by 
. Furthermore, let 
be any point on . The circle 
with center and radius 
intersects 
once more at point 
and the circumcircle once more at point 
. Prove that the points 
and 
lie on a straight line.
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](http://latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
. Prove the inequality
When does equality hold?
denote the sum of the alternating harmonic series. Rearrange this series as follows
Assume as known that this series converges as well and denote its sum by 
. Denote by 
respectively the 
-th partial sums of both series. Prove that
has always integral solutions for and 
whenever 
is a positive integer.
is a sequence of real numbers such that
Prove that if 
converges, then 
for all .
where 
, is cut by the line 
the tangents at the points of intersection are concurrent.
such that 
is a perfect square.
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
does there exist an 
matrix 
whose entries are all in 
, such that 
is the matrix of all ones? has 
times the eigenvalue 0 and the eigenvalue . So the charaterisric polynomial of must be 
. This is only for 
an integer polynomial.
and let 
be the 
-matrix with entries 
. We can choose as the block-matrix 
.
.![$Z=[0,\cdots,0]^T,V=[1,\cdots,1]^T\in \{0,1\}^m$](http://latex.artofproblemsolving.com/0/5/b/05be892fde5a92fb2627f6f1384b3e5e18ae6edf.png)
; then put![$M_0=[V,Z,\cdots,Z],M_1=[Z,V,Z,\cdots,Z],....,M_{m-1}=[Z,\cdots,Z,V]$](http://latex.artofproblemsolving.com/4/5/b/45b6c7cb39f4ac60ce4bbec97cbbb9f41bf44509.png)
.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
and 
.
is a perfect square.
and pairs 
(e.g. via digits in 
iff for the corresponding pairs 
and 
we have 
.
for all 
.
such that 
.
.
.
.
,
,
for some integer 
hence 
is a perfect square.