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.
be a positive integer. For a set 
of real numbers, let 
denote the number of increasing arithmetic progressions of length at least two all of whose terms are in . Prove that, if is a set of real numbers, then![\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]](http://latex.artofproblemsolving.com/c/8/8/c887443e08b9f61e5ce55c3e35fcf2912a6da2a5.png)
, 
is acute, 
, and 
. Let point 
be the intersection of the tangent to the circumcircle of at point 
and the perpendicular bisector of segment 
. Given that 
, find the radius of the circumcircle of 
.
. 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.
positive integers with least common multiple 
and greatest common divisor 
. It is known that is not a perfect square and is not among the initially written numbers. Two players 
and 
play the following game, taking turns alternatingly, with starting first. In a move the player has to write a number which has not been written so far, by taking two distinct integers 
and 
from the board and write LCM
or LCM/. The player who writes or loses. Who has a winning strategy?
. At a maths camp, there are 
children, each of whom selects tables, with children tables, with children and let 
be a convex polygon in the plane. Let 
be the set of all midpoints 
of segments 
where 
. Assume that all of these midpoints are distinct, i.e. consists of 
elements. Dissect the polygon 
into triangles so that the following hold:
for some pairwise distinct indices 
.
.
be an acute triangle with 
and 
. Let 
be the midpoint of the side 
. The circumcircle of triangle 
intersects the side 
again at 
and the circumcircle of triangle 
intersects the side 
again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
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)
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.
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)
.
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.