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 
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 
.
, Elbek writes 
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
- Greatest common divisor of numbers 
and 
board, there is initially a tree of height 
. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
. Determine the largest 
such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.
+1 w
, find the number of terms with coefficient divisible by 
.
such that the equation
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
, there exists an integer 
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of positive real numbers with sum equal to 2. Let 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of positive real numbers 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
on ![$[1,n]$](http://latex.artofproblemsolving.com/2/2/4/2240bb122bdadbe086c0a6291dcd5a3eb82ad5cf.png)
as a word with 
in some order, hence 
is a permutation on 5 symbols. We write 
for the wellness of namely the amount of pairs 
that appear in the same order in the word. Here 
.
for the number of 
with wellness 
. Hence we want to determine 
. hence 
. The symbol k adds 
to the wellness hence now we count the permutations on ![$[1,7]\setminus \{k\}$](http://latex.artofproblemsolving.com/2/a/3/2a3fb0361b0937dbfba193cae8ae422eb3328c31.png)
with wellness . Since we do not consider the symbol anymore, we may as well count the number of permutations on ![$[1,6]$](http://latex.artofproblemsolving.com/a/b/0/ab036e7da17bee704b3b7ee0ca5ea79f424465e5.png)
with wellness . This equals 
.
.. If the first symbol is then it contributes 
to the wellness hence 
. We find 
. The lower bound may be 
, in that case we start counting at 
. This reduces the problem to finding the numbers 
. Etc.
and then work your way upwards.
of 
by a unique 
-tuple of integers 
such that 
for all valid , where each 
represents the number of pairs of integers 
satisfying 
. Thus, we seek the number of -tuples such that for all valid and 
.
in the expansion of 
. But notice that 
is a root of 
. Hence, the sum of the coefficients of the terms of with degree divisible by is 
. Observe that the coefficients of 
and 
are trivially each 
, and the coefficients of 
and 
are equal due to symmetry. Thus, the coefficient of is simply 
.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
satisfying 
and 
.