Found problems: 2
2016 Saudi Arabia IMO TST, 3
Find the number of permutations $ ( a_1, a_2, . \ . \ , a_{2016}) $ of the first $ 2016 $ positive integers satisfying the following two
conditions:
1. $ a_{i+1} - a_i \leq 1$ for all $i = 1, 2, . \ . \ . , 2015$, and
2. There are exactly two indices $ i < j $ with $ 1 \leq i < j \leq 2016 $ such that $ a_i = i $
and $ a_j = j$.
2017 NIMO Problems, 8
For each nonnegative integer $n$, we define a set $H_n$ of points in the plane as follows:
[list]
[*]$H_0$ is the unit square $\{(x,y) \mid 0 \le x, y \le 1\}$.
[*]For each $n \ge 1$, we construct $H_n$ from $H_{n-1}$ as follows. Note that $H_{n-1}$ is the union of finitely many square regions $R_1, \ldots, R_k$. For each $i$, divide $R_i$ into four congruent square quadrants. If $n$ is odd, then the upper-right and lower-left quadrants of each $R_i$ make up $H_n$. If $n$ is even, then the upper-left and lower-right quadrants of each $R_i$ make up $H_n$.
[/list]
The figures $H_0$, $H_1$, $H_2$, and $H_3$ are shown below.
[asy]
pair[]sq(int n){pair[]a;
if(n == 0)a.push((.5,.5));
else for(pair k:sq(n-1)) { pair l=1/2^(n+1)*(1,(-1)^(1+(n%2)));a.push(k+l);a.push(k-l); }
return a;}
void hh(int n,real k){
pair[] S=sq(n);real r=1/2^(n+1);
for(pair p:S)filldraw(shift(p+(k,0))*((r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle));
label("$H_"+string(n)+"$",(k+.5,-.3));}
size(7cm);
for(int i=0;i<=3;++i)hh(i,1.6*i);
[/asy]
Suppose that the point $P = (x,y)$ lies in $H_n$ for all $n \ge 0$. The greatest possible value of $xy$ is $\tfrac{m}{n}$, for relatively prime positive integers $m, n$. Compute $100m+n$.
[i]Proposed by Michael Tang[/i]