be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
be the midpoint of 
.
is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
and 
be positive integers. The cells of an 
grid are colored amber and bronze such that there are at least 
amber cells and at least 
bronze cells. Prove that it is possible to choose 
chosen cells lie in the same row or column.
and 
are erected outside an acute triangle 
Suppose that ![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
-
of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let 
?
of identical dominoes is a maximal grid-aligned configuration on the board if 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: 
be the number of distinct maximal grid-aligned configurations obtainable from 
Y by Adventure10
A positive integer divisor of 
is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as 
, where 
and 
are relatively prime positive integers. What is 
?
is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as 
, where 
and 
are relatively prime positive integers. What is 
?
This post has been edited 1 time. Last edited by LauraZed, Jan 31, 2020, 6:14 PM
Reason: fixed phrasing to match test, added answer choices
Reason: fixed phrasing to match test, added answer choices
divisors of 
,
factors. However, if 
,
,
factors. Thus the answer is 
giving 
.
by prime factorizing. @OP please include the answer choices.
case.
divisors. Use prime factorization to take a square root but round up exponents that don't divide by 
evenly. It's 
for 
.
.
total divisors. All perfect square divisors of 
or 
.
choices for the power of 
,
for the powers of 
and 
.
.
.
![\[12!=2^{6+3+1}\cdot3^{4+1}\cdot5^2\cdot7\cdot11=2^{10}\cdot3^5\cdot5^2\cdot7\cdot11,\]](http://latex.artofproblemsolving.com/7/2/1/7219fb8204eb06284bc409a4e57814a5c06662e6.png)
which has 
divisors.
.
,
possible values of ![\[\frac{36}{11\cdot6\cdot3\cdot4}=\frac{1}{22}\implies\boxed{\textbf{(E)}\ 23}\]](http://latex.artofproblemsolving.com/5/d/8/5d8e0c83098aa8fd03106e04006911acfaa56d16.png)
.
So, it has 
we take the floor function when each exponent is divided by 
so 
Now finding all the divisors, 
So, 
ive found the prime factorization but im confused how to go on...
(this is a formula, look up "formula for amount of divisors" if you don't know it). Our divisor is in the form 
,
.
for 
and one way for 
,
desirable cases. Answer 
.
