and so on to create any number!
are perfect squares or perfect cubes (or both)?![\[
\begin{aligned}
\text{Exponent of } 2
&= \left\lfloor \frac{12}{2} \right\rfloor
+ \left\lfloor \frac{12}{4} \right\rfloor
+ \left\lfloor \frac{12}{8} \right\rfloor
= 6 + 3 + 1 = 10 \\
\text{Exponent of } 3
&= \left\lfloor \frac{12}{3} \right\rfloor
+ \left\lfloor \frac{12}{9} \right\rfloor
= 4 + 1 = 5 \\
\text{Exponent of } 5
&= \left\lfloor \frac{12}{5} \right\rfloor
= 2 \\
\text{Exponent of } 7
&= \left\lfloor \frac{12}{7} \right\rfloor
= 1 \\
\text{Exponent of } 11
&= \left\lfloor \frac{12}{11} \right\rfloor
= 1
\end{aligned}
\]](http://latex.artofproblemsolving.com/0/8/9/08995e5c6a19a236ebb6e7dcc2dbc993c58419f4.png)
![\[
12! = 2^{10} \cdot 3^5 \cdot 5^2 \cdot 7 \cdot 11
\]](http://latex.artofproblemsolving.com/0/5/d/05d49158382df85362e933443eb03ec708d96cff.png)
is the exponent of each prime factor, we find the number of perfect square divisors by taking 
:
is used to find the total perfect cubes:
t
?
can be divided into divisors 
and 
.
and 
,
t
,
,
,
opposite the sides 
,
,
,
and 
,
.
be the points with 
,respectively. Let 
be a point such that 
is the altitude from 
.
.
.
.
,
.
.
.
,
.
given that 
has 
factors.
can be factored out in the following expression, resulting in:
,
,
,
,
,
.
.
be the point of intersection of the altitude from 
and 
.
,
is shared by both 
and 
,
and 
,
through HL Theorem.
and 
are corresponding sides, we can substitute 
in
.
are the adjacent and hypotenuse of 
,
.
are corresponding angles, 
.
.
and 
be geometric sequences with the same common ratio. It is given that 
.
where 
are relatively prime positive integers, find 
.
and 
intersect at the point 
.
,
.
?
?
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
.
,
lies on diagonal 
such that 
.
.
has 
,
,
.
.
Y by RainbowJessa, Sharkship, RollingPanda4616, huajun78, PikaPika999, HacheB2031, GodGodGodGodGoose, Soupboy0, jkim0656, booking, dragon888, lendsarctix280, ahxun2006
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! 
Please hide solutions and answers, hints are fine though!
Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!
Please hide solutions and answers, hints are fine though!
Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!
This post has been edited 7 times. Last edited by Leeoz, Apr 14, 2025, 4:51 AM
and 
.
So, 
.
are 6 cases, which add up t 
.
contribute a total of 
.
contribute 
.
contribute 
.
and 
are the same number, as are 
and 
,
and 
,
.
in the product (prime factorization) although it's still kind of a bash
Since each is relatively prime, I can do the simple calculation 
it must have the die rolls 
and 
and 
and 
This gives 
and 
This gives 
gives 
The other person made a mistake
/ 
is divided by 2025?
.
.
.
.
and that when 
,
.
so 
is a solution.
,
,
or 
it has absolute value greater than 
so the answer is 
.
, so if x = 7 then the right hand side will become 
which we know is equal to 720 . That may seem like the only answer , but we also have 
is also equal to 720 so 0 and 7 are our solutions. There are other complex solutions but for now (0,7) is our answer.
