we define a \textit{step} as either doubling the value of the integer or subtracting 3 from it. What is the minimum number of steps required to obtain 25 from 11?![\[n^3<2025<3^n.\]](http://latex.artofproblemsolving.com/8/9/e/89eed4daec7a19210ede13accfadb4cd1ec65757.png)
3. A rectangle with length 20 units and height 16 units is divided into 10 smaller congruent rectangles. Let 
be the largest possible perimeter of one of these small rectangles; compute the value of 
satisfy the system![\[\left\{\begin{array}{l}
x^2+y^2=z^2+22\\
y^2+z^2=x^2+76\\
x^2+z^2=y^2-4.
\end{array}
\right.\]](http://latex.artofproblemsolving.com/b/7/d/b7dc2e00e17c3e1bf3108bfd8b4f722e006f5430.png)
What is the value of 
into the interior of regular hexagon 
whose sides are mirrors, and hits side 
at 
It then reflects and hits 
at 
and finally reflects and hits 
at 
If 
then how many degrees are in 
![\[\frac{\log_2(\log_2 3)}{\log_4(\log_4 9)}?\]](http://latex.artofproblemsolving.com/5/2/6/52675b044e3c0d279b6fb5815449abf4fafda07d.png)
8. The area of equiangular octagon 
with 
and 
can be written in the form 
,
.
and 
denote Don and Kam’s final scores, respectively. How many ordered pairs 
are possible?
denote the sum of the remainders when 
and 
For example, when 
we have 
Compute the integer 
for which ![\[\sum_{n=1}^{N}s(n)=2025.\]](http://latex.artofproblemsolving.com/6/a/a/6aaed99ed4cf8de4c87a57cd90062dca9da70166.png)
11. For complex numbers 
we define the function ![\[f(z)=\frac{z+3}{z-2i}.\]](http://latex.artofproblemsolving.com/7/e/d/7ed1763ea1084dd79f0330cc66c2ba566406a923.png)
Over all values of 
for which 
is real, the minimum possible value of 
can be written in the form 
for positive integers 
Compute the value of 
and 
.
and 
and 
respectively. Compute the area of 
given that the area of 
is 
.
is divided by 2025?
uniformly at random and tells you the sum of its digits. (Leading zeros are allowed, so your friend can choose any password from 0000, 0001, and so on to 9999.) Then, you select a digit 
your friend tells you the password if and only if 
is the median of the set 
be the set of all such passwords. Suppose you select 
![\[f(x)=\lceil{1+\sqrt{x+1}}\rceil+\lfloor{1-\sqrt{x-1}}\rfloor.\]](http://latex.artofproblemsolving.com/b/6/e/b6e0a20edc81ef9cdfcacf77b342583ab5509ae3.png)
Compute the 
s
for which 
.
with 
satisfy the congruence ![\[2^{2^x+2^y}\equiv 1\pmod{101}?\]](http://latex.artofproblemsolving.com/e/3/b/e3bc13aacc8e0869e5f3ce9a509bb26974d0a05c.png)
17. 
has circumcircle 
and incenter 
is extended to intersect 
and 
is extended to intersect 
at 
If 
and 
is a cyclic quadrilateral, then compute 
is constructed as follows:![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
.
at first and leave ![$\left[0, \frac{1}{3}\right]$](http://latex.artofproblemsolving.com/b/0/a/b0a58f8e9a43b655d29a17bf96fceb4ddbe182ed.png)
and ![$\left[\frac{2}{3}, 1\right]$](http://latex.artofproblemsolving.com/e/8/1/e81f52db805dbffa1d98108fadd9f713c6f2aa95.png)
.
,
,
an element of the Cantor set?
satisfying 
,
can be expressed in the simplest form of 
,
.
.
meet 
at 
,
.
as 
respectively. Compute the area of 
.
and 
is square-free, compute 
.
and 
.
Equality holds when 
Equality holds when 
Equality holds when 
are proper subsets of 
and they are closed for addition and subtraction. Try to prove that:
Y by
Consider a teapot that holds 
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a 
% tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a 
% tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?This post has been edited 1 time. Last edited by smartvong, Mar 26, 2025, 8:44 AM
![\[
\begin{aligned}
&\text{If you notice, we can have at most } (1000 \cdot n\%) \text{ ml of tea.} \\[10pt]
&\text{So we just need to discard } \left(\frac{n}{m} \cdot 1000\right) \text{ ml from our teapot and refill it with water.} \\[10pt]
&\text{This will turn the contents into:} \\
&\text{Tea: } 10m, \quad \text{Water: } 1000 - 10m. \\[10pt]
&\text{After discarding:} \\
&\text{Tea: } 10n, \quad \text{Water: } \frac{1000n}{m} - 10n. \\[10pt]
&\text{Adding back water until the teapot is filled:} \\
&\text{Tea: } 10n, \quad \text{Water: } 1000 - 10n. \\[10pt]
&\text{There is now } n\% \text{ of tea.} \\[10pt]
&\text{So the amount of tea served will be: } 10n.
\end{aligned}
\]](http://latex.artofproblemsolving.com/a/c/b/acb97e68d9f81b0822885d3ee4f8849ab253315e.png)
