Prove that 
Equality holds at 
and its cyclic permutations.
Y by
1. For an integer 
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?
2. Find the sum of all integer values of
that satisfy the inequality chain ![\[n^3<2025<3^n.\]](//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 
4. Positive integers
satisfy the system
![\[\left\{\begin{array}{l}
x^2+y^2=z^2+22\\
y^2+z^2=x^2+76\\
x^2+z^2=y^2-4.
\end{array}
\right.\]](//latex.artofproblemsolving.com/b/7/d/b7dc2e00e17c3e1bf3108bfd8b4f722e006f5430.png)
What is the value of 
5. \begin{problem}
Four husband-wife couples go ballroom dancing one evening. The husbands' names are Henry, Peter, Steve, and Roger, while the wives' names are Elizabeth, Keira, Mary, and Anne. At a given moment, Henry's wife is dancing with Elizabeth's husband, who is not Henry; Roger and Anne are not dancing; Peter is playing the trumpet; and Mary is playing the piano. Given that Anne's husband is not Peter, how many different letters are in the name of Roger's wife?
6. A laser is fired from vertex
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 
7. What is the value of the expression
![\[\frac{\log_2(\log_2 3)}{\log_4(\log_4 9)}?\]](//latex.artofproblemsolving.com/5/2/6/52675b044e3c0d279b6fb5815449abf4fafda07d.png)
8. The area of equiangular octagon 
with 
and 
can be written in the form 
, find 
.
9. A toe-wrestling tournament between Don and Kam consists of three matches. In each match, the winner is the first person to reach five points. After the three matches, each person’s score is the number of matches they won, plus the sum of the points they earned during all of their matches. Let
and 
denote Don and Kam’s final scores, respectively. How many ordered pairs 
are possible?
10. For each positive integer, let 
denote the sum of the remainders when is divided by 
and 
For example, when 
we have 
Compute the integer 
for which ![\[\sum_{n=1}^{N}s(n)=2025.\]](//latex.artofproblemsolving.com/6/a/a/6aaed99ed4cf8de4c87a57cd90062dca9da70166.png)
11. For complex numbers 
we define the function ![\[f(z)=\frac{z+3}{z-2i}.\]](//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 
and 
Compute the value of 
12. In convex quadrilateral
and 
. Let 
and be the midpoints of 
and 
respectively. Compute the area of 
given that the area of 
is 
.
13. What is the remainder when
is divided by 2025?
14. Your friend plays a prank on you by changing your phone's password. Your friend chooses a password consisting of 4 decimal digits
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 
\null
Now, your friend picks a password whose digits sum to 20; let
be the set of all such passwords. Suppose you select such that the probability that your friend tells you the password, given this information, is maximized. Compute the number of passwords in for which this would not occur, given your choice of 
15. For real numbers we define the function ![\[f(x)=\lceil{1+\sqrt{x+1}}\rceil+\lfloor{1-\sqrt{x-1}}\rfloor.\]](//latex.artofproblemsolving.com/b/6/e/b6e0a20edc81ef9cdfcacf77b342583ab5509ae3.png)
Compute the 
smallest integer 
for which 
.
16. How many ordered pairs
with 
satisfy the congruence ![\[2^{2^x+2^y}\equiv 1\pmod{101}?\]](//latex.artofproblemsolving.com/e/3/b/e3bc13aacc8e0869e5f3ce9a509bb26974d0a05c.png)
17. 
has circumcircle 
and incenter 
is extended to intersect at a point 
and 
is extended to intersect 
at 
If 
and 
is a cyclic quadrilateral, then compute 
18. The
is constructed as follows:
i) Start with the closed interval![$[0, 1]$](//latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
.
ii)Remove the open middle third of the interval, so we remove
at first and leave ![$\left[0, \frac{1}{3}\right]$](//latex.artofproblemsolving.com/b/0/a/b0a58f8e9a43b655d29a17bf96fceb4ddbe182ed.png)
and ![$\left[\frac{2}{3}, 1\right]$](//latex.artofproblemsolving.com/e/8/1/e81f52db805dbffa1d98108fadd9f713c6f2aa95.png)
.
iii) Remove the open middle third from each of the remaining closed intervals, and repeat this step infinitely.
For how many integer values of
, where 
, is 
an element of the Cantor set?
19. For complex numbers
satisfying 
, the maximum value of 
can be expressed in the simplest form of 
, find 
.
20. Consider cyclic quadrilateral with all integer side lengths and 
. Let 
meet 
at 
, 
. Denote the centers of the circumcircles of polygons 
as 
respectively. Compute the area of 
. The answer is in the simplest form of 
and 
is square-free, compute 
.
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?2. Find the sum of all integer values of
that satisfy the inequality chain ![\[n^3<2025<3^n.\]](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 
4. Positive integers
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 
5. \begin{problem}
Four husband-wife couples go ballroom dancing one evening. The husbands' names are Henry, Peter, Steve, and Roger, while the wives' names are Elizabeth, Keira, Mary, and Anne. At a given moment, Henry's wife is dancing with Elizabeth's husband, who is not Henry; Roger and Anne are not dancing; Peter is playing the trumpet; and Mary is playing the piano. Given that Anne's husband is not Peter, how many different letters are in the name of Roger's wife?
6. A laser is fired from vertex
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 
7. What is the value of the expression
![\[\frac{\log_2(\log_2 3)}{\log_4(\log_4 9)}?\]](http://latex.artofproblemsolving.com/5/2/6/52675b044e3c0d279b6fb5815449abf4fafda07d.png)
8. The area of equiangular octagon 
with 
and 
can be written in the form 
, find 
.9. A toe-wrestling tournament between Don and Kam consists of three matches. In each match, the winner is the first person to reach five points. After the three matches, each person’s score is the number of matches they won, plus the sum of the points they earned during all of their matches. Let
and 
denote Don and Kam’s final scores, respectively. How many ordered pairs 
are possible?10. For each positive integer
, let 
denote the sum of the remainders when is divided by 
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 
and 
Compute the value of 
12. In convex quadrilateral
and 
. Let 
and be the midpoints of 
and 
respectively. Compute the area of 
given that the area of 
is 
.13. What is the remainder when
is divided by 2025?14. Your friend plays a prank on you by changing your phone's password. Your friend chooses a password consisting of 4 decimal digits
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 
\null
Now, your friend picks a password whose digits sum to 20; let
be the set of all such passwords. Suppose you select such that the probability that your friend tells you the password, given this information, is maximized. Compute the number of passwords in for which this would not occur, given your choice of 
15. For real numbers
we define the function ![\[f(x)=\lceil{1+\sqrt{x+1}}\rceil+\lfloor{1-\sqrt{x-1}}\rfloor.\]](http://latex.artofproblemsolving.com/b/6/e/b6e0a20edc81ef9cdfcacf77b342583ab5509ae3.png)
Compute the 
smallest integer 
for which 
.16. How many ordered pairs
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 at a point 
and 
is extended to intersect 
at 
If 
and 
is a cyclic quadrilateral, then compute 
18. The
is constructed as follows:i) Start with the closed interval
![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
.ii)Remove the open middle third of the interval, so we remove
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)
.iii) Remove the open middle third from each of the remaining closed intervals, and repeat this step infinitely.
For how many integer values of
, where 
, is 
an element of the Cantor set?19. For complex numbers
satisfying 
, the maximum value of 
can be expressed in the simplest form of 
, find 
.20. Consider cyclic quadrilateral
with all integer side lengths and 
. Let 
meet 
at 
, 
. Denote the centers of the circumcircles of polygons 
as 
respectively. Compute the area of 
. The answer is in the simplest form of 
and 
is square-free, compute 
.This post has been edited 1 time. Last edited by Bluesoul, Mar 30, 2025, 10:31 AM
