with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
such that
with domain ![$[0,2025]$](http://latex.artofproblemsolving.com/d/1/7/d17ac41d54c4b1888fb8bb43cf11b372691694a0.png)
satisfying 
and 
is strictly increasing on 
have local extrema at 
respectively. find 
. Here 
are constants .
be the region in the complex plane enclosed by curve 
for 
.
such that 
.
Where 
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
(0<x<2π)
if 
of 
such that 
.
be a positive semi-definite matrix. Prove that the matrix 
,
.
?
such that 
for all positive integers 
and 
be two points that are diametrically opposite to each other on a unit sphere. 
,
,
,
and 
respectively, and the points 
are collinear.
Doing this, we get that 
Now, we can solve this by factoring. By using RRT, we get that 
is a factor. Upon division, we get 
is the factored form. Now, we know that our only positive solutions are 
so our values of 
Summing these gives our answer of 
in or something like that idk my friend explained it pretty wonkily idk if he even solved it right but anyhow we press on. Transformations suck.
Furthermore, note that 
Thus, the answer is just 
so 
Cross multiplying and solving this quadratic, we get that our solution is 
Thus, 
We know from 
that 
so substituting this gives us 
Now, plugging this in to 
and 
gives us 
Solving these equations gives us 
Thus, by Vieta's, the sum of the roots is 
thus by median properties (the 2:1 ratio split thingy), we know that 
(For reference, I missed this part in-contest) Now, we can calculate 
by the Pythagorean Theorem to be 
and thus 
Now, by Pythagorean theorem, we know that the altitude from 
so our area is just 
Thus, simplifying and substituting 
we get that 
Simple guess and check gives us 
as our solutions. My dumb self forgot the former solution. Thus, 
giving us 
as our sum.
we get that 
Now if we try finding 
we will realize that 
this 
pattern continues until we once again reach 
thus 
Now return to the problem statement; 
but each inner function will just become 
hence the answer is 
Thus, simplifying, we get that 
or that 
Plugging this in to the first equation, we get that 
so this means that 
Now, 
thus 
is the diameter, thus we can draw 
and so by Special Right Triangles we do NOT in fact get 
but rather 
.
to be an integer. Plugging in factors of 
we see that there are 
integers that work.
Plugging in numbers gives us 
as our smallest value.
Furthermore, 
so 
meaning that 
.
and thus 
giving us 
Furthermore, we know that 
so our answer is just 
in the summation. Essentially, we have 
.
In other words, we want to find 
.
.
in such a polynomial is 
.
,
.
.
,
.
and 
,
and divide by 
.
and 
in our sum, subtract by 
.
term look kind of weird? Well, we actually notice 
,
.
,
,
is our answer!
,
terms. Drawing it out on the unit circle (which I heavily advise), it cycles between 
and 
.
,
,
.
,
for when 
terms! Anyways, the answer is 
.
so our coefficient is just 
Thus, our answer is just 
.
etc.
we get that it is 
Thus, we note that for an integer 
Since 
we know that all values of 
satisfy our conditions. Thus, our answer is just 
is our solution. For these Pell Equations, all solutions are of the form 
for 
Thus, we plug in 
giving us 
as our answer.
and let 
be our three-digit number that we remove. Thus, we can write 
By our condition, we know that 
or that 
Furthermore, we know that 
does not work, so we try 
We know that 
by the fact that it must be divisible by a 
Noting that 
doesn't work, we try 
or that 
This, miraculously, works, and thus our value of 
Y by Alex-131
Problem 1: Determine the area enclosed by the graphs of 
Hint
Problem 2: Calculate the sum of the real solutions to the equation
Hint
Problem 3: List the two transformations needed to convert the graph
to 
Hint
Problem 4: Let
be positive real numbers such that 
and 
Determine the value of 
Hint
Problem 5: Eve and Oscar are playing a game where they roll a fair, six-sided die. If an even number occurs on two consecutive rolls, then Eve wins. If an odd number is immediately followed by an even number, Oscar wins. The die is rolled until one person wins. What is the probability that Oscar wins?
Hint
Problem 6: In triangle
is on point 
such that 
and 
and 
is a point on 
such that 
and 
Given that 
calculate 
Hint
Problem 7: Determine the sum of the zeroes of the quadratic of polynomial
given that 
Hint
Problem 8:
Hint
Problem 9:
Find the sum of all real solutions to
Hint
Problem 10:
Define the function
![\[f(x) =
\begin{cases}
x - 9, & \text{if } x > 100 \\
f(f(x + 10)), & \text{if } x \leq 100
\end{cases}\]](//latex.artofproblemsolving.com/4/e/5/4e5b4486030a59e74df648819a1a59f82622b3b7.png)
Calculate
.
Hint
Problem 11:
Let
be real numbers such that 
Find 
Hint
Problem 12: Points
are on circle 
such that 
and 
Determine the path length from 
to 
formed by segment 
and arc 
Hint
Problem 13: Determine the number of integers
such that the expression 
is also an integer.
Hint
Problem 14: Determine the smallest positive integer
such that 
is a multiple of 
Hint
Problem 15: Suppose and 
are real numbers such that 
and 
Calculate 
Funnily enough, I guessed this question right in contest.
Hint
Problem 16: A sequence of points
will follow the rules such that
![\[
p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5).
\]](//latex.artofproblemsolving.com/6/0/3/6036ef2619ffb1e91db0ab310388217e472490f2.png)
How many sequences 
are possible such that 
is the only point with equal coordinates?
Hint
Problem 18: (Also stolen from akliu's blog post)
Calculate
Hint
Problem 19: Determine the constant term in the expansion of
Hint
Problem 20:
In a magical pond there are two species of talking fish: trout, whose statements are always true, and \emph{flounder}, whose statements are always false. Six fish -- Alpha, Beta, Gamma, Delta, Epsilon, and Zeta -- live together in the pond. They make the following statements:
Alpha says, "Delta is the same kind of fish as I am.''
Beta says, "Epsilon and Zeta are different from each other.''
Gamma says, "Alpha is a flounder or Beta is a trout.''
Delta says, "The negation of Gamma's statement is true.''
Epsilon says, "I am a trout.''
Zeta says, "Beta is a flounder.''
How many of these fish are trout?
Hint
SHORT ANSWER QUESTIONS:
1. Five people randomly choose a positive integer less than or equal to
The probability that at least two people choose the same number can be written as 
Find 
Hint
2. Define a function
on the positive integers using the rule that for 
For all prime , 
and for all other 
Find the smallest possible value of such that 
Hint
3. How many integers
can be written as the sum of two distinct, non-negative integer powers of 
Huge shoutout to OTIS for teaching me how to solve problems like this.
Hint
4. Let
be the set of positive integers of such that 
for some other positive integer 
Find the only three-digit value of in 
Hint
5. Let be a positive integer and let be the integer that is formed by removing the first three digits from 
Find the value of with least value such that 
Hint
Hint
Problem 2: Calculate the sum of the real solutions to the equation
Hint
Problem 3: List the two transformations needed to convert the graph
to 
Hint
Problem 4: Let
be positive real numbers such that 
and 
Determine the value of 
Hint
Problem 5: Eve and Oscar are playing a game where they roll a fair, six-sided die. If an even number occurs on two consecutive rolls, then Eve wins. If an odd number is immediately followed by an even number, Oscar wins. The die is rolled until one person wins. What is the probability that Oscar wins?
Hint
Problem 6: In triangle
is on point 
such that 
and 
and 
is a point on 
such that 
and 
Given that 
calculate 
Hint
Problem 7: Determine the sum of the zeroes of the quadratic of polynomial
given that 
Hint
Problem 8:
Hint
Problem 9:
Find the sum of all real solutions to
Hint
Problem 10:
Define the function
![\[f(x) =
\begin{cases}
x - 9, & \text{if } x > 100 \\
f(f(x + 10)), & \text{if } x \leq 100
\end{cases}\]](http://latex.artofproblemsolving.com/4/e/5/4e5b4486030a59e74df648819a1a59f82622b3b7.png)
Calculate
.Hint
Problem 11:
Let
be real numbers such that 
Find 
Hint
Problem 12: Points
are on circle 
such that 
and 
Determine the path length from 
to 
formed by segment 
and arc 
Hint
Problem 13: Determine the number of integers
such that the expression 
is also an integer.Hint
Problem 14: Determine the smallest positive integer
such that 
is a multiple of 
Hint
Problem 15: Suppose
and 
are real numbers such that 
and 
Calculate 
Funnily enough, I guessed this question right in contest.
Hint
Problem 16: A sequence of points
will follow the rules such that![\[
p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5).
\]](http://latex.artofproblemsolving.com/6/0/3/6036ef2619ffb1e91db0ab310388217e472490f2.png)
How many sequences 
are possible such that 
is the only point with equal coordinates?Hint
Problem 18: (Also stolen from akliu's blog post)
Calculate
Hint
Problem 19: Determine the constant term in the expansion of
Hint
Problem 20:
In a magical pond there are two species of talking fish: trout, whose statements are always true, and \emph{flounder}, whose statements are always false. Six fish -- Alpha, Beta, Gamma, Delta, Epsilon, and Zeta -- live together in the pond. They make the following statements:
Alpha says, "Delta is the same kind of fish as I am.''
Beta says, "Epsilon and Zeta are different from each other.''
Gamma says, "Alpha is a flounder or Beta is a trout.''
Delta says, "The negation of Gamma's statement is true.''
Epsilon says, "I am a trout.''
Zeta says, "Beta is a flounder.''
How many of these fish are trout?
Hint
SHORT ANSWER QUESTIONS:
1. Five people randomly choose a positive integer less than or equal to
The probability that at least two people choose the same number can be written as 
Find 
Hint
2. Define a function
on the positive integers using the rule that for 
For all prime , 
and for all other 
Find the smallest possible value of such that 
Hint
3. How many integers
can be written as the sum of two distinct, non-negative integer powers of 
Huge shoutout to OTIS for teaching me how to solve problems like this.
Hint
4. Let
be the set of positive integers of such that 
for some other positive integer 
Find the only three-digit value of in 
Hint
5. Let
be a positive integer and let be the integer that is formed by removing the first three digits from 
Find the value of with least value such that 
Hint
This post has been edited 3 times. Last edited by mathnerd_101, Apr 13, 2025, 2:52 PM
= 
Thus 1 minus this is 
so the answer is 
,
.
,
.
.
.
,
.
works. Clearly, nothing less than 16 works, because every number less than 16 can be expressed as 
or 
.
ways to get there
,
,
.
.
