are real numbers satisfying the inequality![\[
(a+3c)(a+2b+3c) < 0,
\]](http://latex.artofproblemsolving.com/a/1/0/a10c3947792e07c09febf72091d64fa838808335.png)
prove that![\[
(2b-3c)^2 > 4a(a+2b+3c).
\]](http://latex.artofproblemsolving.com/c/9/b/c9ba26d545b33c2255544189acaad22155591f0a.png)
then prove 
and 
Prove that
and 
.
Equality holds when 
Equality holds when 
Equality holds when 
which are set of natural numbers and for that 
to be a natural number too.
.
.
.
.
satisfying the equation:![\[
(x^2y - y^2z)(1 - zx) = (xy^2 - zx^2)(1 - yz).
\]](http://latex.artofproblemsolving.com/d/1/c/d1c11cb351b59ae5e79edc1e1997834b5402638a.png)
![\[
x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
\]](http://latex.artofproblemsolving.com/c/1/2/c122da8411c836a38fb2ea639338c3fd9749a619.png)
.
![\[
\begin{aligned}
&\text{Each question is worth 10 marks. If you just provide the answer, you get 5 marks. If you provide sufficient workings, you receive up to 5 marks.} \\[10pt]
&\textbf{Q1.} \text{ Alice and Bob are playing a game where Alice starts first. There is a common positive integer } x \text{ given, and on their turn,} \\
&\text{each player subtracts an integer } n \text{ where } 1 \leq n \leq 9, \text{ such that the common number becomes } (x-n). \text{ Given a target } y, \\
&\text{the player wins when their turn ends with } (x-n) = y. \\[10pt]
&\text{E.g. } x = 25, y = 1: \\
&\text{On Alice's turn, she chooses to subtract 9, so the common number is now 14.} \\
&\text{On Bob's turn, he chooses to subtract 3, so the common number is now 11.} \\
&\text{On Alice's turn, she chooses to subtract 2, so the common number is now 9.} \\
&\text{On Bob's turn, he chooses to subtract 8, so the common number is now 1. Bob wins.} \\[10pt]
&(i) \text{ Assuming } (x, y) \text{ is a pair of integers such that Alice will have a strategy to guarantee a win, find that strategy.} \\
&(ii) \text{ Find all } (x, y) \text{ where Bob will have a strategy to guarantee a win.} \\
&\text{[Modified Intermediate Mathematical Olympiad Maclaurin paper Q2]} \\[20pt]
&\textbf{Q2.} \text{ Given a fair } n\text{-sided die, where the sides are } 1, 2, 3, \ldots, n-1, n, \text{ find the probability of rolling } n \\
&\text{at least once in } m \text{ rolls.} \\
&\text{[Original question]} \\[20pt]
&\textbf{Q3.} \text{ Determine the smallest natural number } n \text{ such that } n^n \text{ is not a divisor of } 2025!. \\
&\text{[Modified Flanders Math Olympiad 2016 Q2]} \\[20pt]
&\textbf{Q4.} (n+1)^{n-1} = (n-1)^{n+1}. \text{ Find all real } n. \\
&\text{[Original Question]} \\[20pt]
&\textbf{Q5.} a, b, c, d, x \text{ are integers. } 0 \leq a, b, c, d \leq 9. \text{ Find the number of possible } (a, b, c, d) \text{ such that} \\
&7^a + 7^b + 7^c + 7^d = 100x. \\
&\text{Note: } (2, 0, 2, 4) \text{ and } (2, 0, 4, 2) \text{ are 2 separate solutions.} \\
&\text{[Intermediate Mathematical Olympiad Maclaurin paper Q3]} \\[20pt]
&\textbf{Q6.} \text{A sequence is defined as } a_1 = 2025, \text{ and for all } n \geq 2: \\
&a_n = \frac{a_{n-1} + 1}{n}. \\
&\text{Determine the smallest } k \text{ such that } a_k < \frac{1}{2025}. \\
&\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q1]} \\[20pt]
&\textbf{Q7.} \text{There are } n \geq 3 \text{ students in a classroom. Every day, the teacher splits the students into exactly 2 non-empty groups,} \\
&\text{and each pair of students from the same group will shake hands once. Suppose after } k \text{ days, each pair of students} \\
&\text{have shaken hands exactly once, and } k \text{ is as minimal as possible.} \\
&\text{Prove that } \sqrt{n} \leq k - 1 \leq 2\sqrt{n}. \\
&\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q2]} \\[20pt]
&\textbf{Q8.} \text{Given a fair } n\text{-sided die with sides } 1, 2, \ldots, n: \\
&\text{Roll the die. If you roll } n, \text{ you win. Else, roll again.} \\
&\text{HOWEVER, if your roll is not greater than your previous roll, you lose.} \\[10pt]
&\text{E.g. } n = 4: \\
&\text{134: win, } \quad 31: \text{ lose, } \quad 122: \text{ lose, } \quad 24: \text{ win.} \\
&\text{Find the probability that you win for any given } n \text{ without using summation.} \\
&\text{[Original Question]}
\end{aligned}
\]](http://latex.artofproblemsolving.com/c/f/b/cfb583f0d96a9d709a5896acc0f3f8b3938e9acc.png)
with 
.
as 
and let 
at 
respectively. Draw circle 
such that 
and 
and 
,
can be written in the simplest form of 
,
.
are exponential growth for 
and decay for 
still labelled growth although it's decreasing, or is there no label for it, or is it now classified as a decay (similar question for a function such as 
; still a decay, now a growth, or nothing?)
Y by sixoneeight, parmenides51
In the spirit of parmenides51, I guess.
p1. We call a time on a
hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, 
, 
and 
are nice times. How many nice times occur during the course of one day? (We do not consider times of the form 
.)
p2. Along Stanford’s University Avenue are
palm trees which are either red, green, or blue. Let the positive integers 
, 
, 
be the number of red, green, and blue palm trees respectively. Given that
![\[R^3+2B+G=12345,\]](//latex.artofproblemsolving.com/c/0/4/c04b64fcac3fe356acdbd5b85d765d50687ed2dd.png)
compute .
p3.
integers are each selected uniformly at random from the range 
to inclusive and put into a set 
. Each integer is selected independently of the others. What is the expected value of the minimum element of ?
p4. Cornelius chooses three complex numbers
uniformly at random from the complex unit circle. Given that real parts of 
and 
are 
, compute the expected value of the real part of 
.
p5. A computer virus starts off infecting a single device. Every second an infected computer has a
chance to stay infected and not do anything else, a 
chance to infect a new computer, and a 
chance to infect two new computers. Otherwise (a 
chance), the virus gets exterminated, but other copies of it on other computers are unaffected. Compute the probability that a single infected computer produces an infinite chain of infections.
p6. In the language of Blah, there is a unique word for every integer between
and 
inclusive. A team of students has an unordered list of these 
words, but do not know what integer each word corresponds to. However, the team is given access to a machine that, given two, not necessarily distinct, words in Blah, outputs the word in Blah corresponding to the sum modulo of their corresponding integers. What is the minimum 
such that the team can narrow down the possible translations of "" to a list of Blah words, using the machine as many times as they want?
p7. Compute
![\[\sqrt{6\sum_{t=1}^\infty\left(1+\sum_{k=1}^\infty\left(\sum_{j=1}^\infty(1+k)^{-j}\right)^2\right)^{-t}}.\]](//latex.artofproblemsolving.com/6/7/a/67a0c0f264d80449379f40de07c81f9bc91ca627.png)
p8. What is the area that is swept out by a regular hexagon of side length as it rotates 
about its center?
p9. Let
be the the area enclosed by the relation
![\[x^2+y^2\le2023.\]](//latex.artofproblemsolving.com/1/1/3/11369b81a474887a0f45687a8213902f7434030a.png)
Let be the area enclosed by the relation
![\[x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}.\]](//latex.artofproblemsolving.com/3/9/3/39322b9614ce64c3527cf1fa0c75829ac53b123e.png)
Compute the limit of as 
for 
.
p10. Let
be the set of positive integers which are the product of an even number of distinct primes, including . Let 
be the set of positive integers which are the product of an odd number of distinct primes. Compute
![\[\sum_{n\in\mathcal{S}}\left\lfloor\frac{2023}{n}\right\rfloor-\sum_{n\in\mathcal{T}}\left\lfloor\frac{2023}{n}\right\rfloor.\]](//latex.artofproblemsolving.com/4/e/3/4e379a247b8f2ace8c37592801a4a9f0f339f17f.png)
p11. Define the Fibonacci sequence by
, 
, and 
for 
. Compute
![\[\lim_{n\rightarrow\infty}\frac{F_{F_{n+1}+1}}{F_{F_n}\cdot F_{F_{n-1}-1}}.\]](//latex.artofproblemsolving.com/6/4/9/6494b62f9e04be61b87a86afa6f97e7205fe9ff2.png)
p12. Let, , 
, and 
be points in the plane with integer coordinates such that no three of them are collinear, and where the distances 
, 
, 
, 
, 
, and 
are all integers. Compute the smallest possible length of a side of a convex quadrilateral formed by such points.
p13. Suppose the real roots of
are 
(roots
may be repeated). Compute
![\[\sum_{i=1}^k\frac{1}{2-r_i}.\]](//latex.artofproblemsolving.com/1/3/2/132097cb3af840351e816a4f063f8fd0cd078409.png)
p14. A teacher stands at
and has some students, where there is exactly one student at each integer position in the following triangle:
Here, the circle denotes the teacher at and the triangle extends until and includes the column 
. A teacher can see a student 
if there is no student in the direct line of sight between the teacher and the position . Compute the number of students the teacher can see (assume that each student has no width—that is, each student is a point).
p15. Suppose we have a right triangle
where is the right angle and lengths 
. Suppose we have points , 
, and 
on , , and respectively with 
. What is the minimum possible length of 
?
p1. We call a time on a
hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, 
, 
and 
are nice times. How many nice times occur during the course of one day? (We do not consider times of the form 
.)p2. Along Stanford’s University Avenue are
palm trees which are either red, green, or blue. Let the positive integers 
, 
, 
be the number of red, green, and blue palm trees respectively. Given that![\[R^3+2B+G=12345,\]](http://latex.artofproblemsolving.com/c/0/4/c04b64fcac3fe356acdbd5b85d765d50687ed2dd.png)
compute .p3.
integers are each selected uniformly at random from the range 
to inclusive and put into a set 
. Each integer is selected independently of the others. What is the expected value of the minimum element of ?p4. Cornelius chooses three complex numbers
uniformly at random from the complex unit circle. Given that real parts of 
and 
are 
, compute the expected value of the real part of 
.p5. A computer virus starts off infecting a single device. Every second an infected computer has a
chance to stay infected and not do anything else, a 
chance to infect a new computer, and a 
chance to infect two new computers. Otherwise (a 
chance), the virus gets exterminated, but other copies of it on other computers are unaffected. Compute the probability that a single infected computer produces an infinite chain of infections.p6. In the language of Blah, there is a unique word for every integer between
and 
inclusive. A team of students has an unordered list of these 
words, but do not know what integer each word corresponds to. However, the team is given access to a machine that, given two, not necessarily distinct, words in Blah, outputs the word in Blah corresponding to the sum modulo of their corresponding integers. What is the minimum 
such that the team can narrow down the possible translations of "" to a list of Blah words, using the machine as many times as they want?p7. Compute
![\[\sqrt{6\sum_{t=1}^\infty\left(1+\sum_{k=1}^\infty\left(\sum_{j=1}^\infty(1+k)^{-j}\right)^2\right)^{-t}}.\]](http://latex.artofproblemsolving.com/6/7/a/67a0c0f264d80449379f40de07c81f9bc91ca627.png)
p8. What is the area that is swept out by a regular hexagon of side length
as it rotates 
about its center?p9. Let
be the the area enclosed by the relation![\[x^2+y^2\le2023.\]](http://latex.artofproblemsolving.com/1/1/3/11369b81a474887a0f45687a8213902f7434030a.png)
Let be the area enclosed by the relation![\[x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}.\]](http://latex.artofproblemsolving.com/3/9/3/39322b9614ce64c3527cf1fa0c75829ac53b123e.png)
Compute the limit of as 
for 
.p10. Let
be the set of positive integers which are the product of an even number of distinct primes, including . Let 
be the set of positive integers which are the product of an odd number of distinct primes. Compute![\[\sum_{n\in\mathcal{S}}\left\lfloor\frac{2023}{n}\right\rfloor-\sum_{n\in\mathcal{T}}\left\lfloor\frac{2023}{n}\right\rfloor.\]](http://latex.artofproblemsolving.com/4/e/3/4e379a247b8f2ace8c37592801a4a9f0f339f17f.png)
p11. Define the Fibonacci sequence by
, 
, and 
for 
. Compute![\[\lim_{n\rightarrow\infty}\frac{F_{F_{n+1}+1}}{F_{F_n}\cdot F_{F_{n-1}-1}}.\]](http://latex.artofproblemsolving.com/6/4/9/6494b62f9e04be61b87a86afa6f97e7205fe9ff2.png)
p12. Let
, , 
, and 
be points in the plane with integer coordinates such that no three of them are collinear, and where the distances 
, 
, 
, 
, 
, and 
are all integers. Compute the smallest possible length of a side of a convex quadrilateral formed by such points.p13. Suppose the real roots of
are 
(rootsmay be repeated). Compute
![\[\sum_{i=1}^k\frac{1}{2-r_i}.\]](http://latex.artofproblemsolving.com/1/3/2/132097cb3af840351e816a4f063f8fd0cd078409.png)
p14. A teacher stands at
and has some students, where there is exactly one student at each integer position in the following triangle:
and the triangle extends until and includes the column 
. A teacher can see a student 
if there is no student in the direct line of sight between the teacher and the position . Compute the number of students the teacher can see (assume that each student has no width—that is, each student is a point).p15. Suppose we have a right triangle
where is the right angle and lengths 
. Suppose we have points , 
, and 
on , , and respectively with 
. What is the minimum possible length of 
?This post has been edited 2 times. Last edited by peace09, May 3, 2023, 5:38 PM
into a convoluted, intimidating problem—equivalently, an elementary schooler who just learned about the aforementioned equation and overexcitedly turned it into an "original" "thought-provoking" problem
is the probability that the virus dies out eventually from 1 computer. Then, if there are 
computers, then it is 
since the different chains starting from different computers are essentially independent. Thus, 
The roots of this are 
The first solution obviously does not make sense. Thus, either 
(which would give answer 4/5) or 
meaning that the virus is guaranteed to die out eventually (giving answer 0).
so 
.
so its area is 
so 
and 
is equivalent to finding all the fractions with numerator ![$n \in [1, 20]$](http://latex.artofproblemsolving.com/e/7/2/e72458f99ffb64e0f0c9ee250f52870e6ee8687d.png)
and denominator ![$m \in [1,21]$](http://latex.artofproblemsolving.com/4/f/a/4fa2737e3eb137a156691b66ea355ca73228f3f8.png)
such that 
is irreducible. Clearly this is true iff 
.![\[\sum_{k=2}^{21} \phi(k) = 139\]](http://latex.artofproblemsolving.com/3/a/b/3abce0cb9974c14301022fdf58d185f517380d42.png)
valid 
.
to account for 
and add 
to count the student at 
.
.
a
has a sum of digits between 
Hence, the answer is 
From simple bounding noting how the RHS and the LHS changes, we see that 
From basic substition and algebra, we find that 
is a solution.
.
be an integer mod 99.
,
mod 99 must be distinct, which isn't true if 
to the word actually given by 
is also valid (checkable that 
and 
where roots may be repeated. We have to find the value of 
We have to find an equation whose roots are 
by transformation of roots we get 
putting this value in the given equation we get, 
after calculate we gte the answer as 
and after cancellation we get our answer as 
becomes 
