![\[
\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]}
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\]](http://latex.artofproblemsolving.com/c/f/b/cfb583f0d96a9d709a5896acc0f3f8b3938e9acc.png)
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?)
being a natural number, let 
be a divisor of 
satisfying 
.
as the sum of the digits of a natural number 
.![\[
\{x\} \text{ as the fractional part of } x.
\]](http://latex.artofproblemsolving.com/b/f/7/bf7b4cc20cbdcee755c73f967c6d231a0896b4b4.png)
The smallest natural number ![\[
\left\{\frac{n}{s(n)}\right\} = \frac{1}{6}
\]](http://latex.artofproblemsolving.com/e/9/3/e9324df5f1fe12512254c2a8e273629b6b824126.png)
is 
.
,
,
are 
positive numbers that are not all necessarily different but all 
that contain at least 3 elements and do not have any two elements whose difference is exactly 7 is ...
be an odd positive integer satisfying 
with exactly 2 distinct prime factors. Prove that 
![\[
19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100
\]](http://latex.artofproblemsolving.com/d/3/d/d3d8657e2ab69cdc0886e2498ab03f7fd959056f.png)
Y by
A set 
is called a 
if it satisfies the following conditions:
contains at least two elements.
For all 
with 
, we have 
, and among the two numbers 
and 
, exactly one is rational.
For all 
, 
is irrational.
What is the maximum number of elements that can have?
is called a 
if it satisfies the following conditions:
contains at least two elements.
For all 
with 
, we have 
, and among the two numbers 
and 
, exactly one is rational.
For all 
, 
is irrational.What is the maximum number of elements that
can have?
