[/quote]
,
.
people and 
colors, where each person has one ball of each color. For each person, their 
such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed 
and 
.
Equality holds when 
Equality holds when 
![\[
\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 tangents to 
at 
.
and 
and 
,
can be written in the simplest form of 
,
.
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
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?)
be a positive integer. On the table, we have 
ornaments in 
different colours.
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in 
,
has side lengths 
,
,
inside 
,
,
and denote them as 
,
,
.
?
.
be a point on side 
.
be the circumcenter of 
,
satisfying 
.
Y by Sedro
Given a fair 
-sided die with sides 
, consider the following game:
1. Roll the die. If the roll results in, you win immediately.
2. Otherwise, roll again. However, if the second roll is not greater than the previous roll, you lose.
3. Continue rolling until either:
- You roll, in which case you win.
- Or, your current roll is not greater than your previous roll, in which case you lose.
For example, when
:
- Rolls
: Win
- Rolls
: Lose
- Rolls
: Lose
- Rolls
: Win
Find a formula to find the probability of winning for any given.
-sided die with sides 
, consider the following game:1. Roll the die. If the roll results in
, you win immediately.2. Otherwise, roll again. However, if the second roll is not greater than the previous roll, you lose.
3. Continue rolling until either:
- You roll
, in which case you win.- Or, your current roll is not greater than your previous roll, in which case you lose.
For example, when
:- Rolls
: Win- Rolls
: Lose- Rolls
: Lose- Rolls
: WinFind a formula to find the probability of winning for any given
.
be the probability of winning with an 
,
:
,
:
,
:
,
:
.![\[
P(\text{subset}) = \left(\frac{1}{n}\right)^{\text{size of subset}}.
\]](http://latex.artofproblemsolving.com/b/e/5/be51b61c021a32194a5cf0b338f3ca463a049352.png)
,
numbers from 
.![\[
\binom{n-1}{a-1}.
\]](http://latex.artofproblemsolving.com/2/5/a/25a57b2befc8280ff4361f895e67cdc4ea60f0a1.png)
![\[
W(n) = \sum_{a=1}^{n} \binom{n-1}{a-1} \left(\frac{1}{n}\right)^a.
\]](http://latex.artofproblemsolving.com/f/4/4/f44d7da3bae4fe176f3073d201ba54fcb0255e88.png)
![\[
\begin{aligned}
W(n) &= \sum_{a=1}^{n} \binom{n-1}{a-1} \left(\frac{1}{n}\right)^a \\
&= \sum_{a=0}^{n} \binom{n-1}{a-1} \left(\frac{1}{n}\right)^a \quad (\text{the term for } a=0 \text{ does not exist as } \binom{n-1}{-1} \text{ is undefined}) \\
&= \sum_{a=0}^{n} \binom{n}{a} \left(\frac{1}{n}\right)^a - \sum_{a=1}^{n} \binom{n-1}{a} \left(\frac{1}{n}\right)^a \quad (\text{using Pascal's identity: } \binom{n}{a} = \binom{n-1}{a-1} + \binom{n-1}{a}) \\
&= \left(1 + \frac{1}{n}\right)^n - \sum_{a=1}^{n-1} \binom{n-1}{a} \left(\frac{1}{n}\right)^a \quad (\text{as } \binom{n-1}{n} \text{ does not exist}) \\
&= \left(1 + \frac{1}{n}\right)^n - \left(1 + \frac{1}{n}\right)^{n-1} \\
&= \left(\frac{n+1}{n}\right)^{n-1} \cdot \left(1 + \frac{1}{n} - 1\right) \\
&= \left(\frac{n+1}{n}\right)^{n-1} \cdot \frac{1}{n} \\
&= \frac{(n+1)^{n-1}}{n^n}.
\end{aligned}
\]](http://latex.artofproblemsolving.com/6/d/f/6df01dc9daa9603fdb843999cfae843e3d234193.png)
