AoPS Academy
+2 w
is defined by the conditions
and 
for a set of indices 
.
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
is a postive integer.
be a function that is at least twice differentiable on an open interval containing ![\( [0, 2\pi] \)](http://latex.artofproblemsolving.com/9/7/7/977adaa4f83dc8164975a3e78539f27cbb354adf.png)
. Given that![\[
f(0) = f(2\pi) = f'(0) = f'(2\pi) = 0
\]](http://latex.artofproblemsolving.com/d/9/9/d998bdd53ffec1fa4c88688d7402ad25efa31112.png)
and![\[
f(x) + f''(x) \geq 0, \quad \forall x \in [0,2\pi].
\]](http://latex.artofproblemsolving.com/a/1/c/a1c24260aa02cd989630295d9de8741af659e958.png)
Prove that 
for all ![\( x \in [0,2\pi] \)](http://latex.artofproblemsolving.com/d/c/5/dc59f105fb3b9f6be175d2c035ba2ead4b9d984a.png)
.
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a % tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?
, with help of 
and Moivre's formula.
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
is the midpoint of 
and 
is the midpoint of 
. Suppose 
is similar to 
. Find the length of .
satisfying ![\[|z -a | + |z + a| = 2|c|\]](http://latex.artofproblemsolving.com/4/b/d/4bdafe46f8d3ed1c8945a1553fc3bc0d23fe11e2.png)
if and only if 
If this condition is fulfilled, what are the smallest and largest values of 
and radius 
and 
.
![\[
\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)
