8 question contest for fun :)

Art of Problem Solving

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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8 question contest for fun :)

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#1 h Mar 26, 2025, 8:49 AM

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$\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}$

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#2 h Mar 26, 2025, 9:19 PM

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problem 8 is just binomial theorem
p(n) = (1 + 1/n)^(n-1) / n

This post has been edited 1 time. Last edited by scannose, Mar 26, 2025, 9:20 PM
Reason: whoops

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mathprodigy2011

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#3 h Mar 26, 2025, 10:05 PM

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mathprodigy2011

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#4 h Mar 26, 2025, 10:13 PM

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This post has been edited 1 time. Last edited by mathprodigy2011, Mar 26, 2025, 10:57 PM
Reason: 3 isnt equal to 4

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