Summary of "[PMO Tutorial #4] Listing the Possibilities - Combinatorics, Algebra"

Summary of [PMO Tutorial #4] Listing the Possibilities - Combinatorics, Algebra

This tutorial focuses on problem-solving strategies involving combinatorics and algebra, specifically through examples from the Philippine Mathematical Olympiad (PMO). The main lessons revolve around counting possibilities, analyzing sequences, and applying inequalities to probability problems.

Main Ideas and Concepts

1. Introduction to the Philippine Mathematical Olympiad (PMO)

The PMO is the oldest and most prestigious nationwide math competition for high school students in the Philippines, officially launched in 1986.
It serves as a selection ground for the country’s representatives to the International Mathematical Olympiad.
Organized by the Mathematical Society of the Philippines (MSP) and the Department of Science and Technology - Science Education Institute (DOST-SEI).

2. Problem-Solving Approach: Listing Possibilities and Case Analysis

Not all problems can be solved by direct formula evaluation; sometimes, enumerating possibilities or considering different cases is necessary.
Example problems from past PMO stages illustrate this approach.

Detailed Problem Walkthroughs and Methodologies

Problem 1: Probability of Forming a Geometric Sequence from Three 12-sided Dice Rolls

Setup

Three fair 12-sided dice (numbered 1 to 12) are rolled simultaneously.
Find the probability that the resulting numbers can be arranged to form a geometric sequence.

Key Points

Total outcomes = (12^3 = 1728).
Count ordered triples ((a, b, c)) such that they can be rearranged into a geometric sequence.
Common ratio (r) can be integer or fractional.
Constraints:
- (a, b, c \in {1, \ldots, 12}).
- (r^2 = \frac{c}{a}) must be a perfect square ratio.
- (r) cannot be too large (e.g., (r=4) fails because (4^2 \times a > 12)).

Methodology

Assume (a \leq b \leq c) for simplification.
Consider possible values of (r):
- (r=1): (a=b=c), all 12 values possible.
- (r=2): (a, 2a, 4a) with (4a \leq 12) → (a \leq 3).
- (r=3): (a, 3a, 9a) with (9a \leq 12) → (a=1).
- (r= \frac{3}{2}): (a, \frac{3}{2}a, \frac{9}{4}a), (a) divisible by 4, (a=4) valid.
Count permutations for each valid triple.
Sum counts to get numerator = 42.
Probability = (\frac{7}{288}) after simplification.

Problem 2: Probability of Forming an Arithmetic Sequence from Three 6-sided Dice Rolls

Similar to Problem 1 but with arithmetic sequences.
Total outcomes = (6^3 = 216).
Numerator also 42 (invited to try as exercise).
Focus on identifying possible common differences.

Problem 3: Probability that a Point Lies Above a Given Curve

Setup

Choose two numbers (a, b) independently and uniformly from ({1, \ldots, 10}).
Consider the curve (y = x^3 - b x^2).
Find the probability that the point ((a, b)) lies above this curve.

Key Points

Total pairs = (10^2 = 100).
Condition for ((a,b)) above the curve: [ b > a^3 - b a^2 ]
Rearranged inequality: [ b > \frac{a^3}{a^2 + 1} ]
After algebraic manipulation and integer constraints, simplified to: [ b > a^2 - 1 ]
Evaluate for (a = 1, 2, 3) (since for (a \geq 4), (b) would exceed 10).
Count valid (b) values for each (a):
- (a=1): (b > 0) → 10 choices.
- (a=2): (b > 3) → 7 choices.
- (a=3): (b > 8) → 2 choices.
Total valid pairs = 19.
Probability = (\frac{19}{100}).

Additional Notes

The tutorial emphasizes the importance of:
- Breaking down complex problems into manageable cases.
- Using algebraic manipulation to simplify inequalities.
- Understanding the meaning of conditions in combinatorial probability.
Encourages active problem solving by inviting viewers to try similar problems themselves.

Speakers / Sources Featured

Narrator / Presenter: An unnamed instructor or tutor affiliated with the Mathematical Society of the Philippines, explaining PMO problems.
Mathematical Society of the Philippines (MSP): Organizer of the PMO and promoter of mathematics education in the Philippines.
Department of Science and Technology - Science Education Institute (DOST-SEI): Partner organization in organizing the PMO.

Summary

The video tutorial uses PMO problems to demonstrate how to list possibilities and analyze cases in combinatorics and algebra. It covers probability problems involving geometric and arithmetic sequences from dice rolls and an inequality problem involving points above a curve. The key lesson is to combine logical reasoning, algebraic manipulation, and careful counting to solve complex probability questions.