Problem 1-9: Triangle Side Expression

Problem Posts

 [Difficulty 4] [PDF]

QUESTION
(Hong Kong Team Selection Test 2009) Let $a$, $b$, $c$ be the sides of a triangle. Determine all possible values of $$\frac{a^2+b^2+c^2}{ab+bc+ac}$$

Laconic Solution Sketch
Apply Triangle Inequality, or Ravi Transformation.

Problem 1-7: Back-Engineering

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[Difficulty 2] [PDF]

QUESTION
(Adapted from UVA 10784) The number of diagonals of an $n$-gon is not less than $N$. Find a closed-form expression for the minimum possible value of $n$.

Laconic Solution Sketch
A polygon has an integral side number. Use bounds.

Problem 1-5 Trivial Inequality

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Problem 1-5 (Algebra) [Difficulty 2] [PDF]

QUESTION
(Sipnayan 2011 High School Elims, Average Question 2) Suppose $x$ and $y$ are real numbers such that $$2x^2+y^2-2xy+12y+72\le0$$. What is the value of $x^2y$?

Overview/Laconic Solution Sketch
Group into squares and apply the Trivial Inequality.