Problem Post 2-6: How Many Shapes, Part 3: Rectangles in Not-Rectangles

		Problem Posts
		Part 3 of 4 of How Many Shapes?

[Level 2-3]

While it would be far more fun for me if I jumped from topic to topic, I believe I owe it to you, dear readers, to flog this horse until it's brain dead. Completeness is a virtue.

So last time we finished off the case where we count rectangles in rectangles, even when some 'matchsticks' or grid wires are missing. Now onwards to rectangles in non-rectangles.

CAN WE DO BETTER?

• What if the original grid were a rectangle, instead of a square? Or an irregular figure?
• What if we were counting rectangles instead of squares?
• What if some of the grid 'wires' are missing?
• What if we're working on a triangular grid, counting triangles? Does our logic still apply?

2 Rectangles in Non-Rectangles

This is ad hoc land. The base technique would again be to count one-by-one, but as we will see there are tricks for certain special irregular figures.

Problem Post 2-5: How many Shapes, Part 2: Rectangles in Rectangles

		Problem Posts
		Part 2 of 4 of How Many Shapes?

[Level 2-3]

So in my previous Miscellaneous post, we solved a fairly simple Internet puzzle, counting the number of squares in a square grid. But now, we ask ourselves if we can do more:

CAN WE DO BETTER?

• What if the original grid were a rectangle, instead of a square? Or an irregular figure?
• What if we were counting rectangles instead of squares?
• What if some of the grid 'wires' are missing?
• What if we're working on a triangular grid, counting triangles? Does our logic still apply?

Problem 1-8 Meticulous Manipulations

Problem Posts

[Difficulty 2] [PDF]

QUESTION
(2012 PMO Area Stage) If $x+y+xy=1$, where $x$, $y$ are nonzero real numbers, find the value of $$xy+\frac{1}{xy}-\frac{y}{x}-\frac{x}{y}$$

Laconic Solution Sketch
Manipulate.

Problem 1-7: Back-Engineering

Problem Posts

[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

Problem Posts

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.