Problem Solving

Talkativetree wrote:I love these hard problems. Oddly enough I think they're fun. Is there anywhere else I could find similar problems?

You might try Magoosh.
Of the 500 questions there, I'd say that about 100-150 of them are in the 650+ range. (some of them have already been posted in the other forums)

hey all.

regarding the "right triangle ABC" problem
(found here: https://www.beatthegmat.com/problem-solv ... 30325.html)

the solutions above would be nice in a geometry course; a lot of them are very pretty.

but - let's assume that this is a gmat problem. (NOTE: this would not conceivably be a gmat problem - it's too obscure/difficult to show up on the test.

if this were a gmat problem:
* it would be multiple choice, with five NUMBERS as answers - say, 30, 35, 40, 45, 50.
therefore:
* that NUMBER would be the same, regardless of the particular shape of the right triangle ABC.
therefore:
* if you can answer the problem for ANY SINGLE EXAMPLE of ABC, then you've got THE answer.

so:

we can just let ABC be a 45-45-90 triangle.
if that's the case, then BO will be a horizontal line, and it will be clear (from symmetry) that BO cuts right angle ABC into two 45-degree angles.
that's all you need.

again, BECAUSE THE PROBLEM WOULD BE MULTIPLE-CHOICE, there's no need to produce such elaborate general solutions; finding just one solution is sufficient.

lunarpower wrote:hey all.

regarding the "right triangle ABC" problem
(found here: https://www.beatthegmat.com/problem-solv ... 30325.html)

the solutions above would be nice in a geometry course; a lot of them are very pretty.

but - let's assume that this is a gmat problem. (NOTE: this would not conceivably be a gmat problem - it's too obscure/difficult to show up on the test.

if this were a gmat problem:
* it would be multiple choice, with five NUMBERS as answers - say, 30, 35, 40, 45, 50.
therefore:
* that NUMBER would be the same, regardless of the particular shape of the right triangle ABC.
therefore:
* if you can answer the problem for ANY SINGLE EXAMPLE of ABC, then you've got THE answer.

so:

we can just let ABC be a 45-45-90 triangle.
if that's the case, then BO will be a horizontal line, and it will be clear (from symmetry) that BO cuts right angle ABC into two 45-degree angles.
that's all you need.

again, BECAUSE THE PROBLEM WOULD BE MULTIPLE-CHOICE, there's no need to produce such elaborate general solutions; finding just one solution is sufficient.

Great point, Ron - if you understand this, you can find simple solutions to some of the harder GMAT problems -- Q10, for example, in the diagnostic test at the beginning of OG12 (the one with the pentagram diagram). The solution given in the book introduces 10 unknowns, which is a bit crazy. But since the answers are all numbers (we don't have 'cannot be determined' in the answer choices), that guarantees that the answer is the same no matter how skew or symmetric the diagram; a GMAT question can only have one correct answer, after all. So you can freely assume the pentagon is regular, and thus has interior angles of 108, making all of the triangles isosceles with angles 72, 72 and 36, from which the answer comes immediately; 5*36 = 180.

Note, however, that you could not use this approach on a question like Q179 in the Problem Solving section of OG12; there one of the answers is 'cannot be determined', which means that it is at least possible that the answer may be different depending on what numbers you start with (and indeed, in that question at least, it is different for different starting numbers).