Daily pair of puzzles - perceived difficulty

I'll be posting a lot of puzzles/questions over the next few weeks and I thought I'd address a few comments.

First, a little background.  These are from a yearly contest put out by the University of Vermont for any in-state student. The math department up there has been doing this for years and I've been giving it for years. Each year I add to our (me and the Mrs.) collection but I just recently acquired a notebook of tests going back to something like '72. These are great questions for students and I figured you all would appreciate if I'd share - judging from the response, you do.

The test itself is 41 questions.  The students have 2 hours and all the scrap paper they need. No calculators, no computers, no Wolfram Alpha, no smartphones, no collaboration; it's just you and this piece of 11x17 paper folded to a four page booklet ... and a pile of scrap paper. Afterwards, they'll give out a Best in School and Best in Region, along with money prizes to the top six in the state.. Best in School is usually 12-15 correct. I tell the PreCalc kids that 6-10 is an achievement.

So, time is a consideration, "brain freeze" is rampant, the occasional quiet muttered "You bastard, I gotcha" and you can't discount the missing calculator. It's a challenge and every question is "just on the tip of their brain and then everything cancels and 'viola'."  That "aha!" moment happens often. The dope-slap happens too, when they forget to reduce 14/10 to 7/5.

For me as a teacher, it's a perfect vehicle for breaking them loose from tech, for giving them a sense of perspective, for giving them a challenge and setting them up against the best of the state. Most importantly, it's a single test that incorporates EVERYTHING that they know into one single set of questions. To solve each one requires a complete knowledge base and access to both trivia and method, and theorem and idea. Most questions are asked in an odd way that's a fun twist on what you've already done. It's a way for kids to realize that they actually DO know something and that getting a tough answer is really satisfying.

The test format: The first one is always some kind of compound fraction and page one (10 - 13 questions) usually contains questions that are "easy" for the students in that they have SOME idea of where to go. They'll make computational mistakes here.

Pages 2-3 are medium difficulty in that the students really struggle to get anywhere that seems "right" and the questions tend to be wordy and confusing.

Page four are the ones that students rarely get. As math teachers, we look at an infinite series as easy, but the students don't ... when you have it as a infinite series of triangles, they wilt in the heat.

And don't forget that 2 hour time limit.
And that other problem that took an entire page of algebra.
And "I forgot the square of 14 was 196, not 216, so that problem needs fixing. Oh wow, all this cancels now. Is that the answer? It seems too simple."

And these are juniors and seniors. In high school.

So the difficulty isn't so much a true rating for us, but I'll include it as a reference point.