Help! Help! Help!?!?!?!
I’m having a lot of trouble with math lately. It’s just TOOOOO hard!!!???!!!
I have to take a test soon, and I want to at least get a 67% on the test. If I don’t, I will not pass. How do I study??!!??!!??
Like this geometry problem?!?! It’s so easy, but I cannot solve. Triangle ABC has point P on side BC. Angle bisector of angle APB intersects circumcircle of APB at X and of APC at Y. Let M be midpoint of BC. Show that XMY is a right triangle.
Help! Help! Help!?!?!?!
@RinTohsaka As a junior in college majoring in mathematics, I can attest to that fact: math is oftentimes hard.
My geometry’s a bit rusty, and I haven’t been able to solve this one yet. A good first step might be to use the fact that quadrilaterals AXBP and AYPC are cyclic (obviously) and do some angle chasing - you get a couple surprising things.
OK, I know it’s early and I’ve been grading final exams forever.
But how does the bisector of <APB intersect <APC anywhere but P?? <s APB and APC are supplements-- my diagram is showing PX as a tangent to circumcircle APC.
Again, I’m tired. And my head was spinning with numbers before I opened this thread.
But here’s a bigger question: What final are you studying for? And do you know for a fact that this question will be on it?
I always advise my kids to start studying with topics from easiest to hardest. Create a checklist of the topics/ long problems you expect to see. And cross them off, one by one, from easiest to hardest. Create a list of things you know you can do, accumulate those points. If you don’t get the hardest question, that’s OK, as long as you have enough of those easier-- for you-- questions to get you the grade you need.
Is this a regular geometry course? Take a look at www.regentsprep.org It’s run by a school system in upstate NY, in Oswego (home of one of the SUNY campuses) to prepare its kids for NY’s Regents exams. I love their review site-- it’s a great mix of solid info and practice questions.
@bjkmom I don’t think this is necessarily the case - however the part I’m a little confused is, if P and M are the same point (i.e. P is the midpoint of BC), then triangle XMY should be degenerate.
@RinTohsaka I’m a little doubtful of this problem. Using an example similar to above, if we let triangle ABC be equilateral and P and M be the same point, then X, Y, and M are collinear (with no two points at the same location). Where is this problem from?
sorry - I meant
angle bisector of angle APB intersects circumcircle of APB at X other than P
angle bisector of angle APC intersects circumcircle of APC at Y other than P
was being lazy oops
but if ABC is equilateral, then angle XMA = 45 degrees since angle BMA = 90 degrees since midpoint is also altitude of triangle, and also angle YMA = 45 degrees (similarly), so angle XMY is 90 degrees so that particular case should be fine
also problem is from a handout
darn also this problem: if p(x) is an integer polynomial of degree n, show that p^k(x) = x has at most n integer solutions (for some fixed k; e.g. p^3(x) = p(p(p(x))))
darn geometry and algebra are too difficult
@RinTohsaka I interpreted it as “angle bisector of APB intersects circumcircle of APB at X and angle bisector of APB intersects circumcircle of APC at Y,” which doesn’t seem to work. So basically we have to prove that XMPY is cyclic.
We’d have to assume n > 1; otherwise p(x) = x is a counterexample.
This problem is 2006 IMO #5, there is a solution on AoPS. I tried inducting on k a few ways but didn’t get far…
@BenzeneRings a stroll who posts IMO #5’s and failed SAT math - I know at least one person who did that…
@RinTohsaka the problems are very easy. For instance, anyone who has taken Algebra II can solve the second problem. I will present short solution here if you have not solved yet.
Let a1, a2, …, ak be the fixed points, so we have ai-aj | P(ai)-P(aj) | … | Q(ai)-Q(aj) = ai-aj, so |ai-aj| = |P(ai)-P(aj)|. Then, it is easy to see that P(ai) = ai + k or P(ai) = -ai + k for some constant k. Then, a_i are roots of the polynomial T(x) = P(x) - x - k or T(x) = P(x) + x - k (one or the other, not both), so there are at most n roots so k <= n. Does require some thinking, but if you cannot solve, then you should be more worried about qualifying for AIME than even medaling at the IMO.
@kirito69 might want to distinguish k’s. Also, there is a full solution on AoPS.
sadly, other that the T roll part, I don’t understand anything you folks are saying:). I’m exaggerating of course, I understand triange too 
@MITer94 oh right good point. thanks!