Putnam/Olympiad problem of the day

[http://math.harvard.edu/putnam/ No.4675484]

An acute-angled triangle ABC is given in the plane. The circle with diameter AB intersects altitude CC' and its extension at points M and N and the circle with diameter AC intersects altitude BB' and its extensions at P and Q. Prove that the points M, N, P, Q lie on a common circle.

[Anonymous No.4675517]

Here's a diagram.

[Anonymous No.4675605]

M and N are equally distanced from A.
Q and P are also equally distanced from A.
^These two can be proven via perpendicularity of altitudes on diameters, I think.
Now all that's left is to prove that one of the first two is distanced the same as one of the second two, which is what I can't figure out.

[Anonymous No.4676732]

AP^2 = AB' . AC = AC' . AB = AN^2

I am completely disappointed. Although I can no longer take part in it, I've heard that there will no longer be a geometry problem in IMO. And THIS 8th-grade problem counts as the Putnam problem of the day? Plato is turning in his grave.

[Anonymous No.4676772]

>>4676732
is geometry being removed from olympiads permanently? i hope this is the case. the stuff that's often required to solve olympiad geometry problems is largely irrelevant to higher mathematics, and a waste of time in my opinon.

the problem authors should replace plane geometry with analysis or abstract algebra.

[Anonymous No.4676953]

>>4676772
Try learning physics or optics. Or even just vectors and coordinate systems (which naturally lead to linear algebra and transformation). Try proving <span class="math"> \cos(10^{\circ}) + \cos(130^{\circ}) + \cos(250^{\circ}) = 0. [/spoiler]. There is a deep connection between all the fields of mathematics. I remember the awe I felt when I learned how vectors would generalize a whole bunch of problems like Menelaus' theorem, some complicated inequalities would rely on geometry for the proof, or how a combinatorial result like Sperner's lemma would lead to Brouwer's fixed point theorem.

Your argument is the same as all the arguments against IMO itself. The simple answer is that you don't know what the heck you're talking about. Students untrained in geometry simply lack spatial reasoning and imagination required to understand the relation between objects in a plane or multi-dimensional space. That's why I said Plato is turning in his grave. The world is full of plebeians untrained in geometry.

[Anonymous No.4677295]

>>4676772
what do you mean by higher mathematics? lel

>>4676953
stop exaggerating, 1/10

[Anonymous No.4677676]

>>4677295
>look at me I can troll lel

[Anonymous No.4677783]

I like how this is supposedly mathematical pablum, and yet nobody has solved it yet.

[Anonymous No.4677802]

>>4677783
I will give you 10 minutes to realize your idiocy and feel bad about yourself.

[Anonymous No.4677820]

>>4677759

[Anonymous No.4677855]

>>4676953
>The world is full of plebeians untrained in geometry
I couldn't agree more with you since I don't understand where does
>AP^2 = AB' . AC
come from

Anyone care to explain ? (you can even laugh at me if you want)

[Anonymous No.4677945]

>>4677855
APC is a right-angled triangle. See
http://en.wikipedia.org/wiki/Right_triangle#Altitude

[Anonymous No.4678025]

So to the gents solving these problems:

How long do you work on each problem?
How many do you actually solve?
How many do you make zero progress with?
And what level is your mathematical education at?

As a non-Mathematician who cannot even begin solving these I was just wondering how hard they actually are.

[Anonymous No.4678121]

>>4678025
I doubt anyone goes on /sci/ every day. I'm only around whenever I'm back home at weekends, for example.

I myself am a newfag on this board (not 4chan), 1st year in college, and have personally solved all the problems I see (5 of them). There was one that took more than an hour. This one barely a minute. So there's a discrepancy.

I don't design these problems so I don't know what someone needs to solve all of them. But this particular problem is one of the elementary math problems that require very little advanced mathematics, so even a non-mathematician usually has enough tools to solve them (unless you didn't learn geometry where you live). The point is to make you think and use what you know, not overwhelm you with hidden, advanced tools. Whether they have enough imagination to solve them is another story.

[Anonymous No.4678171]

>>4678025
Usually I can solve these problems in less than an hour, 30 minutes in average. However, some problems are really hard which need more than hour, but this last one is one of the easiest geometry problems (took only a few minutes).
If I can't solve a problem in an hour and I don't find it particularly interesting then I stop thinking on it.

I have an MS in (pure) mathematics so I don't participate in math contests anymore (i went to two IMOs). However, I had better problem solving skills at age 18 when I was more prepared in these kind of problems.

There is a huge difference between doing an actual mathematical research and solving a problem in a contest.
In the contest you can be sure that there exists a solution which can be found in less than an hour our two, and this information often gives you some hints how to start the solution.

[Anonymous No.4678203]

>>4678171

I very much agree. Doing my Masters in math at the moment, and I was never good at these kinds of problems. But with harder questions that took time and dedication to a problem that may not have a solution I'm pretty decent at. The two are different types of mathematics, and neither should be put on a pedestal.

[Anonymous No.4679280]

>>4677855
he's trollin', you can't take seriously anyone who uses the word pleb and all its derivatives, bad troll is bad