The boiling point of liquid oxygen is 90K, so it's easily condensed by liquid nitrogen.
I have personally made LOX by pumping air through a glass U tube immersed in liquid nitrogen, so I can confirm it works.
Later:
As discussed in the comments, what condenses is a mixture of liquid oxygen and nitrogen rather than pure liquid oxygen. The dew point for air is about 82K, far enough above the boiling point of liquid nitrogen for a condensate to form, and the condensate is about 50% liquid oxygen. This article includes the relevant phase diagram.
From personal experience I can say the condensate is blue, though I did not test its magnetic properties or the violence of its reaction with organic materials.
Why do we boil water to cook food? It's not actually because there's anything magic about the boiling of water, or that the physical process of boiling in particular does anything. Usually it's because we want a constant-temperature heat bath. Say you are boiling vegetables. You boil water, and you know that water is at 100 degrees. Water actually cannot get any hotter than this--it stays at that temperature or it becomes steam and leaves the pot. Then you put the vegetables into the boiling water, and therefore you know that they are in a 100 degree environment. Then, you know you need to leave them in there for however long--let's say five minutes.
Suppose, though, you were at high altitudes and water boiled at 95 degrees. Well, now when you put your vegetables into boiling water, they are only in a 95 degree environment, so the cooking time has changed. Your recipe no longer works correctly, and you will have to boil the food for longer.
Or maybe not, actually. The other possibility is that you are putting food in an oven, say something that's supposed to be 200 degrees. In this case, the water that's probably in the thing you're baking actually keeps the food cooler for longer. It reaches 100 degrees and will stay there until it boils off. However, if it boils off at 95 degrees, then again your cooking parameters will have changed. Since the water boils off faster, your dish spends more time at higher temperatures and can burn.
The common factor here is that boiling water is very energy-intensive, more so than just getting it to 100 C. It is therefore quite easy (and common) to get water to 100 C and have it stay there for some time, and we use this in cooking to calibrate our recipes. If pressure changes, than all the temperatures we assume to hold in cooking change, and procedures may need to be adapted as well.
Best Answer
For elevations less than about 100 km (for reference, the peak of Mt. Everest is about 8.8 km above sea level), the relative concentration of oxygen in the air is fairly constant at about 21%.
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It's true that there's less oxygen (more specifically,the partial pressure of oxygen is lower) with increasing altitude - and this is simply because there is less gas overall.
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The reason it's difficult to breathe at higher altitudes is that the ability of your lungs to oxygenate your blood depends on the partial pressure of O$_2$ in your lungs when you take a breath. At sea level and under ordinary conditions, the partial pressure of O$_2$ in your lungs is approximately $21\% \times 100\ \mathrm{kPa} \approx 21\ \mathrm{kPa}$. This defines normal, at least in a limited sense. If you're breathing pure oxygen, then you could potentially have an O$_2$ partial pressure of $100\ \mathrm{kPa}$, which can help compensate for damage to the lungs (e.g. from scarring) which reduces their ability to oxygenate blood - though as user Arsenal points out in a comment, under ordinary circumstances this would induce hyperoxia, which is bad news.
On the other hand, at the top of Mt. Everest the partial pressure in your lungs would drop to approximately $21\% \times 30\ \mathrm{kPa} \approx 6\ \mathrm{kPa}$ - nowhere near enough to sustain for extended periods, especially under increased physical stress. Breathing pure oxygen from a tank boosts this number to closer to $30\ \mathrm{kPa}$, which is why most climbers take their own oxygen with them.
However, above an altitude of 12 km (roughly the altitude at which commercial airliners fly) the pressure drops below $20\ \mathrm{kPa}$, which means that even breathing pure oxygen won't give you the normal required partial pressure of O$_2$, and you will risk hypoxia. That is why planes which fly above this altitude have positive-pressure respirators for pilots in case of emergency. Rather than being simply a higher concentration of oxygen, the gas in a positive-pressure respirator is (as the name suggests) actively pressurized above the ambient atmospheric pressure to force the required amount of oxygen into your lungs. See e.g. page 4 of this booklet from the US FAA and the associated regulations.