The Putnam Competition is insane. Citing a factual statement, the median score on the competition is 1 out of a possible 120.
Let's put this into perspective. You have some 4000 of the world's brightest undergraduate mathematicians, who were sifted through the upper deciles of college-entering students from high school, then filtered through some self-selection within the universities. They sit down for 6 hours to concentrate their best on effectively a 120 mark test.
And at least half of them score 1/120 and below, i.e. the majority of the world's brightest mathematicians score either 0/120 or 1/120 out of this test. To make it more dramatic, on the upper end, there are people who score 120/120 once in a while.
Imagine the feeling that you'd get if you spent 6 hours and scored zero... that must be one of the rare experiences that the Latin proverb, "Amici, diem perdidi," finds an application for. Of course, saving the dramatization, not all of the best mathematicians-to-be take part in the competition, and how good one is at mathematics competitions doesn't reflect one's academic aptitude/productivity. Competitions differ from research.
Some questions on the competition have deceptively simple solutions. For example, find
$\int_{2}^{4}\dfrac{\sqrt{\ln\left(9-x\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x+3\right)}}\, dx$ --------- (1987-B1)
(if you worked this out in full within 6 hours on test day, you would have scored 10, probably placing you ahead of at least 2000 people!)
My first impression was to assault it with differentiation under the integral sign:
Let $\phi\left(x\right)=\int_{u_{1}}^{u_{2}}f\left(x,\alpha\right)\, dx$
then $\dfrac{d\phi}{d\alpha}=\int_{u_{1}}^{u_{2}}\dfrac{\partial f}{\partial\alpha}\, dx+f\left(u_{2,}\alpha\right)\dfrac{\partial u_{2}}{\partial\alpha}-f\left(u_{1},\alpha\right)\dfrac{\partial u_{1}}{\partial\alpha}$
Or, this looking rather convoluted, if $u_1, u_2$ are constants,
$\dfrac{d\phi}{d\alpha} = \int_{u_{1}=a}^{u_{2}=b}\dfrac{\partial f}{\partial\alpha}\, dx$
This of course, could brute-force some definite integrals with logarithmic reciprocals, and isn't taught very much nowadays since we can evaluate definite integrals numerically now. As told in Feynman's autobiography, back when they used punchcards for calculations when they were developing the atomic bomb, he managed to solve the integral in one of their calculations which had gone incomplete for a month, saving much precious time.
The problem though, is to think of an auxillary function $f\left(x,\alpha\right)$ that will render the single variable function that we're given susceptible to this method. Often, we exploit the case $\alpha =1$ i.e. $\phi \left(1\right)$ for this. This could be very time consuming if you aren't familiar with the form of the given function... or perhaps be a fruitless effort at the end of it.
Here comes the beauty of the solution, as I've said: simply note from substituting $x = 6-t$ that
$\int_{2}^{4}\dfrac{\sqrt{\ln\left(9-x\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x-3\right)}}\, dx=\int_{2}^{4}\dfrac{\sqrt{\ln\left(x+3\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x-3\right)}}\, dx$
Hence,
$\int_{2}^{4}\dfrac{\sqrt{\ln\left(9-x\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x+3\right)}}\, dx$
$=\int_{2}^{4}1-\dfrac{\sqrt{\ln\left(x+3\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x-3\right)}}\, dx$
$\implies2\int_{2}^{4}\dfrac{\sqrt{\ln\left(9-x\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x+3\right)}}\, dx=2$
$\therefore\int_{2}^{4}\dfrac{\sqrt{\ln\left(9-x\right)}}{\sqrt{\ln\left(9-x\right)}+\sqrt{\ln\left(x+3\right)}}\, dx=1$
Pretty. On another hand, I recently received an email from the head of the mathematics department hoping to help me consider majoring in math. I thought of it for a moment - I think I am better at mathematics than physics; and I'm bad at both.
But I'm not sure of defecting from physics... one aspect of physics that is really appealing is that its applications are very real; that its phenomena are very direct experiences. I remember writing in my admissions essay that I didn't have a noble reason for choosing physics; neither could I trace it to a childhood dream. But I find physics more naturally compelling. Its imaginative landscape is richer than a fantasy world. You could cultivate applications for its fertile land, speak the elegant mathematics of which its native language is constructed, and seek the ontological implications that underlie its explanations. So physics makes a greedy choice for me that still exposes me to some pretty, advanced mathematics (note the comma).
After all, much of modern mathematics developed from the efforts to formulate physics: calculus as a problem of variations instead of mere slopes; vector and tensor algebra from the the mechanics of motion and general relativity, and so on.
And I think my mathematical intuition is rather poor to do research. Some will argue that intuition is learned (sounds contradictory), but seriously, there are some people whom I know who just can never make it past how to use a graphing calculator... (I'm one of them.)
Anyway, if I were to pry a little deeper, why do I like mathematics? I think that mathematics is like music. We could replicate everyday sounds with musical instruments - but nobody likes listening to that. Similarly, the notions of mathematics are in fact wonderful in that they could map the experiences of daily life but there's another, "unrealistic" perspective that we could appreciate through mathematics. The straight line exists only as a perfection on a coordinate plane, something that doesn't manifest in our life: the closest that we can get to a 'straight line', is in fact many dots coming together (we call them atoms), or the history of a photon's motion when we give it the shortest instance.
The conclusion to the film adaptation of the Pulitzer Prize-winning play, Proof goes,
"If I go back to the beginning, I could start it over again. I could go line by line, try and find a shorter way. I could try to make it... better."
which draws us to a very beautiful aspect of mathematics and its contrast to everyday life. "If I go back to the beginning, I could start it over again", as opposed to, "If I could go back to the beginning, I would start it over again."
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