Archive for March, 2009

2009 AIME I

The 2009 AIME I answers are posted here on AoPS.

By my standards, I failed with 3 (or 4?) careless mistakes:

  1. Incorrect. Sloppy thinking! 964 is geometric as well, not just 931.
  2. Correct. Silly algebra.
  3. Correct. Silly algebra. :)
  4. Correct. Silly geometry.
  5. Correct. Silly geometry!
  6. Correct. Close call, but caught myself thinking 4^3 = 4^4.
  7. Incorrect. Careless! I got to the end surprisingly quickly, finding that a_{n+1} = log_5{3n+5}, but in my excitement, instead of finding n for which 3n+5 equaled a power of 5, I found n = 5 for which 3n+5 equaled a multiple of 5 instead. ERGH.
  8. Incorrect. Careless. I screwed up simple addition.
  9. Incorrect. Counting problems murder me. Overcounted.
  10. Incorrect. This one wasn’t really a careless mistake. I knew that MVE had to repeat, so I thought about MVEMVEMVEMVEMVE and MMMMMVVVVVEEEEE but never considered something like MMVVEEMMMVVVEEE. eh I was just stupid.
  11. Correct. Easy for a #11.
  12. Unsolved.
  13. Unsolved.
  14. Unsolved.
  15. Unsolved.

for a total of 6 correct.

After crunching through the first 11 problems and checking (shame on me), I had 30 minutes left. Unfortunately, this was when a loud IT guy came in the room and tried to explain to the proctor how to reinstall MS Office on his MacBook; my concentration subsequently fell apart.

So an index of 100.5 (AMC 12A) + 60 = 160.5 this year. Progress from my 8th grade index of 139.5 (AMC 10B) + 70 = 209.5 is something I’d rather not think about anymore. :(

(but I will persevere and receive 138 + 110 = 248 next year!)

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Supercritical Water Reactors (SCWRs)

My dad is a nuclear engineer, and he likes fluid dynamics. :) Last night he explained to me how the specific heat of a supercritical fluid sharply peaks at a certain pressure range (this relates, in some way too complicated for me, to the delta function). Okay, I didn’t understand all of it, but this is the idea behind supercritical water reactors (SCWRs) currently under investigation in 13 countries around the world. If the pressure of the coolant can be maintained within that range, we could get more efficient, simpler, and generally safer reactors because there would be no worry of such things as steam bubbles (different density = problems), pumps, or condensers.

In light water reactors (most reactors in the world are LWRs), water enters the reactor core at 290°C and exits at around 315°C—only a 35°C difference due to water’s high specific heat, but this could be improved by using supercritical fluids.

Wikipedia has better writers:

Advantages

  • Supercritical water-cooled reactors promise to have thermal efficiencies of approximately 45% versus the current 33% of light water reactors.
  • The supercritical coolant has a high specific enthalpy.
  • The SCWR design is far simpler than current designs, eliminating circulation pumps, pressurizers, steam generators, steam separators and dryers.

Disadvantages

  • Many of the materials needed for the SCWR are either expensive, rare, or do not exist.
  • Specific start-up procedures required to avoid instability.
  • Unknown chemistry.