Short Questions vs. Long Questions

Transcription

1 Strategy in Answering Short Questions Short Questions vs. Long Questions In local mathematics contests, questions are mainly divided into two types: long questions and short questions. Long questions are problems which require full working steps, and most of them are proving problems. On the other hand, short questions are problem that only answers are needed and you don t need to hand in how you tackle with the problem. Short-question-based contests usually require contestants to complete all the problems within a short time. Here is a table comparing time needed for long-question-based and short-question-based contests. Contest Questions Time allowed Time/Question/Person IMO 6 9 hours 1.5 hours APMO 5 4 hours 1.25 hours CHKMO 4 3 hours 45 minutes HK Team Selection 7 3 hours 25 minutes IMOPHK 20 3 hours 9 minutes HKMHASC 17 ~ 20 2 hours 6 ~ 7 minutes PCMSIMC /I hours 4.5 minutes PCMSIMC /G minutes 9 minutes ISMC /I hours 4.5 minutes ISMC /G 63 1 hours 3.8 minutes HKMO /HI minutes 4 minutes HKMO /HG minutes 8 minutes HKMO /FI 4 5 minutes 1.3 minutes HKMO /FG 4 5 minutes 5 minutes Long-question-based Short-question-based As presented here, short-question-based contests only give you about 5.6 minutes per question on average. This is little time you can work on, and therefore short questions should be handled differently with long questions. 97

2 Incomplete Induction In short questions, you have to forget about all rigorousness in mathematics. Don t think about without loss of generality. Answers in short questions are often derived from (several) special cases. This is unacceptable in long questions, but must be applied in short questions it s just because you don t have enough time. Example A: (HKMO 1994/HI Q7) Find the last digit of the number Solution A: This is impossible to evaluate , and assume you know nothing about modular arithmetic. What you can do is to derive the answer from special cases: k 3 k From the value of k = 0 to 6, we see a repeating pattern Therefore, we may conjecture that the unit digit of is the same as 3 1 by the repeating pattern, which is 3. 98

3 Assumption With Loss of Generality For short questions, the original problems are usually designed that the answer is invariant for every different available cases (if not, either the problem or you are wrong). Therefore, we can pick a special case that is easy to deal with and get the answer quickly. Example B: (HKMO 2001/FI1 Q1) a, b and c are lengths of the opposite sides of A, a b B and C of ABC respectively. If C = 60 and + = P, find the value b+ c a+ c of P. Solution B: Please notice that an equilateral triangle has all its interior angles equal to 60. Thus, with loss of generality we assume that ABC is an equilateral triangle. Moreover, 1 1 we assume the side length of this triangle is 1. Therefore P= + =

4 Direct Measurement For geometrical problems, a convenient tool is to measure the result by a marked ruler and protractor. However, most figures provided in the question paper are not drawn to scale. Thus you should copy one by your own before you measure. Example C: (JSMQ 1999/FG Q15) Refer to the figure, given that KF = 8 and FL = 3. Find the length of LG. D A M C B K F L G Solution C: This is the figure that is drawn to scale, and LG is measured to be 6.6. D A M C B G K 8 cm F 3 cm L 6.6 cm There is a disadvantage with measuring that the precision is very low. For example, in example C we claimed the answer is 6.6 by measurement, but we don t know if it is 20 really 6.6 or = or something else. Moreover, the figure drawn with 3 hand is not perfect either. There may be little errors in the construction that result in big change in the final answer. 100

5 Guessing One final approach in doing short question is to guess the answer. One big rule in doing short questions is do not leave any unanswered questions. It is because if you leave the problem blank, you will score nothing. But if you make some random guess, you still have a small possibility that make it correct. Thus, when, and only when, the contest is going to be ended, you must fill in a guess. Note that you should not do this if the overall performance will be affected because of wrong answers, like the presence of accuracy factor or deduction for wrong answers, etc. 101

6 References: MathsWorld2001 [ Original documents: Direct Measurement: Chapter 9 Geometry Section