Text Contents 읽기(2013 Software경진대회 문제 및 답안 - Maple 답안)

수학 한마당

Software 경진대회

<문제 1> 영역 R={| ≥ ,  ≥ , 



 



≤ }에서 적분



_







  



 



  



 



을 극좌표로 변수변환하여 계산하시오. (단,





  



  이다.)

<Sol>

이분법(bisection method)은 중간값 정리를 이용하여 구간 [a,b]를

반분해 가면서   이 되는 점 를 찾아내는 방법이다. 함수

가 구간 [a,b]에서 연속이고   이면, 

_

 , 

_

  라

하고 a와 b의 중점을 

_

_{ }





_

 

_

이라 하자. 만일 

_

 ≠  이면



_



_

   또는 

_



_

  이다. 실근 는 구간 

_

 

_



또는 

_

 

_

사이에 있을 것이다. 만일 

_



_

  이면, 이번에는



_

 

_

, 

_

 

_

이라 하고, 그 중점 

_

_{ }





_

 

_

에서 함수 값



_

를 계산하고 앞 단계와 같은 방법으로 

_

의 부호와 

_

 및



_

의 부호를 비교하여 실근 가 구간 

_



_

 및 

_

 

_

의 어느

구간에 속하는지를 결정한다. 이 과정을 계속하면 실근 가 속해 있는

구간이 점차 좁혀지면서 실근 를 알아낼 수 있게 된다.

<문제 2>

  



 



  일 때,   ,   , 

_

 

_{  }

  

 

가 되는 

_

를

구하여라.

※ Maple에서 프로시져 Bisection을 사용하면 아래와 같은 결과를

얻는다.

<Sol>

<문제 3> 성균관대 공과대학 학생이 명이 있다. 게임을 하기 위해 팀을

나누는데 5명씩 한 팀으로 묶으면 3명이 남고, 7명씩 한 팀으로 묶으면

4명이 남고, 11명씩 한 팀으로 묶으면 5명이 남는다. 공과대학 학생 은

모두 몇 명인가? (단, 는 4000명을 넘는 최소정수)

예를 들어, Maple 프로그램에서 “chrem()”, SAGE 프로그램에서 “crt()”,

Mathematica 프로그램에서 “ChineseRemainder()”을 이용하여 문제를

풀 수 있다.

위의 명령어를 사용하지 말고 문제를 해결해보아라.

<Sol 1>

<문제 4> 다음 그림을 보고 여자가 원하는 꽃의 모양을 그려라.

<문제 5> 아래의 그림 1에 의해 만들어지는 것을 Sierpinski Triangle

이라 한다. 주어진 Sierpinski Triangle의 코드를 이용하여 그림 2를

그려보아라.

그림 1. Sierpinski Triangle를

만드는 과정

그림 2.

Sage :

# Sierpinski Triangle at step 4 def sierpinski(S,A,B,C,N): T1 = Graphics() T2 = Graphics() T3 = Graphics() if (N>0): M_AB=[(A[0]+B[0])/2,(A[1]+B[1])/2] M_AC=[(A[0]+C[0])/2,(A[1]+C[1])/2] M_BC=[(B[0]+C[0])/2,(B[1]+C[1])/2] S += polygon([M_AB,M_AC,M_BC],color='white') if (N>1): T1 += sierpinski(T1,A,M_AB,M_AC,N-1) T2 += sierpinski(T2,M_AB,B,M_BC,N-1) T3 += sierpinski(T3,M_AC,M_BC,C,N-1) S += T1+T2+T3 return S A=[.5,sqrt(3)/4] B=[0,0] C=[1,0] S = Graphics() S += polygon([A,B,C],color='black') S += sierpinski(S,A,B,C,4) S.show(aspect_ratio=1, axes=False)

Mathematica : sierpinski

<문제 6> 음대생 모차르트와 공대생 장영실이 절대음감 놀이를 하고

있었다. 전화 통화 상으로 휘파람 소리를 내어 어떤 음인지 맞추는

것이었다. 하지만 모차르트가 문제를 낼 때에 통신 상태가 좋지 않아

많은 잡음이 발생하였고, 장영실은 난관에 봉착했다. 하지만 공대생인

장영실은 통화 내용을 녹음하고 있었고 다행히 아래와 같은 데이터를

얻게 됐다.

(a) Curve fitting을 활용하여 위의 데이터를

복원하여라. (단, x축은 millisecond, y축은

amplitude이고, 휘파람 소리는 sine함수

(      ) 모양을 갖는다.)

(b) 오른쪽 표를 참고하여 모차르트가 낸 음이 무엇인지 구하여라. (단

주파수는 주기의 역수이고 단위는 Hz이다.)

Sage data

[(0, 0.238726557599306), (0.100000000000000, -0.00106430080664327), (0.200000000000000, -0.552415997359500), (0.300000000000000, -0.490860805333598), (0.400000000000000, 0.344921255536726), (0.500000000000000, 0.218712318034386), (0.600000000000000, 0.226865872313974), (0.700000000000000, 0.227304368920393), (0.800000000000000, -0.0215306181333410), (0.900000000000000, -0.304304570611808), (1.00000000000000, 0.133203673484065), (1.10000000000000, 0.109875578432012), (1.20000000000000, 0.295359224807616), (1.30000000000000, 0.464391414788198), (1.40000000000000, 0.931476586336045), (1.50000000000000, 0.815129683376910), (1.60000000000000, 0.392269919166651), (1.70000000000000, 0.266317154908948), (1.80000000000000, 0.759173470403515), (1.90000000000000, 1.02961921410618), (2.00000000000000, 0.818988541770028), (2.10000000000000, 1.33058170479361), (2.20000000000000, -0.0106304924967900), (2.30000000000000, 0.794277338584339), (2.40000000000000, 1.06834524752293), (2.50000000000000, 0.429875503997622), (2.60000000000000, 1.30696381521764), (2.70000000000000, 0.804957379259014), (2.80000000000000, 0.783441034361311), (2.90000000000000, 1.39562029919050), (3.00000000000000, 1.16102415014942), (3.10000000000000, 1.59187717915691), (3.20000000000000, 0.729730642366330), (3.30000000000000, 1.17791618373868), (3.40000000000000, 0.390608741273900), (3.50000000000000, 1.52226464376289), (3.60000000000000, 0.923784476889042), (3.70000000000000, 0.829670766129800), (3.80000000000000, 0.932669307546741), (3.90000000000000, 1.14234598321232), (4.00000000000000, 0.500710697558121), (4.10000000000000, 0.592618043593119), (4.20000000000000, 1.15676394341977), (4.30000000000000, 0.819356114742084), (4.40000000000000, 0.807804693513561), (4.50000000000000, 1.55933385330901), (4.60000000000000, 0.901606542122742), (4.70000000000000, 1.00984786195597), (4.80000000000000, 0.679324065592616), (4.90000000000000, 1.02068771838068), (5.00000000000000, 0.977818575096133), (5.10000000000000, 1.03475205238557), (5.20000000000000, -0.103910649907259), (5.30000000000000, 0.981842104139987), (5.40000000000000, 0.422688552202437), (5.50000000000000, 0.00766764362708416), (5.60000000000000, 0.487608273270939), (5.70000000000000, 0.433274732392297), (5.80000000000000, 0.298650410459409), (5.90000000000000, 0.862757806937851), (5.99999999999999, 0.192085805686460), (6.09999999999999, 0.837009672553558), (6.19999999999999, 0.106075673880802), (6.29999999999999,

0.597851605513888), (6.39999999999999, 0.0434199164319980), (6.49999999999999, 0.289840209720640), (6.59999999999999, 0.220846924939167), (6.69999999999999, 0.0306726118665809), (6.79999999999999, 0.115690519740589), (6.89999999999999, -0.388870016582899), (6.99999999999999, -0.645554176764036), (7.09999999999999, -0.282525705913652), (7.19999999999999, -0.178591270193534), (7.29999999999999, -0.0602052584824280), (7.39999999999999, -0.478067215534890), (7.49999999999999, -0.763129810139659), (7.59999999999999, 0.124196545985498), (7.69999999999999, -0.678370540558398), (7.79999999999999, -0.628233276573054), (7.89999999999999, -1.26013377006657), (7.99999999999999, -0.200628135487097), (8.09999999999999, -0.836921702115714), (8.19999999999999, -1.06529203565071), (8.29999999999999, -1.52409284914189), (8.39999999999999, -1.10126537644158), (8.49999999999999, -0.513427445098486), (8.59999999999999, -0.295219145574798), (8.69999999999999, -0.175066810272216), (8.79999999999998, -1.35972529465098), (8.89999999999998, -0.535484283326289), (8.99999999999998, -0.804176144790150), (9.09999999999998, -1.28894765058573), (9.19999999999998, -1.70551308457956), (9.29999999999998, -0.871144616678133), (9.39999999999998, -0.787299855514117), (9.49999999999998, -0.702233592726486), (9.59999999999998, -1.34812882361573), (9.69999999999998, -1.34099056778808), (9.79999999999998, -0.789621139608874), (9.89999999999998, -1.05201289151652), (9.99999999999998, -0.221165280189115), (10.1000000000000, -0.548981192771710), (10.2000000000000, -1.19655932671510), (10.3000000000000, -1.33198673618446), (10.4000000000000, -1.00288818453729), (10.5000000000000, -1.50467980466325), (10.6000000000000, -1.39419963000100), (10.7000000000000, -0.691338104555618), (10.8000000000000, -1.87675235413828), (10.9000000000000, -0.0714228791604897), (11.0000000000000, -1.00698026996393), (11.1000000000000, -0.597026152328669), (11.2000000000000, -0.913991269764388), (11.3000000000000, -0.985580462035566), (11.4000000000000, -0.227817034172203), (11.5000000000000, -0.171811536357626), (11.6000000000000, -0.347117378286799), (11.7000000000000, -0.497875034993729), (11.8000000000000, -0.165172149433396), (11.9000000000000, -0.250483410204926), (12.0000000000000, 0.120949717574794), (12.1000000000000, -1.00760224590490), (12.2000000000000, -0.383531331169237), (12.3000000000000, -0.319299262770709), (12.4000000000000, -1.13073983297892), (12.5000000000000, 0.00173137316717853)]

Maple data

X =0, 0.100000000000000, 0.200000000000000, 0.300000000000000, 0.400000000000000, 0.500000000000000, 0.600000000000000, 0.700000000000000, 0.800000000000000, 0.900000000000000, 1.00000000000000, 1.10000000000000, 1.20000000000000, 1.30000000000000, 1.40000000000000, 1.50000000000000, 1.60000000000000, 1.70000000000000, 1.80000000000000, 1.90000000000000, 2.00000000000000, 2.10000000000000, 2.20000000000000, 2.30000000000000, 2.40000000000000, 2.50000000000000, 2.60000000000000, 2.70000000000000, 2.80000000000000, 2.90000000000000, 3.00000000000000, 3.10000000000000, 3.20000000000000, 3.30000000000000, 3.40000000000000, 3.50000000000000, 3.60000000000000, 3.70000000000000, 3.80000000000000, 3.90000000000000, 4.00000000000000, 4.10000000000000, 4.20000000000000, 4.30000000000000, 4.40000000000000, 4.50000000000000, 4.60000000000000, 4.70000000000000, 4.80000000000000, 4.90000000000000, 5.00000000000000, 5.10000000000000, 5.20000000000000, 5.30000000000000, 5.40000000000000, 5.50000000000000, 5.60000000000000, 5.70000000000000, 5.80000000000000, 5.90000000000000, 5.99999999999999, 6.09999999999999, 6.19999999999999, 6.29999999999999, 6.39999999999999, 6.49999999999999, 6.59999999999999, 6.69999999999999, 6.79999999999999, 6.89999999999999, 6.99999999999999, 7.09999999999999, 7.19999999999999, 7.29999999999999, 7.39999999999999, 7.49999999999999, 7.59999999999999, 7.69999999999999, 7.79999999999999, 7.89999999999999, 7.99999999999999, 8.09999999999999, 8.19999999999999, 8.29999999999999, 8.39999999999999, 8.49999999999999, 8.59999999999999, 8.69999999999999, 8.79999999999998, 8.89999999999998, 8.99999999999998, 9.09999999999998, 9.19999999999998, 9.29999999999998, 9.39999999999998, 9.49999999999998, 9.59999999999998, 9.69999999999998, 9.79999999999998, 9.89999999999998, 9.99999999999998, 10.1000000000000, 10.2000000000000, 10.3000000000000, 10.4000000000000, 10.5000000000000, 10.6000000000000, 10.7000000000000, 10.8000000000000, 10.9000000000000, 11.0000000000000, 11.1000000000000, 11.2000000000000, 11.3000000000000, 11.4000000000000, 11.5000000000000, 11.6000000000000, 11.7000000000000, 11.8000000000000, 11.9000000000000, 12.0000000000000, 12.1000000000000, 12.2000000000000, 12.3000000000000, 12.4000000000000, 12.5000000000000 Y=0.238726557599306, -0.00106430080664327, 0.552415997359500, -0.490860805333598, 0.344921255536726, 0.218712318034386, 0.226865872313974, 0.227304368920393, 0.0215306181333410, -0.304304570611808, 0.133203673484065, 0.109875578432012, 0.295359224807616, 0.464391414788198, 0.931476586336045, 0.815129683376910, 0.392269919166651, 0.266317154908948, 0.759173470403515, 0.02961921410618, 0.818988541770028, 1.33058170479361, -0.0106304924967900, 0.794277338584339, 1.06834524752293, 0.429875503997622, 1.30696381521764, 0.804957379259014, 0.783441034361311, 1.39562029919050, 1.16102415014942, 1.59187717915691, 0.729730642366330, 1.17791618373868, 0.390608741273900, 1.52226464376289, 0.923784476889042, 0.829670766129800, 0.932669307546741, 1.14234598321232, 0.500710697558121, 0.592618043593119, 1.15676394341977, 0.819356114742084, 0.807804693513561, 1.55933385330901, 0.901606542122742, 1.00984786195597, 0.679324065592616, 0.02068771838068, 0.977818575096133, 1.03475205238557, -0.103910649907259, 0.981842104139987, 0.422688552202437, 0.00766764362708416, 0.487608273270939, 0.433274732392297, 0.298650410459409, 0.862757806937851, 0.192085805686460, 0.837009672553558, 0.106075673880802, 0.597851605513888, 0.0434199164319980, 0.289840209720640, 0.220846924939167, 0.0306726118665809, 0.115690519740589, -0.388870016582899, -0.645554176764036, -0.282525705913652,-0.178591270193534, -0.0602052584824280, -0.478067215534890, -0.763129810139659, 0.124196545985498, -0.678370540558398, -0.628233276573054, -1.26013377006657, -0.200628135487097, -0.836921702115714, -1.06529203565071, -1.52409284914189, -1.10126537644158, -0.513427445098486, -0.295219145574798, -0.175066810272216, -1.35972529465098, -0.535484283326289, -0.804176144790150, -1.28894765058573, -1.70551308457956, -0.871144616678133, -0.787299855514117, -0.702233592726486, -1.34812882361573, -1.34099056778808, -0.789621139608874, -1.05201289151652, -0.221165280189115, -0.548981192771710, -1.19655932671510, -1.33198673618446, -1.00288818453729, -1.50467980466325, -1.39419963000100, -0.691338104555618, -1.87675235413828, -0.0714228791604897, -1.00698026996393, -0.597026152328669, -0.913991269764388, -0.985580462035566, -0.227817034172203, -0.171811536357626, -0.347117378286799, -0.497875034993729, -0.165172149433396, -0.250483410204926, 0.120949717574794, -1.00760224590490, -0.383531331169237, -0.319299262770709, -1.13073983297892, 0.00173137316717853

Mathematica data

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<Sol>

(a) >with(plots): >data:=[주어진 값을 벡터 형식으로 입력] >with(Statistics): >X:=Vector([x값에 해당하는 것을 벡터 형태로],datatype=float) >Y:=Vector([y값에 해당하는 것을 벡터 형태로],datatype=float)