Maple Treasure Hunt (a) What is the first nonzero digit after the decimal point in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW5HRiQ2JVEiMkYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSZtZnJhY0dGJDYoLUYvNiVRIjFGJ0YyRjUtRiM2Ji1GLzYlUSUyMDEwRidGMkY1RjIvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dGPy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZOLyUpYmV2ZWxsZWRHRjRGMkZFRkgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUlc2l6ZUdRIzEyRidGMkY1 ? evalf(2^(1/2010));

JCIrNFxNKzUhIio=

The answer is 3. (b) What is the 2010th digit after the decimal point in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW5HRiQ2JVEiMkYnLyUlc2l6ZUdRIzE4RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjpGPUY6Rj1GK0Y6Rj0= ? You need to use 2011 digits, since one is before the decimal point. evalf(sqrt(2), 2011);

JCJmaHIjWzZcVyQzZ2pDZ1t5djVaa15lbmFfSExbTCZlTW46aCczUCQqMztzcGo0NnQ/SHBPeFxVIVxuPUBfc2hkeU0yKSk9I2YsKyc+VlB5WVpPUjx3R1pIUUspM3VAS15lMWpVQGVaUkFQQCxuKip6J2VkUyZvIVswc0xqOmQrJVxReiQ+PyY9MUwpenBKNVcnPlFjdiNcdURta1FYUiopR2EnNENsNEVYJClHZXYvKlFEXCZbIiopb0AmSCdceiVwTlpNWEdLUiopPndkXyRlbSo+RG1ceHB5aiZHIylHblEmKikqKSpIJCkqbzF1N0IoUl5YPip6UXhARTlvYHh4LiU+VSpmXEZ5aGpMRWcpeUQpZkA8bU92KSpSI0huIz5seid6bHlYTyUpUkxCYGMhNGxVcVM/OT1Cczk9JjNkTTNTJ1skZiIpKTRhPCopcF9pPEBsIyoqZlJlOnInM0FpVUZbaEFqUXNqTWZEPFpmUnZERW5KRj1Aa2w/bDRWaTwtKT0qKipROXVcJkdpdnJkIipccUomW0hPVl9nPzgiUUJIOVIpMy9ebWQ6JSpROUpLKClvLicqbyQpNFhzVztdYl5ma2ZvbyZwZWwrVkAqUUh3JHlHX1NveUImPSlvPHAkcDoreU07KT0mUUpjI0gnKT1gXDwwSVMtWnBSbC11NzwhbzlqXytjKEciUWIvOk4sXjooM0NnIkg2dmglcFJLbG5BYVNeal9mdyMpR2MiKkcnUWolcCozcyUpWz0mSCNHQDojZTsoZSg+JSopb3MiZTtSbzY2PDVaSnBHPTR0MnJMZy5lNXk5YEc8TVQlXC03ImZdQCUpZlcrIj1fIWZbN3AjSDFVdUMhb0shPSgqKkdQI1tsdi9RVHAqSHdnIUhaTyp6RDJYbyRvKXAkPVNdbiozIlFkXCFHRk9kRjYlcEEreEEiNFgtL0F0NmNfJilSUEoiKW9qc10vVDFIeXMqWz5VLlsnPS5TcWVbZissaiMzZCJva0MpUTYuIT1mRTRQV1IqKSlSZTlfbVdlJjMtZ0FkRSNcWmUyaSwpZWUsWmFbRS1zYHducHFsYlQoWyZlVnIyc0NvOWdnbyZHL1VMPVU6eFxcNydbRTQmKUdOc0IoKSpvJj1jLDhJcm9tZjZKWXkwZUAnPXAkSFxvSSN5Q2FrSiF6RlNhSV0iWysqcHEqKipvUWdyJSkpXDNNZ3VWTjZxJj1gUC5BdmIkKUg8JypmeEddZHVhXVJfJGVfaD8xa0gpMz1MWWxSYHNvKil6U15fKnppLkV3UjNYQFZyM1ZbSSNIQmI0Jyk+Kikzaz11XkdaLGlvZk1YRnFmZ3IqZiY0LFpyMDEjeV82XV0qKnpiRmYwdiNmbUFLVEplJCoqNChcQFRrN3N0XWUwT1sjXC0oSDc0QllRLE5GZFR3S012UV0pUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIiElNT8=

The answer is 2. (c) What is the 2010'th prime number? It may take some digging, but searching for "prime" in Text will eventually find you the proper command: ithprime. ithprime(2010);

IiZ4dSI=

The answer is 17477. (d)Find integers a and b such that 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 expand((a+b*sqrt(2010))^2);

LCgqJEkiYUc2IiIiIyIiIiooRiRGJ0kiYkdGJUYnIiU1PyNGJ0YmRiYqJEYpRiZGKg==

Evidently we want 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 and 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. solve({a^2 + 2010*b^2 = 73321, 2*a*b = -372});

NiU8JC9JImFHNiIiI0ovSSJiR0YmISInPCQvRiUhI0ovRikiIic8JC9GJSwkLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiQsJiokSSNfWkdGNSIiIyIlNT8hIiIiIiIvSSZsYWJlbEdGNUkkX0wzR0YmISZnPyIvRiksJEYzRic=

We could take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY8LUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIzMxRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRidGNS1GLDYlUSJiRidGL0YyRjVGT0Y1LUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZgby1GVjYkUSI2RidGOUY1LUY2Ni9RI29yRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZbcEY7Rj5GQEZCRkRGRkZIRkpGTUY1RitGT0Zcb0ZVRllGNUZpbkZPLUZWNiRRIzYuRidGOUY5 (e) What is the first digit of the 2010'th Bernoulli number? More digging in the help system will find the bernoulli command. bernoulli(2010);

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

It's not an integer: note the /LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEpMjg4MDE1NDJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHRi8=at the end. evalf(%);

JCIrNUNjekoiJWJU

The first digit is 3. (f) Find the limit of 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 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJnNyYXJyO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Ri9GPA==. Hint: the exponential function in Maple is exp. limit(x^3/(exp(x) - exp(sin(x))), x = 0);

IiIn

The answer is 6.