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Difference between revisions of "PUMAC 2008-2009 Number Theory A problems"

(New page: 1. (2 points) How many zeros are there at the end of 792! when written in base 10? 2. (3 points) Find all integral solutions to <math>x^y-y^x=1</math>. 3. Find the largest integer <math>...)
 
 
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2. (3 points) Find all integral solutions to <math>x^y-y^x=1</math>.
 
2. (3 points) Find all integral solutions to <math>x^y-y^x=1</math>.
  
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3. Find the largest integer <math>n</math>, where <math>2009^n</math> divides <math>2008^(2009^(2010))+2010^(2009^(2008))</math>.
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3. (3 points) Find the largest integer <math>n</math>, where <math>2009^n</math> divides <math>2008^{2009^{2010}}+2010^{2009^{2008}}</math>.
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4. (3 points) <math>f(n)</math> is the sum of all integers less than <math>n</math> and relatively prime to <math>n</math>. Find all integers <math>n</math> such that there exist integers <math>k</math> and <math>l</math> such that <math>f(n^k)=n^l</math>.
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5. (4 points) If <math>f(x)=x^{x^{x^{x}}}</math>, find the last two digits of <math>f(17)+f(18)+f(19)+f(20)</math>.
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6. (4 points) What is the largest integer which cannot be expressed as <math>2008x+2009y+2010z</math> for some positive integers <math>x</math>, <math>y</math>, and <math>z</math>?
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7. (5 points) Find the smallest positive integer <math>n</math> such that <math>32^n=167x+2</math> for some integer <math>x</math>.
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8. (5 points) Find all sets of three primes <math>p</math>, <math>q</math>, and <math>r</math> such that <math>p+q=r</math> and <math>(r-p)(q-p)-27p</math> is a perfect square.
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9. (7 points) Find the number of positive integer solutions of <math>(x^2+2)(y^2+3)(z^2+4)=60xyz</math>.
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10. (7 points) What is the smallest number <math>n</math> such that you can choose <math>n</math> distinct odd integers <math>a_1, a_2,...a_n</math>, none of them 1, with <math>\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_n}=1</math>?

Latest revision as of 22:39, 4 February 2009

1. (2 points) How many zeros are there at the end of 792! when written in base 10?

2. (3 points) Find all integral solutions to $x^y-y^x=1$.

3. (3 points) Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}}+2010^{2009^{2008}}$.

4. (3 points) $f(n)$ is the sum of all integers less than $n$ and relatively prime to $n$. Find all integers $n$ such that there exist integers $k$ and $l$ such that $f(n^k)=n^l$.

5. (4 points) If $f(x)=x^{x^{x^{x}}}$, find the last two digits of $f(17)+f(18)+f(19)+f(20)$.

6. (4 points) What is the largest integer which cannot be expressed as $2008x+2009y+2010z$ for some positive integers $x$, $y$, and $z$?

7. (5 points) Find the smallest positive integer $n$ such that $32^n=167x+2$ for some integer $x$.

8. (5 points) Find all sets of three primes $p$, $q$, and $r$ such that $p+q=r$ and $(r-p)(q-p)-27p$ is a perfect square.

9. (7 points) Find the number of positive integer solutions of $(x^2+2)(y^2+3)(z^2+4)=60xyz$.

10. (7 points) What is the smallest number $n$ such that you can choose $n$ distinct odd integers $a_1, a_2,...a_n$, none of them 1, with $\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_n}=1$?

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