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Stage 6 · Number theory

10 problems · National olympiad, first round · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.

   Determine the maximum integer $n$ with the property that for each positive integer $k \leq \frac{n}{2}$ there exist two positive divisors of $n$ with difference $k$.

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2. An integer consists of 7 different digits, and is a multiple of each of its digits.
   
   What digits are in this nubmer?

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3. Find the number of rational solutions of the following equations (i.e., rational $x$ and $y$ satisfy the equations)
   $$x^2+y^2=2$$$$x^2+y^2=3$$

   1. A$~2\text{}$ and }2$
      $
   2. B$~2\text{}$ and }0$
      $
   3. C$~2\text{}$ and infinitely many}$
      $
   4. D$~\text{Infinitely many and }0$

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4. $S\subset\mathbb N$ is called a square set, iff for each $x,y\in S$, $\text{xy 1}$ is square of an integer.
   a) Is $S$ finite?
   b) Find maximum number of elements of $S$.

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5. $p, q, r$ are distinct prime numbers which satisfy
   $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$
   for natural number $A$. Find all values of $A$.

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6. Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and
   $$\frac{a+b}{a+c}=\frac{b+c}{b+a}.$$

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7. Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers.
   
   Valentin Vornicu, Romania

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8. Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.

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9. Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.

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10. Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $kp+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.

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Answer key — Stage 6 · Number theory

Worked solutions for every problem are on the site, one page per problem.

1. $not found$ open
2. $1, 2, 3, 6, 7, 8, 9$ open
3. $\text{(B)}~2\text{ and }0$ open
4. $3$ open
5. $1980$ open
6. $(1, 1, 1)$ open
7. $\left(\frac{1}{2}, \frac{1}{2}, 4\right), \left(\frac{1}{2}, 1, 2\right), (1, 1, 1)$ open
8. $168$ open
9. $375$ open
10. $p-2$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.