Difference between revisions of "1977 Canadian MO Problems"

(Problem 7)
(Problem 7)
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twice. Show that the number <math>f(m,n)</math> of different paths  she can take to work satisfies <math>f(m,n)\le 2^{mn}</math>.
 
twice. Show that the number <math>f(m,n)</math> of different paths  she can take to work satisfies <math>f(m,n)\le 2^{mn}</math>.
  
<math>\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c| }
+
<math> \underbrace{\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c| }
 
     \hline
 
     \hline
 
     &&&&&&&&&& \\ \hline
 
     &&&&&&&&&& \\ \hline
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     &&&&&&&&&& \\ \hline
 
     &&&&&&&&&& \\ \hline
 
     &&&&&&&&&& \\ \hline
 
     &&&&&&&&&& \\ \hline
\end{tabular}</math>
+
\end{tabular}}_{m}}</math>
  
 
[[1977 Canadian MO Problems/Problem 7 | Solution]]
 
[[1977 Canadian MO Problems/Problem 7 | Solution]]

Revision as of 02:27, 7 October 2014

The seven problems were all on the same day.

Problem 1

If $\displaystyle f(x)=x^2+x,$ prove that the equation $\displaystyle 4f(a)=f(b)$ has no solutions in positive integers $\displaystyle a$ and $\displaystyle b.$

Solution

Problem 2

Let $\displaystyle O$ be the center of a circle and $\displaystyle A$ be a fixed interior point of the circle different from $\displaystyle O.$ Determine all points $\displaystyle P$ on the circumference of the circle such that the angle $\displaystyle OPA$ is a maximum.

CanadianMO-1977-2.jpg


Solution

Problem 3

$\displaystyle N$ is an integer whose representation in base $\displaystyle b$ is $\displaystyle 777.$ Find the smallest positive integer $\displaystyle b$ for which $\displaystyle N$ is the fourth power of an integer.

Solution

Problem 4

Let \[p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0\] and \[q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots +b_1x+b_0\] be two polynomials with integer coefficients. Suppose that all of the coefficients of the product $p(x)\cdot q(x)$ are even, but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.


Solution

Problem 5

Solution

Problem 6

Let $0<u<1$ and define \[u_1=1+u\quad ,\quad u_2=\frac{1}{u_1}+u\quad  \ldots\quad  u_{n+1}=\frac{1}{u_n}+u\quad ,\quad n\ge 1\] Show that $u_n>1$ for all values of $n=1,2,3\ldots$.

Solution

Problem 7

A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives on the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n)\le 2^{mn}$.

$\underbrace{\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c| }

    \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline
    &&&&&&&&&& \\ \hline

\end{tabular}}_{m}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

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