1977 Canadian MO Problems

The seven problems were all on the same day.

Problem 1

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


Problem 2

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



Problem 3

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


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.


Problem 5

A right circular cone of base radius $1$ cm and slant height of $3$ cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path?

[asy] path p1=yscale(.25)*arc((0,0),1,0,180); path p2=yscale(.25)*arc((0,0),1,0,-180); path q1=shift(-.25,.4)*rotate(30)*xscale(.85)*p1; path q2=shift(-.25,.4)*rotate(30)*xscale(.85)*p2; draw(p2,black);draw(q2,black); draw(p1,dashed);draw(q1,dashed); draw((-1,0)--(-.5,2.4)--(1,0)); MP("P",(-1,0),W);MP("V",(-.5,2.4),N); draw((-.2,2.5)--(1.2,.2),arrow=ArcArrow()); draw((1.2,.2)--(-.2,2.5),arrow=ArcArrow()); draw((0,0)--(1,0),arrow=ArcArrow()); draw((1,0)--(0,0),arrow=ArcArrow()); MP("1 cm",(.5,.04),S);MP("3 cm",(.5,1.35),NE); [/asy]


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 $n=1,2,3\ldots$.


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 $f(m,n)\le 2^{mn}$.

\[\underbrace{ \left. \begin{array}{|c|c|c|c|c|c|c|c|c|c|c| }      \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline      &&&&&&&&&& \\ \hline \end{array} \right\}n}_m\]