CMO 1977

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

Let $$p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0$$ and $$q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0$$ be two polynomials with integer coefficients. Suppose that all 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.

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

import graph;

unitsize(1 cm);

filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen);
draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed);
draw(yscale(0.3)*(arc((0,0),1.5,180,360)));
draw((1.5,0)--(0,4)--(-1.5,0));
draw((0,0)--(1.5,0),Arrows);
draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360)));

label(" $V$ ", (0,4), N);
label("1 cm", (0.75,-0.5), N);
label(" $P$ ", (-1.5,0), SW);
label("3 cm", (1.7,2));