Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 133

the automaton remains inactive, producing the value 0. Because only the rightmost dig-
its of the numbers count, for any value ofyand any intervalI⊂[ 0 , 1 ], one can find a
numberx∈Isuch thatf(x)=y. Hence the functionfmaps any subinterval of[ 0 , 1 ]
onto[ 0 , 1 ]. It satisfies the intermediate value property trivially. And because any neigh-
borhood of a point is mapped to the entire interval[ 0 , 1 ], the function is discontinuous
everywhere. 

As the poet Paul Valéry said: “a dangerous state is to think that you understand.’’ To
make sure that youdounderstand the intermediate value property, solve the following
problems.

394.Letf:[a, b]→[a, b]be a continuous function. Prove thatfhas a fixed point.

395.One day, a Buddhist monk climbed from the valley to the temple up on the mountain.
The next day, the monk came down, on the same trail and during the same time
interval. Prove that there is a point on the trail that the monk reached at precisely
the same moment of time on the two days.

396.Letf:R→Rbe a continuous decreasing function. Prove that the system

x=f(y),

y=f(z),

z=f(x)

has a unique solution.

397.Letf:R→Rbe a continuous function such that|f(x)−f(y)|≥|x−y|for all
x, y∈ R.Prove that the range offis all ofR.

398.A cross-country runner runs a six-mile course in 30 minutes. Prove that somewhere
along the course the runner ran a mile in exactly 5 minutes.

399.LetAandBbe two cities connected by two different roads. Suppose that two cars
can travel fromAtoBon different roads keeping a distance that does not exceed
one mile between them. Is it possible for the cars to travel the first one fromAto
Band the second one fromBtoAin such a way that the distance between them is
always greater than one mile?

400.Let

P(x)=

∑n

k= 1

akxk and Q(x)=

∑n

k= 1

ak

2 k− 1

xk,

wherea 1 ,a 2 ,...,anare real numbers,n≥1. Show that if 1 and 2n+^1 are zeros of

the polynomialQ(x), thenP(x)has a positive zero less than 2n.