Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely. Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property: For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.

On February 13 [i]The Oshkosh Northwester[/i] listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57 \textsc{am}$, and the sunset as $8:15 \textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? $\textbf{(A)}\hspace{.05in}5:10 \textsc{pm} \quad \textbf{(B)}\hspace{.05in}5:21 \textsc{pm} \quad \textbf{(C)}\hspace{.05in}5:41\textsc{pm} \quad \textbf{(D)}\hspace{.05in}5:57 \textsc{pm} \quad \textbf{(E)}\hspace{.05in}6:03 \textsc{pm} $

A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

A train passes an observer in $t_1$ sec. At the same speed the train crosses a bridge $\ell$ m long. It takes the train $t_2$ sec to cross the bridge from the moment the locomotive drives onto the bridge until the last car leaves it. Find the length and speed of the train.

Worms grow at the rate of $1$ metre per hour. When they reach their maximal length of $1$ metre, they stop growing. A full-grown worm may be dissected into two not necessarily equal parts. Each new worm grows at the rate of $1$ metre per hour. Starting with $1$ full-grown worm, can one obtain $10$ full-grown worms in less than $1$ hour?

It takes a steamer $5$ days to go from Gorky to Astrakhan downstream the Volga river and $7$ days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?

Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to $2$ minutes? (S . Fomin , Leningrad)

When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?

A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour? (NN Konstantinov, Moscow)

In a country, one can get from some point $A$ to any other point either by walking, or by calling a cab, waiting for it, and then being driven. Every citizen always chooses the method of transportation that requires the least time. It turns out that the distances and the traveling times are as follows: $1$ km takes $10$ min, $2$ km takes $15$ min, $3$ km takes $17.5 $ min. We assume that the speeds of the pedestrian and the cab, and the time spent waiting for cabs, are all constants. How long does it take to reach a point which is $6$ km from $A$?

Two men, $A$ and $B$, set out from town $M$ to town $N$, which is $15$ km away. Their walking speed is $6$ km/hr. They also have a bicycle which they can ride at $15$ km/hr. Both $A$ and $B$ start simultaneously, $A$ walking and $B$ riding a bicycle until $B$ meets a pedestrian girl, $C$, going from $N$ to $M$. Then $B$ lends his bicycle to $C$ and proceeds on foot; $C$ rides the bicycle until she meets $A$ and gives $A$ the bicycle which $A$ rides until he reaches $N$. The speed of $C$ is the same as that of $A$ and $B$. The time spent by $A$ and $B$ on their trip is measured from the moment they started from $M$ until the arrival of the last of them at $N$. a) When should the girl $C$ leave $N$ for $A$ and $B$ to arrive simultaneously in $N$? b) When should $C$ leave $N$ to minimize this time?

One day I went for a walk in the morning at $x$ minutes past $5'O$ clock, where $x$ is a 2 digit number. When I returned, it was $y$ minutes past $6'O$ clock, and I noticed that (i) I walked for exactly $x$ minutes and (ii) $y$ was a 2 digit number obtained by reversing the digits of $x$. How many minutes did I walk?

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?