A puzzle of cycling

English version: Contrarreloj ciclista

The men's individual time trial of the UCI Road World Championship takes place today.

The trial participants start out one at a time

We know that among the first 30 participants, 15 are Americans and the other 15 are Europeans.

How many cyclists have to leave so that we make sure that there are 2 of the same continent racing the trial?

Simply 3 cyclists.

If the first 2 are from the same continent, either European or American, then they'll have fulfilled the condition of having 2 riders on the same continent performing the test.

And if they are from different continents, one European and one American, the third cyclist will necessarily coincide with one of the 2 first cyclists, since we know that in the first 30 participants there're no runners from other continents.

The only difficulty with this problem is to be aware of what is being asked. There are people who answer that we need 16 cyclists racing on the road, as they interpret the question the wrong way, and they believe that the question is: how many cyclists must be out to ensure that at least 2 cyclists from different continents are racing the trial?

 

So we conclude that if we want to answer the question correctly, we must pay a care attention about  which is exactly the question.

The basket of balls

Versión española: Contrarreloj ciclista

Before the final of the 2012 European Championship, the Spanish team is practicing penalty shots.

The assitants pick all the balls and put them into a basket beside the penalty spot.

Xabi Alonso is the first to train his shots, and he uses half of the balls in the basket plus one ball more.

Fernando Torres goes in second place, and he also uses half of the remaining balls into the basket, plus one ball more.

Afterwards, Cesc tries kicking, and he uses half of the reamining balls plus one.

Then, Iniesta does the same, and he also uses one half of the balls in the basket plus one.

Finally it's up to Sergio Ramos, but he can not kick any ball as there's no ball in the basket.

How many balls where there in the beginning?

To solve it the best possible way, we start counting from the last player to kick.

We know that Sergio Ramos found no ball in the basket, so Iniesta necessarily kicked 2 balls. That way he took half of the balls (2/2=1) plus 1.

We can calculate it as follows: we take the balls he leaves in the basket (0), we add 1, and multiply the result by two:
 

(0+1) · 2 = 2

When Fabregas takes half the balls remaining in the basket there must be the 2 balls Iniesta kicked later and the balls Cesc used. That is, there were 6 balls: he took half the balls (6/2=3) plus 1, and left the 2 balls Iniesta kicked. We can calculate it as in the previous case:
 

(2+1) · 2 = 6

Following the same scheme, Fernando Torres will find 14 balls: he picks half of them (7) plus one, and leaves 6 to Cesc. If we calculate backwards:
 

(6+1) · 2 = 14

And finally we come to Xabi Alonso. Following the reasoning, he will initially have:
 

(14+1) · 2 = 30 balls

And this is the correct answer. In the beginning, we got 30 balls in the basket!

= 30 balls