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# The Tour De Clochemerle

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Challenge Level

The Tour de Clochemerle is not yet as big as the rival Tour de France.

This year there were five riders, Arouet, Barthes, Camus, Diderot and Eluard, who took part in five stages. The winner of each stage got 5 points, the runner up 4 points and so on down to the last rider who got 1 point. The total number of points acquired over the five stages was the rider's score. Each rider obtained a different score overall and the riders finished the whole tour in alphabetical order with Arouet gaining a magnificent 24 points. Camus showed consistency by gaining the same position in four of the five stages and Eluard's rather dismal performance was relieved by a third place in the fourth stage and first place in the final stage.

Explain why Eluard must have received 11 points in all and find the scores obtained by Barthes, Camus and Diderot.

Where did Barthes come in the final stage?

*Clochemerle is a French satirical novel by Gabriel Chevallier (1895–1969) first published in 1934. It centres on personal rivalries and local politics in the fictional village of Clochemerle and concerns a dispute over the construction of a vespasienne (public urinals) near the village church. The term Clochemerle has entered French as a term to describe "petty, parochial
squabbling".*

*STEP Mathematics II, 1995, Q3. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.*

This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.

Can you find a three digit number which is equal to the sum of the hundreds digit, the square of the tens digit and the cube of the units digit?

How many numbers are there less than $n$ which have no common factors with $n$?