Semaine Européenne Concours de mathématiques Math Contest Les problèmes posés par chaque lycée

BIECZ

PUZZLE NO. 1
You are given 12 identical-looking weights, and an old-fashioned pair of balance scales. Either side of the scales is large enough to hold up to 12 weights. 
Now, eleven of the weights are indeed identical, but one is different from the rest - either heavier or lighter, but you do not know which. 
Your task is to use the balance scales to discover 
        (a)  which of the weights is the odd one out, 
               and 
        (b) whether it is heavier or lighter than the others. 
Your challenge is to discover this information in the smallest possible number of weighings. What is the minimum number of weighings required? Can you explain why? 

The answer from Montargis

PUZZLE NO. 2: WHERE IS NUMBER THERE IS BEAUTY

The answer from Montargis

TAMPERE

UPS AND DOWNS
Smith and Jones started working for different firms at the same salary.
Last year Smith had a raise of 10 per cent and Jones had a drop in pay of 10 per cent. This year Smith had the 10 per cent drop and Jones the 10 per cent raise. Who is making more now?

Three answers

LANCASTER

FARMER SMITH AND HIS SHEEP
Farmer Smith has two fields which are next to one another. In one field are the black sheep and in the other are the white sheep.
To help him remember how many sheep he has he knows that if he squares the number of black sheep and adds on 37 this will give him the square of the number of white sheep.
How many sheep of each colour does he have?
Can you find a quick way of working this out?

The answer from Montargis

RENDSBURG

In Deutschland werden in der Schule Noten von 1 bis 6 verteilt: Die beste Note ist die 1.
Ein Lehrer hat Arbeiten von drei Schülerinnen und Schülern durchgesehen und benotet, dann aber zu Hause vergessen. An die Noten erinnert er sich nur noch undeutlich. (Es handelt sich also sicher um einen Mathematik-Lehrer – die nämlich können mit Zahlen meist wenig anfangen!). Nun sagt der Lehrer den dreien, die Anne, Bernd und Detlef heißen (Achtung: typische deutsche Namen):
„Ihr habt alle unterschiedliche Noten erhalten: Ich habe die Noten 2, 3 und 4 vergeben. Detlef hat keine 4 und Bernd keine 3, aber ich glaube, dass Anne die 3 hat."
Zu Hause stellt der gute Mann dann fest, dass er nur einer Person eine richtige Auskunft gegeben hat, sich bei den beiden anderen aber geirrt hat. (Ganz sicher ein Mathe-Lehrer: Auch die Erinnerung an Zahlen fällt schwer!)
Welche Noten haben die Schülerin und die Schüler bekommen?

Nur wenn das zu schnell geht, gibt es noch eine Zusatzaufgabe, die auch in das Survival-booklet gehört:

Typischer Brief nach Hause. Jeder Buchstabe steht für eine andere Ziffer. Wie groß ist dann die Summe? Gibt es mehrere Lösungen?

Die offizielle Lösung

At German schools test papers of students are usually marked 1 to 6 ( 1 = ecellent, 6 = insufficient).
A teacher has corrected Anne’s, Bernd’s and Detlef’s (typically German names) papers but unfortunately left them at home. He is not quite sure which marks he gave them (must be a maths teacher because most of them do not know what to make of numbers), so he tells them:
"You have all received different marks; altogether I gave you mark 2, 3 and 4. Detlef has not got mark 4 and Bernd not mark 3, but I think Anne got mark 3.
Back at home he finds out that he had given the students one correct information only and that he had been wrong with the other two ( Typical of maths teachers again, they can’t remember figures)..
Which marks did Anne, Bernd and Detlef get?

Additional task only if the first one could be solved quickly. To be taken up into the Survival-booklet:

A typical letter home. Each letter stands for a different number. How large is the sum? Are there more solutions?

MONTARGIS

La traversée du désert
L'explorateur "VAPARTOU" désire traverser le désert "DURAPASSER" de la ville "PORTETOU" jusqu'à la ville "DURAATEINDRE". En préparant son voyage, il a appris qu'il faut 6 jours pour traverser ce désert. Il sait en plus qu'un homme peut porter jusqu'à 4 rations journalières. Il décide donc d'employer des porteurs, mais étant économe, il désire en employer le moins possible.
Sachant que l'explorateur va lui aussi porter des rations, et que les porteurs souhaitent revenir vivant du désert, quel est le nombre minimum d'hommes à employer ?

Crossing the desert
Explorer TRAVELS-EVERYWHERE wants to cross a desert called TOUGH-TO-CROSS starting from CARRY-EVERYTHING City to NOT-EASY-TO-REACH City.
While preparing his trip, TRAVELS-EVERYWHERE has realized that it takes six days to cross the desert. Moreover, TRAVELS- EVERYWHERE knows that a man can carry along with him up to 4 daily rations. That's why TRAVELS-EVERYWHERE has decided to hire bearers but being thrifty, TRAVELS-EVERYWHERE intends to hire as few bearers as possible.
Given the fact that 'TRAVELS-EVERYWHERE' will be carrying some rations itself and that the bearers want to come back alive, what is the minimum number of bearers that will have to be hired?

The answer from Lancaster
La réponse officielle - the official answer

PLOIESTI

Jean is telling John:
"In 12 years time, my son's age will be a third of my age.
In 2 years time, my son's age will be 13 times less than mine.
Could you, please, tell me where my wife is right now?"

The answer from Montargis