Klarar du mattekluringarna?

Victor Ufnarovski, professor vid Matematik LTH och numerisk analys, låter oss så här i semestertidernas slut ta del av sju matematiska problem.

Mattekluringarna är på engelska, och svaren presenterar vi i början av september på appen LTH Now.

Misströsta inte om matematikkunskaperna tryter eller om hjärnan inte ännu är i bästa form:

- Det här ska vara en utmaning, säger Victor Ufnarovski.

1. 

A mouse eats his way through a 3 X 3 cube of cheese by tunneling through all of the 27 
1 X 1 X 1 sub-cubes. If he starts at one of the corner sub-cubes and always moves onto an uneaten adjacent sub-cube can he finish at the centre of the cube? (Assume that he can tunnel through walls but not edges or corners.)

2. 

After elections every member of parliament (PM) has his own absolute rating. When the parliament is set up, he enters a group and gets a relative rating. The relative rating is the ratio of his own absolute rating to the sum of the absolute ratings of the PMs in the group. A PM can move from one group to another only if his relative rating in the new group is higher. In a given day, only one PM can change his group. Show that only a finite number of group moves is possible. (Remark: a rating is a positive real number.)

3. 

Prove that given any 2n-1 natural numbers it is possible to choose n of them such that their sum will be divisible by n.

4. 

A herd consists of 101 cows. Any 100 of them can be split into two groups of 50  cows each such that the weights of the two groups are equal. Prove that all the cows have the same weight.

5. 

A young man walks into a "7-eleven" store and asks for four items. The shop assistant tells him that his bill is 7,11 euros, since the product of the four prices is exactly 7,11. The young man explains indignantly that one is supposed to add together the four prices, not to multiply them. "Oh dear!", exclaims the shop assistant and sums the four numbers. But, can you imagine, the right sum turns out to be 7,11 too! How much did each item cost?

6. 

A polygon inscribed in a circle is triangulated by drawing non-intersecting diagonals. Prove that the sum of the radii of the circles inscribed in all resulting triangles does not depend on the choice of triangulating diagonals.

7. 

Solve the equation 8x(1-2x²)(8x⁴-8x²+1)=1.

Enkelt eller svårt? Här på LTH väntar i alla händelser en höst av lärande.

Tiina Meri

Här finns övningsmaterial i matematik:

http://www.maths.lth.se/matematiklth/personal/andersk/website/index.html

Här hittar du ledtrådar till lösningarna:

http://www.maths.lth.se/matematiklth/personal/ufn/Problempage/solhintsklur.pdf

Och här kan du, till slut, ta del av lösningar:

http://www.maths.lth.se/matematiklth/personal/ufn/Problempage/solklur.pdf