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MATHPROBLEMS ON A BUDGET

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1 The hexagon ABCDEF is in a circlewhere AF=31, and the other sides have the lenght of 81. Calculate the sum of the lenghts in the hexagons diagonals that starts from A.

  1. Leta,b,c be lenghts of the sides in a triangle, show that:
    (a+b-c)(a-b+c)(-a+b+c) ≤ abc.
    Whendoesequalityoccur?

3.Ten kids went to pluckmushrooms in the woods. Nobodycamehomewithnothing, and all toghetertheyhadcollected 54 mushrooms. Show that at leasttwo of the kids picked the same amount of mushrooms.

4.A box in a corner in a 3×3 grid is painted black, the otherboxesarewhite. In onemove, youareallowed to switch color on all boxes in a row or column. Canyouafter anumber of suchmoves make all boxes black?

5.Little Aaron har removedseven pages of a notebook. Can the sum of the removed pages sidenumbers be equal to 300?

6.Inside a 5x5x5 cube, 124 pointsare chosen. Show that inside thiscube, youcanplace a 1x1x1 cubethatwon’tcontainany of the chosen points (none of the pointsare inside the smallercube. It can be placedanywhere, even on the slope).

7.Peter’s squarepatio is coveredwithconcrete-plates. Some of the plateshave the shape of 22 bricks, and somehaveshape of 14 bricks. One of the plates is broken, but as a replacement Peter onlyhave a singleplate of the other kind. Canhereplace the broken platewith a the onehe has by possiblyrearranging the plates on the patio?

8.Show that the number A=(5^99)+(11^99)+(17^99) can be dividedwith 33

9.Show that 7 is a divider to 2222^5555 + 5555^2222

10.Which rest do you get when 9^2014 divideswith 11?

11.When Anna splits the candypieces from herbigeaster egg in piles of 4 pieces, she gets 3 left over. Whenshe splits it in piles of 5 pieces, she gets 4 left over. Whenshe splits it in piles of 6 pieces, she gets 5 left over. Howmanypiecesarethere in the easter egg ifweknowthatits less than 100?

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