Paper, Order, or Assignment Requirements
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.
- Leta,b,c be lenghts of the sides in a triangle, show that:
(a+b-c)(a-b+c)(-a+b+c) ≤ abc.
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?