1.Showthatforall a;npositive integers:
nj n 1
X
i=0 a
(i;n)
where(k;l) denotesthegreatest ommondivisor ofk and l.
2. Let n be a positive integer. For k =1;2;:::;n we de ne a
k
= 1
( n
k )
and b
k
= 2 k n
. Show that
equalityholds:
a
1 b
1
1 +
a
2 b
2
2
+:::+ a
n b
n
n
=0
3.Showthatif a;b; are positive,thenan inequalityholds:
a
p
ab+b 2
+ b
p
b + 2
+
p
a+a 2
3
p
2
4.Given is square hessboard 1919. Stri tlyinside this hessboardwe spe ifyasmaller square
hessboard 17 17. Determine, if the smaller hessboard an be overed using disjoint horizontal
re tangles 51 andverti al14pla edinside thebigger hessboard.
5.Findallfours of integersx;y;z;t satisfyingasystem ofequations:
(
xz 2yt=3
xt+yz=1
6.Show,that there existssu h an integer k,thatk2 n
+1is omposite forevery n1.
7.LetP(x);Q(x);R (x);S(x) besu hreal polynomials,thatforevery real x equation holds:
P(x 5
)+xQ(x 5
)+x 2
R (x 5
)=(x 4
+x 3
+x 2
+x+1)S(x)
Show,that P(1)=0.
8. Point P lies inside a triangle ABC.Let D;E;F be perpendi ular proje tions of P onto lines
BC ;CA;AB. Let O be the ir um entre of triangle DEF, and r be radius of ir um ir le of this
triangle. Showthat:
area(ABC)3r q
3r 2
3jOPj 2
9.A ir le ois given and two trianglesins ribedinto it. The ommon part of themis a hexagon.
Showthat maindiagonalsofthishexagon interse tin one point.
10. Median AM of triangle ABC, in whi h AB 6=AC, interse ts ins ribed ir le ! in points K
and L. Linesparallel to BC throughK and L respe tivelyinterse t ! on emore in X and Y.Lines
AX and AY interse t sideBC inpointsP andQ respe tively.Show, thatBP =CQ.
11.Ontheplanethere isgivenadiskwithradius1.It hasbeen overed withasquarewithaside
of length 2. The square has been divided into 2007 parallel stripes and one has been thrown away.
Showthat thedisk annot be overed usingremainingstripes.
