youngestgroup
Sunday,23September2007
1. Two hundred ontestants took part in Mathemati al Olympiad. They were asked
to solve 6 problems. It is known, that ea h problem has been solved by at least 120
ontestants.Showthattherearesu htwo ontestants,thateveryproblemhasbeensolved
by atleast one of them.
2.Howmany(atmost)disjointsquares22man anpla esonthesquare hessboard
nn in su h a way that the sides of the squares were situated along the lines dividing
grids?
3.Positivenumbers a
1
;a
2
;:::;a
n
sum up to 1.Showthat an inequality holds:
a
1 a
2 +a
2 a
3
+:::+a
n 1 a
n
¬ 1
4
4.Asquare ABCDisgiven.PointsE andF layonthesidesAB andBC respe tively
and BE =BF.PointS isaperpendi ularproje tionof pointB ontolineEC.Showthat
an angle DSF is right.
youngergroup
Sunday,23September2007
1. Two hundred ontestants took part in Mathemati al Olympiad. They were asked
to solve 6 problems. It is known, that ea h problem has been solved by at least 120
ontestants.Showthattherearesu htwo ontestants,thateveryproblemhasbeensolved
by atleast one of them.
3.Positivenumbers a
1
;a
2
;:::;a
n
sum up to 1.Showthat an inequality holds:
a
1 a
2 +a
2 a
3
+:::+a
n 1 a
n
¬ 1
4
4.Asquare ABCDisgiven.PointsE andF layonthesidesAB andBC respe tively
and BE =BF.PointS isaperpendi ularproje tionof pointB ontolineEC.Showthat
an angle DSF is right.
6.FindthenumberofsubsetsM ofthesetf1;2;:::;36gsatisfyingfollowing ondition:
For every two distin t x;y2M also jx yj2M.
oldergroup
Sunday,23September2007
1. Two hundred ontestants took part in Mathemati al Olympiad. They were asked
to solve 6 problems. It is known, that ea h problem has been solved by at least 120
ontestants.Showthattherearesu htwo ontestants,thateveryproblemhasbeensolved
by atleast one of them.
3.Positivenumbers a
1
;a
2
;:::;a
n
sum up to 1.Showthat an inequality holds:
a
1 a
2 +a
2 a
3
+:::+a
n 1 a
n
¬ 1
4
5. In a ute-angled triangle ABC bise tor of
∠
ABC interse ts side AC in point D.PointsEandF areperpendi ularproje tionsofpointsAandC ontolineBDrespe tively.
Point M is a perpendi ular proje tion of point D onto line BC. Show that
∠
EMD =∠
FMD.6.FindthenumberofsubsetsM ofthesetf1;2;:::;36gsatisfyingfollowing ondition:
For every two distin t x;y2M also jx yj2M.
oldestgroup
Sunday,23September2007
5. In a ute-angled triangle ABC bise tor of
∠
ABC interse ts side AC in point D.PointsEandF areperpendi ularproje tionsofpointsAandC ontolineBDrespe tively.
Point M is a perpendi ular proje tion of point D onto line BC. Show that
∠
EMD =∠
FMD.6.FindthenumberofsubsetsM ofthesetf1;2;:::;36gsatisfyingfollowing ondition:
For every two distin t x;y2M also jx yj2M.
7.Show that forpositivea;b; ;d;e an inequality holds:
a
e+a+b +
b
a+b+ +
b+ +d +
d
+d+e +
e
d+e+a
¬2
8.Showthat therearetwosquaresof onse utivepositiveintegers,su hthat between
them there are at least 2007 prime numbers.
