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2000 Italy TST

2000 Italian TST

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yurtman
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2K views1 page

2000 Italy TST

2000 Italian TST

Uploaded by

yurtman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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AoPS Community 2000 Italy TST

Italy TST 2000


www.artofproblemsolving.com/community/c5503
by WakeUp, outback

1 Determine all triples (x, y, z) of positive integers such that


13 1996 z
+ 2 =
x2 y 1997

2 Let ABC be an isosceles right triangle and M be the midpoint of its hypotenuse AB. Points
D and E are taken on the legs AC and BC respectively such that AD = 2DC and BE = 2EC.
Lines AE and DM intersect at F . Show that F C bisects the ∠DF E.

3 Given positive numbers a1 and b1 , consider the sequences defined by


1 1
an+1 = an + , bn+1 = bn + (n ≥ 1)
bn an
√
Prove that a25 + b25 ≥ 10 2.

4 On a mathematical competition n problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored 7 points;
(ii) for each pair of problems, exactly one contestant scored 7 points on both problems.
Prove that if n ≥ 8, then there is a contestant who got 7 points on each problem. Is this state-
ment necessarily true if n = 7?

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