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2001 Italy TST

The document discusses 4 math problems: 1) proving two angles are equal given properties of a quadrilateral, 2) proving an inequality involving 3 variables, 3) finding pairs of prime numbers with related properties, 4) determining the minimum number of balloons that can be adjusted each step to eventually equalize 2001 balloons.

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yurtman
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17K views1 page

2001 Italy TST

The document discusses 4 math problems: 1) proving two angles are equal given properties of a quadrilateral, 2) proving an inequality involving 3 variables, 3) finding pairs of prime numbers with related properties, 4) determining the minimum number of balloons that can be adjusted each step to eventually equalize 2001 balloons.

Uploaded by

yurtman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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AoPS Community 2001 Italy TST

Italy TST 2001


www.artofproblemsolving.com/community/c5504
by outback, WakeUp

1 The diagonals AC and BD of a convex quadrilateral ABCD intersect at point M . The bisector
of ∠ACD meets the ray BA at K. Given that M A · M C + M A · CD = M B · M D, prove that
∠BKC = ∠CDB.

2 Let 0 ≤ a ≤ b ≤ c be real numbers. Prove that

(a + 3b)(b + 4c)(c + 2a) ≥ 60abc

3 Find all pairs (p, q) of prime numbers such that p divides 5q + 1 and q divides 5p + 1.

4 We are given 2001 balloons and a positive integer k. Each balloon has been blown up to a
certain size (not necessarily the same for each balloon). In each step it is allowed to choose
at most k balloons and equalize their sizes to their arithmetic mean. Determine the smallest
value of k such that, whatever the initial sizes are, it is possible to make all the balloons have
equal size after a finite number of steps.

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