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Bmo 1993

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302 views1 page

Bmo 1993

Uploaded by

ramseven37
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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10-th Balkan Mathematical Olympiad

Nicosia, Cyprus – May 3-8, 1993

1. Let a, b, c, d, e, f be real numbers which satisfy

a + b + c + d + e + f = 10,
(a − 1) + (b − 1) + (c − 1)2 + (d − 1)2 + (e − 1)2 + ( f − 1)2 = 6.
2 2

Find the maximum possible value of f . (Cyprus)


2. A natural number with the decimal representation aN aN−1 . . . a1 a0 is called
monotone if aN ≤ aN−1 ≤ · · · ≤ a0 . Determine the number of all monotone num-
bers with at most 1993 digits. (Bulgaria)
3. Circles C1 and C2 with centers O1 and O2 , respectively, are externally tangent at
point Γ. A circle C with center O touches C1 at A and C2 at B so that the centers
O1 , O2 lie inside C. The common tangent to C1 and C2 at Γ intersects the circle C
at K and L. If D is the midpoint of the segment KL, show that ∠O1 OO2 = ∠ADB.
(Greece)
4. Let p be a prime and m ≥ 2 be an integer. Prove that the equation

xp + yp x+y m
 
=
2 2

has a positive integer solution (x, y) 6= (1, 1) if and only if m = p.


(Romania)

The IMO Compendium Group,


D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com

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