OCTOGON MATHEMATICAL MAGAZINE Vol. 22, No.1, April 2014, pp 304-327
Print: ISSN 1222-5657, Online: ISSN 2248-1893 http://www.uni-miskolc.hu/∼matsefi/Octogon/
304
Math Competitions Corner
Jos´e Luis D´ıaz-Barrero23
No. 3
This section of the Journal offers readers an opportunity to solve interesting mathematical problems appeared previously in High School Mathematical Olympiad and University Competitions or used by trainers and contestants to prepare Math Competitions. Elegant solutions, generalizations of the problems posed and new suitable proposals are always welcomed. Proposals should be accompanied by solutions. The origin of the problems appeared previously will be revealed when the solutions are published.
Send submittal to: Jos´e Luis D´ıaz-Barrero, Applied Mathematics III, BARCELONA TECH, Jordi Girona 1-3, C2, 08034 Barcelona, Spain or by e-mail (preferred) to: <jose.luis.diaz@upc.edu>
Solutions to the problems stated in this issue should be posted before March 15, 2014
PROBLEMS
MC–36. Let nandmbe two positive integers. Show that (
2n+1−1,2m+ 1)
= 1
MC–37. Let p < q be twin primes. Find all positive integersnfor which pn−1+qn−1 dividespn+qn.
MC–38. In a warehouse there are several recipients. The 33 lightest weigh together 8/23 of the total weight, the 30 heaviest weigh together, 22/69 of the total weight. How many recipients are there altogether?
23Received: 21.10.2013
2010Mathematics Subject Classification. 11-06.
Key words and phrases. Contest.
Math Competitions Corner 305 MC–39. Find all triples (x, y, z) of real numbers that are solutions of the following system of equations:
2y= 1 3
(1 + 2x+ 2−x) , 2z= 1
3
(1 + 2y+ 2−y) , 2x= 1
3
(1 + 2z+ 2−z) .
MC–40. Let n, x, y, zbe four positive integers such that (x−y)2+ (y−z)2+ (z−x)2= (2n+ 1)xyz Show that (2n+ 1)(x+y+z) + 6 divides x3+y3+z3.
MC–41. Show that the orthocenter of an acute triangle coincides with the center of the circle inscribed in the triangle formed by the feet of the altitudes.
MC–42. Let a1, a2, . . . , an be positive real numbers. Then prove that the functionf :R→R defined by
f(α) =
(aα1 +aα2 +. . .+aαn n
)1/α
is nondecreasing and the following limits hold
α→−∞lim f(α) = min
1≤i≤n
{ ai}
, lim
α→+∞f(α) = max
1≤i≤n
{ ai}
, and
αlim→0f(α) = √n
a1a2. . . an
MC–43. Suppose thatA1, A2, . . . , An arenpoints on the plane located in such a way that for any pointP on the same plane at least one of the distances|P Ai|, 1≤i≤nis irrational. Find the minimum possible value of n.
306 Jos´e Luis D´ıaz-Barrero
MC–44. Let a1, a2, . . . , an, andb1, b2, . . . , bn be positive real numbers.
Prove that
∑
i<j
aibj
2
≥
∑
i<j
aiaj
∑
i<j
bibj
MC–45. Let Abe the set of lattice points in the plane. For each point P(x, y) inA we call neighbors ofP the points with coordinates
(x−1, y),(x+ 1, y), (x, y−1),(x, y+ 1), respectively. LetB be afinite subset ofA. A bijection f :B →B is calledperfect if for allP ∈B point f(P) is a neighbor of P. Prove that if such function exists, then also exists a perfect functiong:B→B for whichg(g(P)) =P for all P ∈B.
