OCTOGON MATHEMATICAL MAGAZINE Vol. 25, No.2, October 2017, pp 464-473 Print: ISSN 1222-5657, Online: ISSN 2248-1893 http://www.uni-miskolc.hu/∼matsefi/Octogon/
464
Math Competitions Corner
J. L. D´ıaz-Barrero35
Dedicated to the 25 years of Octogon Mathematical Magazine
No. 5
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, BarcelonaTech, 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 July 15, 2018
MC–52. Letp1, p2, . . . , pn+1 denote the first n+ 1 primes. Suppose that {A, B} is a partition of the setX ={p1, p2, . . . , pn}, where
A={q1, q2, . . . , qs}and B ={r1, r2, . . . , rt}. Prove that if m=q1q2. . . qs+r1r2. . . rt< p2n+1,then m is a prime.
MC–53. LetC be a circumference, DD′ a diameter and XX′ a chord perpendicular to diameter DD′. A pointO lying onC is joined to points D, D′, X, andX′,respectively. Show that the sum of the projections ofOD and OD′ overOX isOX and their difference is OX′.
35Received: 23.08.2017
2010Mathematics Subject Classification. 11-06.
Key words and phrases. Contest.
Math Competitions Corner 465 MC–54. Leta, b, c be positive real numbers such thata+b+c= 1.Prove that
(x2+y2+z2)
a3
x2+ 2y2 + b3
y2+ 2z2 + c3 z2+ 2x2
≥ 1 9 holds for all positive realsx, y, z.
MC–55. Letn be a positive integer. Prove that X
0≤k<n/2
n
2k+ 1
13k is divisible by 2n−1.
MC–56. Prove that
1
π Z 1
2−√ 3
e−x2dx
1
π
Z 2+√3
1
e−x2dx
!
< 1 36
MC–57. Letf :R→Rbe a function such that a < b implies f(a)<(f(b))3.Prove that:
1. f(x) + 1≥0 holds for allx∈R.
2. 0< f(x)<1 holds for at most one value ofx∈R.
