OCTOGON MATHEMATICAL MAGAZINE Vol. 24, No.1, April 2016, pp 193-203
Print: ISSN 1222-5657, Online: ISSN 2248-1893 http://www.uni-miskolc.hu/∼matsefi/Octogon/
193
Math Competitions Corner
J. L. D´ıaz-Barrero23
No. 4
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 July 15, 2016
MC–46. Find those positive integers n≤2014 for which there exist positive integersr, s such that gcd(rs(r+s), n) = 1 and ndoes not divider−s.
MC–47. LetABC be an acute triangle and letX be the foot of the altitude drawn fromAandY be the intersection of the perpendicular to AC drawn fromX. If the cicumcircle of triangleABX meetsBY at pointZ (distinct of B) and the extension ofAZ meetsXY at pointP, then prove that
BX·XP =P Y ·XC.
23Received: 18.01.2016
2010Mathematics Subject Classification. 11-06.
Key words and phrases. Contest.
194 J. L. D´ıaz-Barrero
MC–48. Let S={1,2, . . . , n}, n≥2, and letf :S −→S be a bijective function distinct from the identity. Letu= �n
k=1|f(k)−k| and letvbe the number of ordered pairs (a, b) of elements ofS such thata > b and
f(a)< f(b). Show thatv < u≤2v, and that u= 2vif and only if there do not exist positive integers a > b > csuch thatf(a)< f(b)< f(c).
MC–49. Let x, y be two distinct positive reals and letz be their harmonic mean. Show that segments of length x, y, z form a triangle if and only if
1 + min{x, y} max{x, y} >√
2
MC–50. Let a, kbe positive integers and let nbe a nonnegative integer.
Show that (ka2+ 1)2n+1 can be expressed as a sum ofk+ 1 squares and (ka2+ 1)2n+2can be expressed as a sum of (k+ 1)2 squares.
MC–51. We have three numbered boxes and 10000 red balls, 10000 blue ones and 10000 yellow ones. Balls of the same color are undistinguishable.
Determine in how many ways they can be distributed in the boxes satisfying:
• Each box has 10000 balls
• No two boxes have the same amount of balls of the same color
• For every two boxesA andB, there is a color csuch that the number of balls of colorcin Ais exactly 2015 or 2016 bigger than the number of balls of colorcin B.
SOLUTIONS
No problem is ever permanently closed. We will be very pleased considering for publication new solutions or comments on the past problems.
MC–36. Let nandmbe two positive integers. Show that
�
2n+1−1,2m+ 1
�
= 1
Training for OCM – 2014 Solution by Jos´e Luis D´ıaz-Barrero, BarcelonaTech, Barcelona, Spain. Letd be the greatest common divisor of the numbers 22n+1−1 and
