J´ ozsef Wildt International Mathematical Competition

Vol. 17, No.1, April 2009, pp 306-312 ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon

306

J´ ozsef Wildt International Mathematical Competition

The XIX^th Edition, 2009³⁹

The solutions of the problems W.1-W.30 must be mailed before 30.October 2009, to Mih´aly Bencze, Str. H˘armanului 6, 505600 S˘acele-N´egyfalu, Jud. Brasov, Romania, E-mail: benczemihaly@yahoo.com

W.1. Leta, b, c be positive real numbers such thata+b+c= 1. Prove that

3

s 1 +a

b+c

¹⁻_bc^a 1 +b c+a

¹⁻_ca^b 1 +c a+b

¹⁻_ab^c

≥64.

Jos´e Luis Diaz-Barrero, Barcelona, Spain W.2. Find the area of the set A={(x, y)|1≤x≤e,0≤y≤f(x)},where

f(x) =

1 1 1 1

lnx 2 lnx 3 lnx 4 lnx (lnx)² 4 (lnx)² 9 (lnx)² 16 (lnx)² (lnx)³ 8 (lnx)³ 27 (lnx)³ 64 (lnx)³

.

Jos´e Luis Diaz-Barrero W.3. Let Φ and Ψ denote the Euler totient and Dedekind‘s totient,

respectively. Determine all nsuch that Φ (n) divides n+ Ψ (n).

J´ozsef S´andor and Lehel Kov´acs

39Received: 22.02.2009

2000Mathematics Subject Classification. 11-06 Key words and phrases. Contest.

W.4. Let Φ denote the Euler totient function.Prove that for infinitely many kwe has Φ 2^k+ 1

<2^k−1 and that for infinitely manym one has Φ (2^m+ 1)>2^m−1.

J´ozsef S´andor W.5. Letp₁, p₂ be two odd prime numbers and α, nintegers withα >1 and n >1. Prove that if the equation

p2−1 2

p1

+

p2+1 2

p1

=αⁿ does not accept integer solutions in the casep1=p2,then the equation does not also have integer solutions for the casep₁6=p₂.

Michael Th. Rassias, Athens, Greece W.6. Prove that

p(n) = 2 +

p(1) +...+phn 2

i+χ1(n) +

p^′₂(n) +...+p^′[ⁿ₂]−1(n) for everyn∈N with n >2, where χ₁(n) denotes the principal character Dirichlet modulo 2, i.e. χ₁(n) =

1, if (n, k) = 1

0, if (n, k) = 0 with p^′(n) we denote the number of partitions of nin exactlym sumands.

Michael Th. Rassias W.7. If 0< a < b, then

Zb a

x²− ^a+b₂ 2

ln^x_aln^x_bdx (x²+a²) (x²+b²) >0.

Gy¨orgy Sz¨oll˝osy, M´aramarossziget, Romania W.8. Ifn, p, q∈N, p < q then

(p+q)n n

Xn k=0

(−1)^k n

k

(p+q−1)n pn−k

=

=

(p+q)n pn

X[ⁿ₂]

k=0

(−1)^k pn

k

(q−p)n n−2k

Gy¨orgy Sz¨oll˝osy

W.9. Let the series

s(n, x) =X

n≥0

(1−x) (1−2x)...(1−nx) n!

Find a real set on which this series is convergent, and then compute its sum.

Find also lim

(n,x)→(∞,0)s(n, x).

Laurent¸iu Modan, Bucharest, Romania W.10. Let consider the following function set

F ={f|f :{1,2, ..., n} → {1,2, ..., n}}

1). Find|F|

2). For n= 2k, prove that|F|< e(4k)^k 3). Findn,if |F|= 540 andn= 2k.

Laurent¸iu Modan W.11. Find all real numbers msuch that

1−m

2m ∈

x∈R|m²x⁴+ 3mx³+ 2x²+x= 1 .

Cristinel Mortici, Tirgovi¸ste, Romania W.12. Find all functionsf : (0,+∞)∩Q→(0,+∞)∩Qsatisfying the following conditions:

1). f(ax)≤(f(x))^a,for everyx∈(0,+∞)∩Qand a∈(0,1)∩Q 2). f(x+y)≤f(x)f(y),for every x, y∈(0,+∞)∩Q.

Cristinel Mortici W.13. Ifa_k>0 (k= 1,2, ..., n),then prove the following inequality

Xn k=1

a⁵_k

!4

≥ 1 n

2 n−1

5

 X

1≤i<j≤n

a²_ia²_j





5

R´obert Sz´asz, Marosv´as´arhely, Romania

W.14. If the function f : [0,1]→(0,+∞) is increasing and continuous, then for everya≥0 the following inequality holds:

Z1 0

x^a+1

f(x)dx≤ a+ 1 a+ 2

Z1 0

x^adx f(x).

R´obert Sz´asz W.15. Let a triangleABC and the real numbers x, y, z >0. Prove that

xⁿcosA

2 +yⁿcosB

2 +zⁿcosC

2 ≥(yz)ⁿ² sinA+ (zx)ⁿ² sinB+ (xy)ⁿ² sinC.

Nicu¸sor Minculete, Sfˆıntu-Gheorghe, Romania W.16. Prove that

Xn k=1

1

d(k) >√

n+ 1−1,

for everyn≥1, whered(n) is the number of divisors ofn.

Nicu¸sor Minculete W.17. Ifa, b, c >0 andabc= 1, α= max{a, b, c};f, g: (0,+∞)→R, wheref(x) = ^2(x+1)_x ², g(x) = (x+ 1)

√1

x + 12

,then

(a+ 1) (b+ 1) (c+ 1)≥min{{f(x), g(x)} |x∈ {a, b, c} \ {α}}. Ovidiu Pop and Gy¨orgy Sz¨oll˝osy W.18. Ifa, b, c >0 andabc= 1, then

X a+b+cⁿ

a²ⁿ⁺³+b²ⁿ⁺³+ab ≤aⁿ⁺¹+bⁿ⁺¹+cⁿ⁺¹ for all n∈N.

Mih´aly Bencze

W.19. Ifxk>0 (k= 1,2, ..., n),then Xn

k=1

x_k 1 +x²₁+...+x²_k

2

≤ Pn k=1

x²_k 1 +

Pn k=1

x²_k .

Mih´aly Bencze W.20. Ifx∈R\_kπ

2 |k∈Z ,then



 X

0≤j<k≤n

sin (2 (j+k)x)





2

+



 X

0≤j<k≤n

cos (2 (j+k)x)





2

=

= sin²nxsin²(n+ 1)x sin²xsin²2x .

Mih´aly Bencze W.21. Ifζ denote the Riemann zeta function, and s >1, then

X∞ k=1

1

k^s+ 1 ≥ ζ(s) 1 +ζ(s).

Mih´aly Bencze W.22. Ifai >0 (i= 1,2, ..., n),then

a₁ a₂

k

+ a₂

a₃ k

+...+ a_n

a₁ k

≥ a₁ a₂ +a₂

a₃ +...+a_n a₁ for all k∈N^∗.

Mih´aly Bencze W.23. Ifx_k∈R (k= 1,2, ..., n) and m∈N, then

1). P

cyclic

x²₁−x1x2+x²₂m

≤3^m Pⁿ

k=1

x^2m_k 2). Q

cyclic

x²₁−x1x2+x²₂m

≤ ³_n^mm _n P

k=1

x^2m_k n

Mih´aly Bencze

W.24. IfK, L, M denote the midpoints of sides AB, BC, CA,in triangle ABC, then for allP in the plane of triangle, we have

AB P K +BC

P L + CA

P M ≥ AB·BC·CA 4P K·P L·P .

Mih´aly Bencze W.25. LetABCD be a quadrilateral in which Ab=Cb= 90^◦.Prove that

1

BD(AB+BC+CD+DA)+BD²

1

AB·AD+ 1 CB·CD

≥2 2 +√

2 .

Mih´aly Bencze W.26. Ifa_i >0 (i= 1,2, ..., n) and Pⁿ

i=1

a^k_i = 1,where 1≤k≤n+ 1, then Xn

i=1

a_i+ 1 Qn i=1

a_i

≥n^1−k1 +n^nk.

Mih´aly Bencze W.27. Leta, n be positive integers such thataⁿ is a perfect number. Prove that

a^n/µ> µ 2,

whereµ denotes the number of distinct prime divisors ofaⁿ.

Michael Th. Rassias W.28. Letθ and p (p <1) be nonnegative real numbers.

Suppose that f :X→Y is a mapping with f(0) = 0 and

2f

x+y 2

−f(x)−f(y)

_Y ≤ θ kxk^p_X

+kyk^p_X

(1) for allx, y∈Z withx⊥y, whereX is an orthogonality space andY is a real Banach space.

Prove that there exists a unique orthogonally Jensen additive mapping T :X →Y, namely a mapping T that satisfies the so-called orthogonally Jensen additive functional equation

2f

x+y 2

=f(x) +f(y) for all x, y∈X with x⊥y, satisfying the property

kf(x)−T(x)kY ≤ 2^pθ

2−2^p kxk^p_X (2)

for all x∈X.

Themistocles M. Rassias W.29. In all triangleABC holds

X 1− r√

3tgA 2 +√

3tgA 2

! 1−

r√ 3tgB

2 +√ 3tgB

2

!

≥3

Mih´aly Bencze W.30. Prove that

X

0≤i<j≤n

(i+j) n

i n

j

=n

2²ⁿ⁻¹− 2n

n

Mih´aly Bencze