József Wildt International Mathematical Competition

The Edition XXVIIIth , 2018 1

The solution of the problems W.1 - W.60 must be mailed before 26. October 2018, to Mihály Bencze,
str. Hărmanului 6, 505600 Săcele - Négyfalu, Jud. Braşov, Romania, E-mail: benczemihaly@gmail.com;
benczemihaly@yahoo.com

W1. We consider a prime number n ≥ 3, two matrices A, B ∈ Mn−1 (Q) , AB = BA, and

2 (n − 1) π 2 (n − 1) π
ε = cos + i sin
n n
such that

a). det (A − εB) = 0

b).
n−1
X
εn−ki det(1 − ε−k ) det(εk B − A) ≥ 0
k=0

for all i ∈ {0, 1, ..., n − 2}

c).

det B > 0
Show that whatever m ≥ 1 is the natural number and whatever the rational numbers b0 , b1 , ..., bm ,
Pm
white bk Ak B m−k 6= On then
k=0
m
!
X
det bk Ak B m−k 6= 0
k=0

Florin Stănescu

W2. Determine the biggest real number α, with the propoerty that for any function f : [0, 1] → [0, ∞)
which meets the requirements:
i). f it’s convex and f (0) = 0
ii). exist ε ∈ (0, 1) such that f it’s differentiable on [0, ε) and f ′ (0) 6= 0; inequality holds
   
Z1 Z1  
 x2   f (x) 
  dx ≥ (x + α) ·    dx
 Rx   R
1 2 
0 f (t) dt 0 f (t) dt
0
0

Florin Stănescu

W3. Consider the complex numbers a, b, c, d, white module one, which have the following properties:
a).

arg a < arg b < arg c < arg d

2000 Mathematics Subject Classification. 11-06.

Key words and phrases. Contest.
b).
s
p 2
[(a − c) (b − d)]
2 |(b + ai) (c + bi) (d + i) (a + di)| − =4
abcd
Show that:
 
i + 4a2 i + 4b2 4
max 1− ; ≥√
i + 8a (b − d) i + 4b (a − c) 17
Note. If a = r (cos t + i sin t) , t ∈ [0, 2π) , then t = arg z.

Florin Stănescu

W4. Let (Fn )n≥0 and (Ln )n≥0 be the Fibonacci and the Lucas sequence, respectively. Compute the
following limits:
a).
 p p √
lim 2n+2 (2n + 1)!!Fn+1 − 2n (2n − 1)!!Fn n
n→∞

b).
 p p √
2n+2 2n
lim (2n + 1)!!Ln+1 − (2n − 1)!!Ln n
n→∞

D.M. Bătineţu-Giurgiu and Neculai Stanciu

W5. Show that in any triangle ABC (with usual notations) holds the following inequalites:
a).

2

s2 + r2 + 4Rr ≥ 8r (4R + r) s2 − r2 − 4Rr
b).

2


s2 + r2 − 8Rr ≥ 8 8R2 + r2 − s2 s2 − 4R (R + r)

D.M. Bătineţu-Giurgiu and Neculai Stanciu

P
n P
n
W6. For n ∈ N ∗ − {1} , xk ∈ R+
∗
, (∀) k = 1, n, Xn (t) = xtk , t ∈ R, Xn (1) = x k = Xn ,
k=1 k=1
m ∈ [1, ∞) show that
n
X
(n − 1) nm
xk Xn (−m) − x−m
k ≥
k=1
Xnm−1

D.M. Bătineţu-Giurgiu and Neculai Stanciu

W7. Find the sum

+∞
X n!
n=0
1 × 3 × ... × (2n + 1)

Moubinool Omarjee

W8. Consider the real √

sequence a0 = 0, a1 = 1, a2n = an , a2n+1 = an + an+1 . Prove that ∀n ∈ N,
an n 1+ 5
2 ≤ ϕ where ϕ = 2 .

Moubinool Omarjee
W9. Find an example of field K, an integer d ≥ 1, a infinite subgroup G of GLd (K) such that there
exist N ≥ 1, that verify ∀g ∈ G, g N = Id .

Moubinool Omarjee
W10. Let a > 0 be a real number. Find the value of the sum
X n3 an
(n − 1)!
n≥1

(Here, 0! = 1! = 1).

José Luis Díaz-Barrero

W11. Let a ≥ 1 be an integer. Calculate
s   s  !
(n + 1)a + 1 na + 1
lim n+1
Γ n
− Γ ,
n→∞ a a

where Γ is the Euler gamma function.

José Luis Díaz-Barrero

W12. Let A1 , A2 , · · · , An ∈ M2 (C), (n ≥ 2) be the solutions of the equation
 
n 2 3
X = .
4 6
n
X
Prove that T r(Ak ) = 0.
k=1

José Luis Díaz-Barrero

W13. Find in closed form the value of
X∞ ∞ ∞ X∞
(−1)n X (−1)k X (−1)n
+
n=3
2n − 3 2k − 3 n=0 (2k − 1)(2n + 2k + 1)
k=n+1 k=1

Paolo Perfetti
W14. Let a, b, c ≥ 0 and a + b + c = 1. Prove that
p p p  
3 3 3 1
4 + 17a2 b + 4 + 17b2 c + 4 + 17c2 a + 10 − abc ≥5
27
Paolo Perfetti
W15. Evaluate
∞
X Γ( k2 + 1)
(−1)k
k=0
Γ( k+3
2 )

Paolo Perfetti
W16. Let n ∈ N, n ≥ 2 and the numbers ai , bi ∈ R, bi > 0 where i ∈ {1, 2, ..., n} . Prove that
 2 
2 a1 a22 a2n 2
(b1 + b2 + ... + bn ) + 2 + ... + 2 ≥ (a1 + a2 + ... + an ) +
b21 b2 bn
 q
2
+n2 (n − 1)
n
(a1 a2 ...an )

Ovidiu T. Pop

W17. Prove that

z z
+a· ≤ 2, ∀z ∈ C ∗
z z
where a ∈ {−1, 1}.

Ovidiu T. Pop

W18. Let a, b, c be nonnegative real numbers such that ab + bc + ca = 3α2 , α ≥ 0. Find the minimum of

(1 + a2 )(1 + b2 )(1 + c2 )

and the values (a, b, c) where the minimum is attained.

Paolo Perfetti

W19. 1). Prove that the function f : (0, +∞) → R, where

f (x) = ex + ln x
is injective
2). Solve the equation

2x √
e2x + ex ln = e
1 − 2x

Ionel Tudor

W20. Let f : [0, 1] → [0, ∞) be a convex and integrable function, with f (0) = 0. Show that
a).

Z1 Z1
2n 1
x f (x)dx ≥ f (x)dx
n+1
0 0

b).

Z1
1
x2018 ln(1 + x2 )dx >
4040
0

Mihaela Berindeanu

W21. Let G (n, m) be the 1-graphs with n vertices and m edges. Draw these graphs and describe them
about their planarity and convexity, if they have the next characteristic polynomial:

P (λ) = λ4 − 3λ2 + 1.
Find Spec (G) and study if these graphs are cospectral.

Laurenţiu Modan
W22. Let (xn )n be the recurrent sequence:
xn 1
xn+1 = 1 + , n ≥ 0, x0 =
2 2
If we consider the recurent sequece yn = 2 − xn , n ≥ n0 , study the convergence of the series:
X
yn
n≥0

and in the convergence case, compute its sum.

Laurenţiu Modan
   
3 = 4 in (Zn , ·) and ord b
W23. Find the smallest natural number n ≥ 2, so that ord b 3 = 5 in
(Zn , +) . Let P ∈ Zn [x] be the polynomial:

P (x) = b
4x4 − x3 + b
6x2 − b
9
Find the decompsition of P in irreductible factors.

Laurenţiu Modan
W24. Prove that the inequality

L61 L62
√ + 4 √ + ···
(L41+ + L2 )( 2L1 + L2 ) (L2 + L2 L3 + L43 )( 2L2 + L3 )
L21 L22 4 2 2

L6n−1 L6n
+ 4 √ + √
(Ln−1 + L2n−1 L2n + L4n )( 2Ln−1 + Ln ) (L4n + L2n L21 + L41 )( 2Ln + L1 )
√
2−1
≥ (Ln+2 − 1) .
3
Ángel Plaza
W25. Let a, b, c be non negative integer numbers such that a + b = c. Prove that

 a n  n X[ n2 ]      i
b n−i n−i−1 ab
+ = + − 2 .
c c i=0
i i−1 c

Ángel Plaza
W26. Let x, y, z be positive real numbers, and n and m integers. Find the maximal value of the
expression
x + ny y + nz
+ +
(m + 1)x + (n + m)y + mz (m + 1)y + (n + m)z + mx
z + nx
+ .
(m + 1)z + (n + m)x + my

Ángel Plaza
W27. If a1 , a2 , ..., an ≥ e, with the notations:
v
n u n
1X uY n
An := ak , Gn := t
n
ak , Hn := P
n
n 1
k=1 k=1
ak
k=1

prove that:
 AnGn Hn ≤ GnAn Hn ≤ HnAn Gn

Dorin Mărghidanu
W28. If a1 , a2 , ..., an > 0, with the notations:
n
1X
An [a1 , a2 , ..., an ] := ak , Gn [a1 , a2 , ..., an ] :=
n
k=1
v
u n
uY n
= t
n
ak , Hn [a1 , a2 , ..., an ] := P
n
1
k=1
ak
k=1

prove that:
a).
An [a1 ,a2 ,...,an ]
(An [a1 , a2 , ..., an ]) ≤ Gn [aa1 1 , aa2 2 , ..., aann ] ;
b).
h i 1
1/a 1/a
Gn a1 1 , a2 2 , ..., a1/a
n
n
≤ (Hn [a1 , a2 , ..., an ]) H[a1 ,a2 ,...,an ]

Dorin Mărghidanu
W29. Let u : R → R be a continuos function and f : (0, ∞) → (0, ∞) be a solution of differential
R x
equation y ′ (x) − y (x) − u (x) = 0 for any x ∈ (0, ∞). Find (exe+fu(x)
(x))2
dx.

D.M. Bătineţu-Giurgiu and Neculai Stanciu

W30. If xk > 0, k = 1, 2, ..., n, then
v
n n n u n
X X X uX
3t
x2k x4k ≤ x n
k x6 k
k=1 k=1 k=1 k=1

Li Yin
W31. For all n ∈ N, then
 n
ln n
Wn ∼ 1− (2.1)
2n
where Wn := (2n−1)!!
(2n)!! is Wallis product. First of all, we prove the following Lemma.
Lemma. Let an > 0, an → 0 and
 
an
lim n −1 =α (2.2)
n→∞ an+1
Then
√ n
lim (1 − n
an ) =α (2.3)
n→∞ ln n
Li Yin
W32. If x0 = 1 and

x3n+1 + 1 = (xn + 1)3

for all n ≥ 0, then [xn ] = n for all n ≥ 1, when [·] denote the integer part.

Tibor Jakab

W33. Prove that if m, n ∈ N ∗ then:

 π  π 
Z2 Z2  n+m
 n  m  2
 sin xdx  cos xdx  ≥
π
0 0

Daniel Sitaru

W34. Let be
2kπ 2kπ
εk = cos + i sin ; k ∈ 1, 2n; n ∈ N ∗ .
2n + 1 2n + 1
Prove that if |x| < 1; |y| < 1 then:
2n
Y
(x − εi ) (y − εi ) < 4n (n + 1)
i=1

Daniel Sitaru

W35. Compute

n Z1
1X x sin πx
L = lim dx
n→∞ n x + (1 − x) k 1−2x
k=1 0

Daniel Sitaru

W36. Prove that if a, b, c ∈ R; a2 + b2 + c2 6= 0 then:

2 2 2 2
(ab + bc + ca) (b − a) (c − a) (c − b)

6 6 6
≤ 2 a 4 + b4 + c 4 − a 2 b2 − b 2 c 2 − a 2 c 2
a +b +c
Daniel Sitaru

W37. If ABCD its an inscriptible quadrilateral: AB = a, BC = b, CD = c, DA = d, then

 
a+b+c+d 1 1
√ >2 √ +√
abcd ab + cd ad + bc
Daniel Sitaru
W38. Let x, y, z > 0, k > 0 such that

x2 + y 2 + z 2 + xyz = 4
Then
1 1 1 k
k + k + k < 2−k + 21− 2
(x + 2) 2
(y + 2) 2
(z + 2) 2

Marius Drăgan
W39. Compute
n−1
X  
2kπ
cos3 x + , n ∈ N∗
n
k=0

Liviu Bordianu

W40. Let x1 , x2 , y1 , y2 , z1 , z2 > 0. Then

2  
(x1 + x2 ) x21 x2
≤ max , 2
(y1 + y2 ) (z1 + z2 ) y 1 z1 y 2 z2

Marius Drăgan and Sorin Rădulescu

W41. Let A, B be two square matrices from C, a, b, c, d integers such that a < b, q the quotient of
dividing b by a, such that cq 6= d and Aa B c = Ab B d = In . Then there is a strictly integer number û
such that B u = In .

Mihály Bencze and Marius Drăgan

W42. Let be n ≥ 2, n ∈ N and x1 , x2 , ..., xn > 0. Then

 n
1 x1 + x2 + ... + xn
≥
x1 x2 ...xn n
 
1 xi1 + xi2 + ...xik
≥ max
1≤i1 <i2 <...<ik ≤n xi1 xi2 ...xik k

Mihály Bencze and Marius Drăgan

√
W43. Letqp1 , p2 , p3 be positive real numbers and the functions f, g : [0, +∞) → R, f (x) = [ x] ,
2
g (x) = x − [x] . Then is true the inequality:

g (p1 ) · f (p2 ) f (p3 ) + f (p1 ) g (p2 ) f (p3 ) + f (p2 ) f (p1 ) g (p3 ) ≤

≤ f (p1 p2 p3 ) + g (p1 ) g (p2 ) g (p3 )

Mihály Bencze and Marius Drăgan

W44. Let x, y, z > 0 and α > 9 such that

 
1 1 1
(x + y + z) + + =α
x y z
Then
 
5 5 5

1 1 1
x +y +z + 5+ 5 ≥
x5 y z
1

≥ α5 − 25α4 + 230α3 − 950α2 + 1705α − 945
16
Marian Cucoaneş and Marius Drăgan

W45. Let O an interior point of ABC triangle and {M } = AO ∩ BC, {N } = BO ∩ AC,

{P } = CO ∩ AB. We denote S1 = σ [BOM ] , S2 = σ [M OC] , S3 = σ [N OC] , S4 = σ [AON ] ,
S5 = σ [AOP ] , S6 = σ [BOP ] such that S5 ≤ S3 . Then
 2S1 + 2S5 ≤ 3 |S1 − S5 | + S2 + S3 + S4 + S6

Marian Cucoaneş and Marius Drăgan

W46. Compute
π    
2π 4π
cotn + cotn + cotn , n∈N
7 7 7

Marian Cucoaneş and Marius Drăgan

W47. Let ABC be an acute triangle, denote A1 , B1 , C1 the midpoints of sides BC, CA, AB. The lines
OA1 , OB1 , OC1 intersect the circumcircle in points A2 , B2 , C2 . Denote x = A1 A2 , y = B1 B2 , z = C1 C2 .
prove that
1).

x + y + z ≥ 3r
2).

(R − x)(R − y)(R − z) = R3 cos A cos B cos C

Nicuşor Minculete

W48. Let be f : [0, 1] → (0, +∞) a continuously function. Prove that if exist α > 0 for which

Z1 Z1
α n 1
x f (x)dx ≥ ≥ f n+1 (x)dx
(n + 1)α + 1
0 0

where n ∈ N then α is unique.

Mihály Bencze

W49. If xk > 0 (k = 1, 2, ..., n), then

v
n u n
X uY X √ √
xk − n t
n
xk ≤ ( x i − x j )2
k=1 k=1 1≤i<j≤n

Mihály Bencze

W50. Compute
π
Z2
(x3 sin x + λ cos x)dx
√ √
5 + x3 cos x + 5 + x6 cos2 x
−π
2

Mihály Bencze

W51. Prove that

" n
#
X 1
√ √ =n
k=1
k 4 + 2k 3 + 2k 2 + k + 1 − k 4 + 2k 3 − k + 1
where [·] denote the integer part.

Mihály Bencze
W52. Let A1 , A2 , ..., An be a convex polygon. Prove that
n
X X n
1 1
≥ π+A
sin Ak sin n−1k
k=1 k=1

Mihály Bencze
W53. Let a, b ∈ {2, 3, 4, 5, 6, 7, 8, 9} and m ∈ N . Prove that exist n ∈ N such that the first digit of an
are bm .

Mihály Bencze
W54. For all invertable matrices A, B ∈ M2 (C) holds
1).

2T r(AB) + T r(A−1 B)detA + T r(B −1 A)detB = 2T r(A)T r(B)

2).

T r(ABA) + T r(BAB) + T r(A)detB + T r(B)detA = (T r(A) + T r(B))T r(AB)

Mihály Bencze
W55. If a ∈ N ∗ , (a, p) = 1 where p is a prime, then
n 
Y 
k−1
ap (p−1)
−1
k=1
n(n+1)
is divisible by p 2 .

Mihály Bencze
W56. In all tetrahedron ABCD holds:
1).
1 2 1 1 2 1 1
+ + + + + ≤ 2
ha hb ha hc ha hd hb hc hb hd hc hd 2r
2).
1 2 1 1 2 1 2
+ + + + + ≤ 2
ra rb ra rc ra rd rb rc rb rd rc rd r
Mihály Bencze
W57. Prove that
π
Z2 Y
n

π
sin2k+1 x + cos2k+1 x dx ≥ √
; n ∈ N∗
2 2 n2
0 k=1

Daniel Sitaru
W58. Let fn be n − th Fibonacci number defined by recurrence

fn+1 − fn − fn−1 = 0, n ∈ N
and initial conditions f0 =

0, f1 = 1.
Prove that 5n2 + 3n − 2 fn − 6nfn+1 is divisible by 50 for any n ∈ N

Arkady Alt
W59. Let E be a Inner Product Space with dot product − · − and F be proper nonzero
subspace. Let P : E −→ E be orthogonal projection E on F.
a). Prove that for any x, y ∈ E, holds inequality

|x · y − x·P (y) − y·P (x)| ≤ kxk kyk

b). Determine all cases when equality occurs.

Arkady Alt

W60. Let x, a, h be arbitrary real numbers such that x > 0, a ≥ −1 , h > 0 and let sequence (xn )
n+a
defined recursively by x1 = x, xn+1 = xn , n ∈ N ∪ {0} .
n+a+h
P∞
Explore for which h the infinite sum xn converges and find it in the case of convergence.
n=1

Arkady Alt