Open questions

Open questions

OQ. 3890. 1). If a₁, a₂, ..., a₉ ∈ {1,2, ...,9}and a_i6=a_j

(i, j ∈ {1,2,3,4,5,6,7,8,9}) then determine all polynomialsfk(n) =Z →Z (k= 1,2,3,4,5,6,7,8,9) such that

P9 k=1

a^f_k^k(n) is divisible bya₁a₂...a₉ for all n∈N.

We have the following 2²ⁿ⁺¹+ 3²ⁿ⁺¹ ≡0 (mod 5),2ⁿ⁺²+ 3²ⁿ⁺¹ ≡0 (mod 7), 2⁸ⁿ⁺³+ 3ⁿ⁺¹ ≡0 (mod 11),2⁴ⁿ⁺²+ 3ⁿ⁺²≡0 (mod 13),

2⁶ⁿ⁺²+ 3⁴ⁿ⁺² ≡0 (mod 17),2³ⁿ⁺⁴+ 3²ⁿ⁺¹ ≡0 (mod 19), 2⁵ⁿ⁺¹+ 3ⁿ⁺³ ≡0 (mod 29),2⁴ⁿ⁺¹+ 3⁶ⁿ⁺⁹≡0 (mod 31)

2). Determine all polynomials f_k(n) =Z →Z (k= 1,2,3,4,5,6,7,8,9) for which

P9 k=1

a^f_k^k(n) is divisible by 45 for alln∈N.

Mih´aly Bencze OQ. 3891. 1). Solve inQ the equation

(x+ 1) (y+ 1) (z+ 1) = (x−1) (y−1) (z−1).If x= ^a+b_a+b, y= ^b+c_b−c, z= ^c+a_c−a, when a, b, c∈Q, then we obtain infinitely many solutions of the given

equation

2). Solve inZ the given equation 3). Solve inQ the equation

Qn k=1

(xk+ 1) = Qn k=1

(xk−1) 4). Solve inZ the previous equation

Mih´aly Bencze OQ. 3892. Ifx_k>0 (k= 1,2, ..., n) such that

Qn k=1

x_k= 1,then determine all a, b, c >0 for which

Pn k=1

1

(a+xk)^b + ¹

1+

Pn k=1

xk

c ≥ _(a+1)ⁿ b +_(n+1)¹ c.If a= 1, b= 2,c= 1, then we obtain a solution forn= 3.

Mih´aly Bencze OQ. 3893. 1). Solve inQ the equationxy+yz+zx=−1. Ifx= ¹_a−b^−ab, y= ¹_b⁻₋^bc_c, z= ¹_c⁻₋^ca_a,where a, b, c∈Q, then we obtain an infinitely many solutions for the given equation.

3). Solve inQ the equation P

cyclic

x₁x₂...x_n−1 =−1 4). Solve the previous equation inZ

Mih´aly Bencze OQ. 3894. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic

xⁿ₁⁻¹+x₂...x_n

xⁿ₂⁻¹+x₃...x₁

≤ _n2n⁴−1

P_n

k=1

x_{k 2n−2}

.

Mih´aly Bencze OQ. 3895. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic

x₁+_x¹

2 x₂+_x¹

3

...

x_n+_x¹

1

≥ _nn²−3

P_n

k=1

x_k+ Pn k=1

1 xk

n−2

.

Mih´aly Bencze OQ. 3896. Ifx_k>0 (k= 1,2, ..., n) then

Pn k=1

xⁿ_k+n² Qn k=1

xk≥ ⁿ⁽ⁿ⁺¹⁾

nⁿ⁻¹² ·

P

cyclic

x1x2

!ⁿ⁺¹

2

Pn k=1

x_k

.

Mih´aly Bencze OQ. 3897. Ifx_k>0 (k= 1,2, ..., n) such that

Qn k=1

x_k= 1,then determine all λ∈R for which

Qn k=1

1 +x^λ_k

≤ _n(n²−ⁿ1)λ

P_n

k=1

xk

_(n−1)λ

.If n= 3 then λ= 2 is a solution.

Mih´aly Bencze OQ. 3898. Ifx_k >0 (k= 1,2, ..., n) such that

Pn k=1

x_k =nthen determine all a, b, λ >0 for whicha

Pn k=1

x^λ_k+b Qn k=1

x_k≥na+b.The casea=n−1, b=n, λ= 2 is a solution.

Mih´aly Bencze

OQ. 3899. Solve inZ the equation _x2+y^xy2+tz² +_y2+z^yz2+tx² +_z2+x^zx2+ty² = _t+2³ . Mih´aly Bencze OQ. 3900. Solve in Z the equation 3 x²+y²+z²

=t²+u²+v²+w².If t=x+y+z, u=x−y, v=y−z, w=z−x, wherex, y, z ∈Z, then the equation have infinitely many solutions inZ.

Mih´aly Bencze OQ. 3901. Solve in Z the equation

x³₁y1+x³₂y2+x³₃y3= 2 z²₁+z₂²+z₃²

+u1t²₁+u2t²₂+u3t²₃.If y1=x2+x3, y₂ =x₃+x₁, y₃ =x₁+x₂, z₁ =x₁x₂, z₂=x₂x₃, z₃ =x₃x₁, u₁ =x₁x₂, u₂ =x₂x₃, u₃=x₃x₁, t₁ =x₁−x₂, t₂ =x₂−x₃, t₃=x₃−x₁, where x1, x2, x3∈Z then the given equation have infinitely many solutions inZ.

Mih´aly Bencze OQ. 3902. Solve in Z the equation

x⁵₁y1+x⁵₂y2+x⁵₃y3=z₁⁴t1+z⁴₂t2+z₃⁴t3+u²₁v1+u²₂v2+u²₃v3.

Ify₁ =x₂+x₃, u₂=x₃+x₁, y₃ =x₁+x₂, z₁=x₁, z₂=x₂, z₃=x₃, t1 =x²₂+x²₃, t2=x²₃+x²₁, t3 =x²₁+x²₂, u1=x1−x2, u2=x2−x3, u3 =x3−x1, v1 =x³₁x2+x²₁x²₂+x1x³₂, v2=x³₂x3+x²₂x²₃+x2x³₃,

v₃ =x³₃x₁+x²₃x²₁+x₃x²₁, wherex₁, x₂, x₃∈Z,then the given equation have infinitely many solutions in Z.

Mih´aly Bencze OQ. 3903. Solve in Z the equation

2 x⁶₁+x⁶₂+x⁶₃

=y₁⁵z1+y⁵₂z2+y₃⁵z3+u²₁v1+u²₂v2+u²₃v3.

Ify₁ =x₁, y₂=x₂, y₃ =x₃, z₁ =x₂+x₃, z₂=x₃+x₁, z₃ =x₃+x₂,

u₁ =x₁−x₂, u₂ =x₂−x₃, u₃=x₃−x₁, v₁=x⁴₁+x³₁x₂+x²₁x²₂+x₁x³₂+x⁴₂, v2 =x⁴₂+x³₂x3+x²₂x²₃+x2x³₃+x⁴₃, v3=x⁴₃+x³₃x1+x²₃x²₁+x3x³₁+x⁴₁ where x₁, x₂, x₃∈Z, then the given equation have infinitely many solutions in Z.

OQ. 3904. Solve in Z the equation 2

x^k+2₁ +x^k+2₂ +x^k+2₃

+ 2 y^k₁z1+y^k₂z2+y^k₃z3

=

= 2

t^k+1₁ u1+t^k+1₂ u2+t^k+1₃ u3

+v²₁w1+v₂²w2+v₃²w3.

Ify₁ =x₁, y₂=x₂, y₃ =x₃, z₁ =x₂+x₃, z₂=x₃+x₁, z₃ =x₂+x₃, t1 =x1, t2 =x2, t3 =x3, u1 =x2x3, u2 =x3x1, u3=x1x2, v1 =x1−x2, v₂ =x₂−x₃, v₃=x₃−x₁, w₁ =−x^k₁ +x^k₂ +x^k₃, w₂=−x^k₂+x^k₃+x^k₁, w₃=−x^k₃+x^k₁+x^k₂,where x₁, x₂, x₃∈Z, then the given equation have infinitely many solutions in Z.

Mih´aly Bencze OQ. 3905. Solve in Z the equationn

Pn k=1

x²_k =y²+ P

1≤i<j≤n

z_ij².If y=

Pn k=1

x_k, z_ij =x_i−x_j, i, j∈ {1,2, ..., n} when x_i ∈Z (i= 1,2, ..., n),then the given equation have infinitely many solutions inZ.

Mih´aly Bencze OQ. 3906. 1). Solve inQ the equation

Pn k=1

1

x²_k = Pnⁿ² k=1

y²_k

+_n¹ P

1≤i<j≤n

z_ij².If y_k=x_k (k= 1,2, ..., n), z_ij = _x^xi

j −^x_x^j_i, where x_i ∈Q (i= 1,2, ..., n) then the given equation have infinitely many solution inQ

2). Solve inZ the given equation

Mih´aly Bencze OQ. 3907. Solve in Z the equation 2

P_n

k=1

x²_k−y1y2−...−yny1

= Pn k=1

z_k². Ify_k =x_k, z_k=x_k−x_k+1,where x_k∈Z (k= 1,2, ..., n), then the equation have infinitely many solutions inZ.

Mih´aly Bencze

OQ. 3908. Solve in Z the equation

x²₁+x²₂+...+x²_k x1+x2+...+xk +^x

2

2+x²₃+...+x²_k+1

x2+x3+...+xk+1 +...+^x

2n+x²₁+...+x²_k−1 xn+x1+...+xk−1 =

n Pn i=1

x²_i Pn i=1

xi

,when k∈ {1,2, ..., n}.The casex1 =x2=...=xn∈Z offer infinitely many solutions.

Mih´aly Bencze OQ. 3909. Solve in Z the equation

(n−1)xⁿ₁⁻¹−x2x3...xn

xⁿ₁⁻¹−(x1x2)ⁿ⁻²¹+xⁿ₂⁻¹ +...+ ^{(n−1)xn−1n −x1x2...xn−1}

xⁿn⁻¹−(xnx1)ⁿ⁻²¹+xⁿ₁⁻¹ =n.

The casex₁=x₂ =...=x_n∈N offer infinitely many solutions in Z.

Mih´aly Bencze OQ. 3910. Ifx_k>0 (k= 1,2, ..., n) such that

Qn k=1

x_k= 1 then determine all a(n), b(n)>0 for which

Pn k=1

x²_k−n≥a(n) Pn k=1

x_k−b(n) P

1≤i<j≤n

x_ix_j.If n= 3, then a(n) =b(n) = 18.

Mih´aly Bencze OQ. 3911. 1). Solve inQ the following system:







 Pn k=1

x_k=n (n−1)

Pn k=1

x²_k+n Qn k=1

x_k=n² 2). Solve inZ the given system.

Mih´aly Bencze

OQ. 3912. Solve in Qthe following equation Qn

k=1

xk+ ⁿ sQn

k=1

x+y_kⁿ

= _n(n^2(x+2)₋₁₎ P

1≤i<j≤nzizj.

Mih´aly Bencze OQ. 3913. Determine all n, k∈N for which n^k+ 1 = aa...a| {z }

m−time

,where a∈ {1,2,3,4,5,6,7,8,9}.An example is 6⁵+ 1 = 7777.

Mih´aly Bencze OQ. 3914. Determine all n, k∈N for which 1^k+¹_k

2^k+¹_k

... n^k+_k¹ is the square of a rational number.

Mih´aly Bencze OQ. 3915. Determine all n, k∈N for which

Φ Φ n^k

+ Φ (Φ (kⁿ)) =n+k,where Φ is the Euler‘s totient function.

Mih´aly Bencze OQ. 3916. Ifx_k∈(0, π) (k= 1,2, ..., n) and

Pn k=1

x_k=π then determine all functions f : (0, π)→(0,+∞) such that

Pn k=1

f(x_k) = P_n

k=1 1 f(xk)

Q_n

k=1

f(x_k).

A solution isf(x) =tgx.

Mih´aly Bencze OQ. 3917. Letf : [a, b]→R be a convex function. Determine all λ∈[0,1]

such that _b−¹_a Rb a

f(x)≤ λ(f(a)+f(b))

2 + (1−λ)f ^a+b₂

. A solution is λ= ¹₂. Mih´aly Bencze OQ. 3918. Let be P(x) =a0xⁿ+a1xⁿ⁻¹+...+an−1x+an.Determine all a₀, a₁, ..., a_n∈C for which all positive rational number ^A_B (A, B∈N^∗) can be represented in following form: _B^A =

Qn i=1

(P(i))^±1.ForP(x) = ^x(x+1)₂ we have a solution.

OQ. 3919. All positive integers mcan be represented in the following forms:

1). m=±d^k(1)±d^k(2)±...±d^k(n) 2). m=±σ^k(1)±σ^k(2)±...±σ^k(n) 3). m=±Φ^k(1)±Φ^k(2)±...±Φ^k(n) 4). m=±Ψ^k(1)±Ψ^k(2)±...±Ψ^k(n) 5). m=±F₁^k±F₂^k±...±F_n^k

6). m=±L^k₁ ±L^k₂ ±...±L^k_n 7). m=± ⁿ_1k

± ⁿ_2k

±...± ⁿ_nk

Mih´aly Bencze OQ. 3920. Determine all a₁, r∈C,a_k=a₁+ (k−1)r, and allp∈N for which all positive integers m can be represented in the following form:

m=±a^k₁±a^k₂±...±a^k_n.

Mih´aly Bencze OQ. 3921. Determine all b1, q∈C, b_k=b1q^k−1 and allp for which all positive integersm can be represented in the following form:

m=±b^k₁±b^k₂±...±b^k_n.

Mih´aly Bencze OQ. 3922. Determine all ak>0 (k= 1,2, ..., n) for which P

cyclic a1−a2

a2+a3 ≥0.

Mih´aly Bencze OQ. 3923. Determine all ak>0 (k= 1,2, ..., n) for which

1 +

Qn k=1

a_k

P

cyclic 1 a1+a2a3...an

!

≥n.

Mih´aly Bencze OQ. 3924. Determine all a_k>0 (k= 1,2, ..., n) such that

Pn k=1

1

ak + 9P ₁

a1+a2+a3 + 25P ₁

a1+a2+a3+a4+a5 +...≥

≥4P ₁

a1+a2 + 16P ₁

a1+a2+a3+a4 +...

Mih´aly Bencze

OQ. 3925. Ifx_k>0 (k= 1,2, ..., n),then Pn

k=1

xⁿ_k+n Qn k=1

x_k ≥ _n−²₁ P

cyclic

x₁...x_n−1(x₁+...+x_n−1).

Mih´aly Bencze OQ. 3926. Ifx_k>0 (k= 1,2, ..., n),then

P

cyclic

x₁+^x

n−1 2

xⁿ₃⁻²

n−1

≥ ^n·2

n−1 Pn k=1

x²ⁿ_k ⁻³ Pn

k=1

xⁿ⁻²_k

.

Mih´aly Bencze OQ. 3927. Ifxk>0 (k= 1,2, ..., n),then determineF(n)>0 such that

Pn k=1

xⁿ_k−n Qn k=1

x_k ≥F(n) (x1−x2) (x2−x3)...(xn−x1).We haveF(1) = 0, F(3) = 4.

Mih´aly Bencze OQ. 3928. Ifxk>0 (k= 1,2, ..., n),then P

cyclic

xⁿ⁺¹₁

xⁿ₁+...+xⁿ_n−1 ≥ _n−¹₁ Pⁿ

k=1

xk. Mih´aly Bencze OQ. 3929. Ifx_k>0 (k= 1,2, ..., n),then

P_n

k=1

x_k Pn k=1

1 xk

≥n+ 2 +

(n−2)(n+1) Pn k=1

xⁿ_k⁻¹ P

cyclic

x1...xn−1 .

Mih´aly Bencze OQ. 3930. Ifxk>0 (k= 1,2, ..., n),then P

cyclic

(x2+...+xn)ⁿ⁻¹

xⁿ₁⁻¹+x2...xn ≥ ⁿ⁽ⁿ⁻₂¹⁾ⁿ⁻¹. Mih´aly Bencze OQ. 3931. Ifxk >0 (k= 1,2, ..., n),then

nⁿ⁻² Pn k=1

x⁽ⁿ⁻¹⁾²_k P_n

k=1

xⁿ⁻¹_k

n−1 +

P

cyclic

x1...xn−1

Pn k=1

xⁿ_k⁻¹ ≥2.

Mih´aly Bencze

OQ. 3932. Ifx_k>0 (k= 1,2, ..., n),then P

cyclic

x2+...+xn

(n−1)xⁿ⁻¹₁ +x2x3...xn ≥ _P^{n(nn −1)}

k=1

xⁿ_k⁻²

.

Mih´aly Bencze OQ. 3933. Ifx_k>0 (k= 1,2, ..., n),then

P

cyclic 1

xⁿ⁻¹₁ +x2...xn ≤

(n−1) P_n

k=1

xk P

cyclic 1 x1+...+xn−1

!

2 P

cyclic

x1...xn−1 .

Mih´aly Bencze OQ. 3935. Ifx_k>0 (k= 1,2, ..., n),then

n(n−1) P

cyclic x1

x2+...+xn ≤ P_n

k=1

x_k Pn k=1

1 xk

.

Mih´aly Bencze OQ. 3936. Ifx_k>0 (k= 1,2, ..., n),then P

cyclic x²₁ x2 ≥ ⁿ⁺¹

s nⁿ

P_n

k=1

xⁿ⁺¹_k

.

Mih´aly Bencze OQ. 3937. Ifxk>0 (k= 1,2, ..., n),then

Pn k=1

x²⁽ⁿ⁻¹⁾_k P

cyclic

x1x2...xn−1 +

n Qn k=1

xk

Pn k=1

xk

≥ ²_n Pⁿ

k=1

xⁿ_k⁻¹.

Mih´aly Bencze OQ. 3938. Ifx_k>0 (k= 1,2, ..., n),then determine alla, b >0 such that

a Pn k=1

xⁿ_k Pn k=1

xⁿ_k⁻¹

+

b P

cyclic

(x1x2)ⁿ⁻¹ Pn

k=1

x²ⁿ_k ⁻³ ≥ ^a+b_n Pⁿ

k=1

x_k.

Mih´aly Bencze OQ. 3939. Ifx_k>0 (k= 1,2, ..., n),then

Pn k=1

xⁿ_k Qn k=1

xk

+

2nⁿ Qn k=1

x_k P_n

k=1

xk

n ≥n+ 2.

OQ. 3940. Ifx_k>0 (k= 1,2, ..., n),then

P_n

k=1

x_k Q_n

k=1

x_k Pn

k=1

xⁿ⁺¹_k

+

(n+1)nⁿ⁻¹ Pn k=1

x

n(n−1) 2

P_n k

k=1

x_k P_n

k=1

x²_k

...

P_n

k=1

xⁿ_k⁻¹

≥n+ 2.

Mih´aly Bencze OQ. 3941.P Ifxk>0 (k= 1,2, ..., n),then

cyclic

1

(x1+x2+...+xn−1)ⁿ⁻¹ ≥ _(n−1)n−1 Pⁿ²

cyclic

x1x2...xn−1.

Mih´aly Bencze OQ. 3942. Ifak>0 (k= 1,2, ..., n),then

P _a1

a2+...+an +

Qn k=1

ak

(n−1) Pn k=1

aⁿ_k ≥ _n(n−1)ⁿ²⁺¹ .

Mih´aly Bencze OQ. 3943. Ifx_k>0 (k= 1,2, ..., n),then P

cyclic

x²₁+...x²_n−1 x1+...+x_n−1 ≤ ⁿ

Pn k=1

x²_k Pn k=1

xk

.

Mih´aly Bencze OQ. 3944. Ifx_k>0 (k= 1,2, ..., n),then

Pn k=1

xⁿ_k⁻¹ P

cyclic

x1x2...xn−1 +

(n−1)ⁿ Qⁿ

k=1

xk

Q

cyclic

(x1+...+xn−1) ≥2.

Mih´aly Bencze OQ. 3945. Ifp_k (k= 1,2, ..., n) are given prime, then solve inN the

equation Qn⁴ⁿ k=1

pk

=

3ⁿ

P

k=1 1 xk.

Mih´aly Bencze

OQ. 3946. If 0< x_k< π, p_k >0,(k= 1,2, ..., n), A_n=

Pn k=1

pkxk

Pn k=1

pk

,

G= Q_n

k=1

x^p_k^k Pⁿ¹

k=1pk

,then ^sin_G^GPn

k=1

pk

≤ Qⁿ

k=1

sinxk

xk

pk

≤ ^sin_A^APn

k=1

pk

. Mih´aly Bencze OQ. 3947. Solve in Qthe equation 2z⁴= x³+y³

x³+ 4y³ .

Mih´aly Bencze OQ. 3948. Solve in Qthe equationx³+nx+n+ 1 = (n+ 1)y² when n∈N is given.

Mih´aly Bencze OQ. 3949. Solve in Z the equationy⁶ = 2x⁴−1

3x⁴−1

5x⁴−1 . Mih´aly Bencze OQ. 3950. Solve in Z the equationa

Pn k=1

x²_k+b P

1≤i<j≤n

xixj =c when a, b, c∈Z are given.

Mih´aly Bencze OQ. 3951. Ifp is a given prime, then solve inZ the equationx²+p=yⁿ.

Mih´aly Bencze OQ. 3952. Solve in Z the equation

x⁴−2y⁴

z⁴−8t⁴

u³−3uv²−v³

= 1.

Mih´aly Bencze OQ. 3953. Solve in Z the equation x³+x²y+axy²+b=cwhen a, b, c∈Z are given.

Mih´aly Bencze

OQ. 3954. Determine all n∈Z for which the equation x⁴+y⁴+nz⁴ = 1 have infinitely many solutions inQ.

Mih´aly Bencze OQ. 3955. Solve in Qthe equation

Pn k=1

x^2p_k =pwhen p∈Z is given.

Mih´aly Bencze OQ. 3956. Ifa, b∈Z are given, then solve inZ the equation

Qn k=1

x²_k+a²

= ay²+b²n

.

Mih´aly Bencze OQ. 3957. Solve in Qthe equation

Pn k=1

x³_k =p,when p is a given prime.

Mih´aly Bencze OQ. 3958. Ifa, b, c, a_k, b_k, c_k∈Z (k= 1,2, ..., n) are given, then solve the equation

Qn k=1

a_kx²+b_ky²+c_kz²

= ax²+by²+cz²n

.

Mih´aly Bencze OQ. 3959. Ifak∈Z (k= 1,2, ..., n) are given, then solve in Q the equation

Pn k=1

akx^2p+1_k = 0.

Mih´aly Bencze OQ. 3960. Solve in Z the equation

Pn k=1

x^2p_k = Pn k=1

y_k^2p whenp∈Z.

Mih´aly Bencze OQ. 3961. Ifak, bk, ck, dk ∈Z (k= 1,2, ..., n) are given, then solve in Qthe equation xy

Qn k=1

akx²+bky²

=zt Qn k=1

ckz²+dkt² .

Mih´aly Bencze

OQ. 3962. Ifp_k∈Z (k= 1,2, ..., n) are given prime, then solve inZ the equation

Pn k=1

p_kx²_k=y³.

Mih´aly Bencze OQ. 3963. Solve in Z the equations

1). x²ⁿ+y²ⁿ=zⁿ 2). x²ⁿ−y²ⁿ=zⁿ

Mih´aly Bencze OQ. 3964. Solve in Z the equationx²ⁿ+y²ⁿ=nzⁿ.

Mih´aly Bencze OQ. 3965. Solve in Z the equationx²ⁿ− ²ⁿ_n

xⁿyⁿ+y²ⁿ=zⁿ.

Mih´aly Bencze OQ. 3966. Solve in Z the equationa bx²+cy²2

+z²= dt²+eu²2

wherea, b, c, d, e∈Z are given.

Mih´aly Bencze OQ. 3967. Ifb, a_k ∈Z (k= 1,2, ..., n) are given, then solve in Z the

equation Pn k=1

a_kx²_k =b Qn k=1

x_k.

Mih´aly Bencze OQ. 3966. Solve in Z the equation

Pn k=0

a_k ⁿ_k

x^n−ky^k=zⁿ wherea_k ∈Z (k= 1,2, ..., n) are given.

Mih´aly Bencze OQ. 3969. Ifp, q are prime, andk∈Z is given, then solve inZ the

equation x²+py² =k z²+qt² .

Mih´aly Bencze

OQ. 3970. LetABC (a6=b6=c) be a triangle. Compute St(a, b, c) = minP_a^t_+b^t

a^t−b^t,when t∈R.We have S1(a, b, c) = 5, S₂(a, b, c) = 3.

Mih´aly Bencze OQ. 3971. Compute S_k= P

1≤i<j≤n 1

(^j_i)^(j−i)^k.

Mih´aly Bencze OQ. 3972. IfSn^p = P

k≥0 n pk

then determine a relation betweenS_n¹, S_n², ..., Sn^p.

Mih´aly Bencze OQ. 3973. IfR^pn= P

k≥0 1

(_pkⁿ) then determine a relation between R_n¹, R²_n, ..., R^pn.

Mih´aly Bencze OQ. 3974. Ifak>0 (k= 1,2, ..., n) andλ≥t >0 then determine all tsuch that

Pn k=1

a^λ+1_k Pn k=1

a^λ_k ≥





Pn k=1

a^λ+1_k Pn k=1

a^λ+1_k ^−t





1 t

. (For t= 2 we have a solution).

Mih´aly Bencze OQ. 3975. Denotean the nearest integer of √^k

n (k≥2).Compute 1).

Pn i=1

a_i 2).

Pn i=1

1 ai

3). _a ¹

1a2...ai +_a ¹

2a3...ai+1 +...+_a ¹

na1...ai−1 wherei∈ {2,3, ..., n−1}.

Mih´aly Bencze OQ. 3976. Ifx_k>0 (k= 1,2, ..., n),then determine alln∈N^∗ for which

P

cyclic

xⁿ₁⁻¹−x2x3...xn

xⁿ₁⁻¹+ (n−1)x2x3...xn

_n−¹₁

≥0.