Open questions

Open questions

OQ. 3154. Ifx_k∈[0,1] (k= 1,2, ..., n), y_k∈[−1,1] (k= 1,2, ..., n) such that Pⁿ

k=1

1 +y_k Qⁿ

i=1

x_i m

≥1.

Mih´aly Bencze OQ. 3155. Ifα, x_k>0 (k= 1,2, ..., n) then

Qn k=1

x²_k+α

≥ ^(α+1)_n ⁿ P

cyclic

x₁x₂

! .

Mih´aly Bencze OQ. 3156. Ifx_k, a_k >0 (k= 1,2, ..., n) then ^√a1x1x2+a_x^{2x2x3+...+anxnx1}

2+...+xn + +^√a1x2x3+a_x^{2x3x4+...+anx1x2}

3+...+x1 +...+^√a1xnx1+a_x^{2x1x2+...+anxn−1xn}

1+...+xn−1 ≥ ^n√a1+a_n−^2+...+a₁ ⁿ. Mih´aly Bencze OQ. 3157. Ifx_k>0 (k= 1,2, ..., n), then

vu ut P

cyclic

xⁿ₁⁻¹x₂

! P

cyclic

x₁xⁿ₂⁻¹

!

≥(n−2) Qⁿ

k=1

x_k+ n

r Q

cyclic

(xⁿ₁ +x₁x₂...x_n).

Mih´aly Bencze and Zhao Changjian OQ. 3158. Ifx_i>0 (i= 1,2, ..., n),then

P

cyclic

x1x2...xk

xk+1(x1+xk+1)(x2+xk+1)...(xk−1+xk+1) ≥ P

cyclic

x1

x2+x3

k−1

,for all k∈ {2,3, ..., n−1}.

Mih´aly Bencze and Shanhe Wu OQ. 3159. Ifx_i>0 (i= 1,2, ..., n),then

P

cyclic

k

qx^k₁ +_x ¹

2+...+xk+1 ≥ ^k q

1 +^nk+1_k .

Mih´aly Bencze and Yu-Dong Wu

OQ. 3160. Ifxi>0 (i= 1,2, ..., n),then determine allα >0 such that the inequality P

1≤i1<...<ik≤n

x_i1x_i2...x_ik ≤α+ ⁿ_k

−αn^k Qⁿ

k=1

x_k is the best possible.

Mih´aly Bencze OQ. 3161. 1). If x, y >0 then determine alla, b >0 such that

1

(x+y)^a +_(x+2y)^{1 a} +...+_(x+ny)^{1 a} ≤ _(x(x+ny))ⁿ b.

2). If x >0, then determine alla, b, c >0 such that Pn

k=1 1

(1+kx)^a ≤n(1 +bnx)^−c.

Mih´aly Bencze OQ. 3162. In all triangleABC denoteN the Nagel point, H is orthocentre, O is circumcentre, I is incentre. Determine all x, y >0 such that

max{(N I)^x(HI)^y; (N I)^y(HI)^x} ≤(x+y) (OI)^x+y.

Mih´aly Bencze OQ. 3163. Ifx_k∈(−1,1) (k= 1,2, ..., n),then

1). Qn ¹ k=1

(1−xk)

+ Qn ¹ k=1

(1+xk) ≥2

2). If y_k, z_k ∈[−1,1] (k= 1,2, ..., n) such that Pⁿ

k=1

y_k= Pⁿ

k=1

z_k= 0,then

n 1 Q

k=1

(1+xkyk)

+ Qn ¹ k=1

(1+xkzk) ≥2.

Mih´aly Bencze OQ. 3164. Ifx_i>0 (i= 1,2, ..., n) andS_k = P

1≤i1<...<ik≤n

x_i1x_i2...x_ik,then

S1

(ⁿ₁) ≥

S2

(ⁿ₂) ¹₂

≥

S3

(ⁿ₃) ¹₃

≥...≥

Sn

(ⁿ_n) _n¹

.

Mih´aly Bencze OQ. 3165. In all triangleABC hold ^9R₂ 2

≤P_a²_b²

c²

Pr²a

a²

≤

9R² 4r

2

.

OQ. 3166. Ifxk>0 (k= 1,2, ..., n),then _n

P

k=1

x_k 2

P

cyclic

x₁x₂

!n−1

≤ ⁿⁿ⁺¹₃ⁿ Q

cyclic

x²₁+x₁x₂+x²₂ .

Mih´aly Bencze OQ. 3167. Ifx_k>0 (k= 1,2, ..., n) andS = Pⁿ

k=1

x_k,then 1). Pⁿ

k=1 xk

S−xk ≤

n Pⁿ

k=1

x²_k (n−1) P

cyclic

x1x2 2). Pⁿ

k=1

xk

S−xk

2

≥

nPⁿ

k=1

x²_k (n−1)² P

cyclic

x1x2

Mih´aly Bencze OQ. 3168. Ifa_,x_k>0 (k= 1,2, ..., n) and

A₁ = _a^{a1x1+a2x2+...+anxn}

nx1+a1x2+...+an−1xn, A₂ = _a^{a2x1+a3x2+...+a1xn}

1x1+a2x2+...+anxn, ..., A_n= _a^{anx1+a1x2+...+an−1xn}

n−1x1+anx2+...+an−2xn,then max{A1, A2, ..., An} ≥1≥min{A1, A2, ..., An}.

Mih´aly Bencze OQ. 3169. Ifx_k∈R (k= 1,2, ..., n) and A⊆ {1,2, ..., n},then

P

i∈A

x_i k

≤ P

1≤i1<...<ik≤n

(x_i1+...+x_i2)...(x_i1 +...+x_ik).

Mih´aly Bencze OQ. 3170. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic

x1(x1−x2)...(x1−xn)≥ ⁽ⁿ⁺¹⁾|^{(x1−x2)n−1(x2−x3)n−1...(xn−x1)n−1}|

Q

cyclic

(x1+x2) 0

@1−

˛˛

˛˛

˛ Q

cyclic x1−x2 x1+x2

˛˛

˛˛

˛

n−11 A

.

Mih´aly Bencze OQ. 3171. Ifx_k, p_k>0 (k= 1,2, ..., n) and

f(pi1, ..., pik;xi1, ..., xik) = ^pi1x_p^{i1+...+pikxik}

i1+...+p_ik ,then Pn

k=1

(−1)^k−1 P

1≤i1<...<ik≤n

lnf(p_i1, ..., p_ik;x_i1, ..., x_ik)≤0.

OQ. 3172. Ifx_k>0 (k= 1,2, ..., n) and Qⁿ

k=1

x_k= 1,then Pn

k=1

q

1 + (a²−1)x²_k≤aPⁿ

k=1

x_k for all a≥1.

Mih´aly Bencze OQ. 3173. Ifx_k>0 (k= 1,2, ..., n) then

2 P

cyclic

x1xⁿ₂⁻¹≥n Qn k=1

x_k+ P

cyclic

xⁿ₁⁻¹x2.

Mih´aly Bencze OQ. 3174. Letf :I →R (I ⊆R) be a convex function, and x_k∈I

(k= 1,2, ..., n). DenoteS_k= ⁿ_k−1 P

1≤i1<...<ik≤n

f_x

i1+...+x_ik k

,then

determine alla_i∈R (i=a,1, ..., m) such that P^m

i=0

(−1)ⁱa_iS_i+1 ≥0.

Mih´aly Bencze OQ. 3175. Ifx_k>0 (k= 1,2, ..., n) then P

cyclic

ax²₁+x2x3

(x1+x2)² ≥ ^n(a+1)₄ for all a >0.

Mih´aly Bencze OQ. 3176. Ifx_k>0 (k= 1,2, ..., n) and

Pn k=1

x_k= 1 then determine all a_k>0 (k= 1,2, ..., n) such that, the inequalities

Pn k=1

x²_k≥ P

cyclic

x₁ a₁+a₂x₂+...+a_nxⁿ₂⁻¹

≥x₁x₂+x₂x₃+...+x_nx₁ are the best possible.

Mih´aly Bencze OQ. 3177. Determine all y_k>0 (k= 1,2, ..., n) for which

Pn j=1

n1 P

i=1

x^yi_j ≥ _P^{n n}

i=1

(√ⁿx1x2...xn)^yi

for allx_k>0 (k= 1,2, ..., n).

Mih´aly Bencze

OQ. 3178. Ifxk∈[0,1] (k= 1,2, ..., n) then determine the minimum and the maximum of the expression

Q

cyclic

(1+x1+x1x2) Q

cyclic

(1+x2+x1x2).

Mih´aly Bencze OQ. 3179. Ifx_k>0 (k= 1,2, ..., n) andS = Pⁿ

k=1

x_k then Pn

k=1

(s−xk) xk(s+xk) ≥

(n−1)2ⁿ

„ _n P

k=1

xk

«n−1

n(n+2) Q

cyclic

(x1+x2) .

Mih´aly Bencze OQ. 3180. Ifx_k>0 (k= 1,2, ..., n) andS = Pⁿ

k=1

x_k, then Pn

k=1 xk

xⁿ_k+(s−xk)ⁿ ≥

Pn k=1

xk

Pn k=1

x³_k+n! P

1≤i<j<k≤n

xixjxk

.

Mih´aly Bencze OQ. 3181. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k= 1 then P

cyclic x1

√1−x2 ≥√ 2 Pⁿ

k=1

√x_k−ⁿ⁺²_n+1.

Mih´aly Bencze

OQ. 3182. Ifx_k>0 (k= 1,2, ..., n) andα≥1, then

s n

Pn k=1

x^2α_k

Pn k=1

x^α_k⁻¹ ≥

Pn k=1

x^α+1_k Pn k=1

x^α_k

.

Mih´aly Bencze OQ. 3183. Ifx_k>0 (k= 1,2, ..., n) then determine alla, b, c >0 such that

P

cyclic x1

(x1+ax2)^b

!¹_b

≥P

x1x2(^xc1+x^c₂)

2

_c+2¹ .

OQ. 3184. If 0< α≤1 andx_k>0 (k= 1,2, ..., n), S= Pⁿ

k=1

x_k, then Pn

k=1

xk

S−xk

α

≥ _(n−ⁿ₁₎^α.

Mih´aly Bencze OQ. 3185. Ifx_k>0 (k= 1,2, ..., n) then determine allα >0 such that

Pn k=1

1

xk ≥ P

cyclic x²₁−αx²₂ x³₁+αx³₂.

Mih´aly Bencze OQ. 3186. Determine all A_k∈M_m(C) (k= 1,2, ..., n) such that

Aⁿ₁⁻¹=A₂A₃...A_n, Aⁿ₂⁻¹=A₁A₃...A_n, ..., Aⁿ_n⁻¹=A₁A₂...A_n−1 for all n, m≥3.

Ifn is odd thenA₁ =A, A₂=εA, A₃ =ε²A, ..., A_n=εⁿ⁻¹A,where ε= cos^2π_n +isin^2π_n is a solution.

Mih´aly Bencze OQ. 3187. Determine all A, B∈M_n(C) for which

rang(AB)−rang(BA) =_n

2

−1,where [·] denote the integer part.

Mih´aly Bencze OQ. 3188. Determine all A, B∈M_n(R) for which

det A^2k+A^2k−1B+...+AB^2k−1+B^2k

≥0 for allk≥1.We have the following result, ifAB=BAthen the affirmation is true.

Mih´aly Bencze OQ. 3189. Ifx_k>0 (k= 1,2, ..., n) then P

cyclic x1

x2 +α

P

cyclic

x1x2

„ _n P

k=1

xk

«₂ ≥n+ ^α_n, whereα >0.

Mih´aly Bencze

OQ. 3190. Ifxk>0 (k= 1,2, ..., n) then P

cyclic

xⁿ₁⁻¹

xⁿ₂+xⁿ₃+x1x2...xn ≥ ⁿ vu ut

Pn k=1

xⁿ_k P

cyclic

xⁿ₁xⁿ₂.

Mih´aly Bencze OQ. 3191. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k=a, Pⁿ

k=1

x²_k=b >1,then 1 + Q^bn⁻¹

k=1

xk

≥ _a¹ P

cyclic x1

x2.

Mih´aly Bencze OQ. 3192. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

√x_k= 1,then P

cyclic

x^r₁+x2x3...xr+1

x1√rx2+...+xr+1 ≥ _n2r^2√−1^rn√²rr,for all r∈ {2,3, ..., n−1}.

Mih´aly Bencze OQ. 3193. Determine all A_k∈M_n(R) (k= 1,2, ..., n) such that

det _m

P

k=1

A^t_kA_k

= 0.

Mih´aly Bencze OQ. 3194. Ifx_k>0 (k= 1,2, ..., n) then

min





Pn k=1

x³_k P

cyclic

x²₁x2;

Pn k=1

x³_k P

cyclic

x1x²₂



+n−1≥ P

cyclic x1+x2

x2+x3.

Mih´aly Bencze OQ. 3195. LetA₁A₂...A_2nA_2n+1 be a convex polygon andB₁, B₂, ..., B_2n+1 are on the sides An+1An+2, An+2An+3, ..., A2nA2n+1, ..., AnAn+1 such that A₁B₁, A₂B₂, ..., A_2n+1B_2n+1 are ceviens of rank p in triangles

A_n+1A₁A_n+2, A_n+2A₂A_n+3, ..., A_2n+1A_nA₁, ..., A_nA_2n+1A_n+1.

Prove that if A₁B₁, A₂B₂, ..., A_2nB_2n are concurent in pointM, then M ∈A_2n+1B_2n+1.

OQ. 3196. If−1≤xk ≤1 (k= 1,2, ..., n) then determine the best constantsm, M >0 such thatm≤ P

cyclic|f(x₁) +f(x₂)−2f(x₃)| ≤M, when f(x) = 4x³−3x+ 1.

Mih´aly Bencze OQ. 3197. DenoteR₁, R₂, R₃ the distances from an arbitrary pointM to the verticesA, B, C of the triangle ABC.

Prove aR²₁+bR₂²+cR²₃

bR²₁+cR²₂+aR²₃

cR²₁+aR²₂+bR²₃

≥

≥ ^abc(a³b+b³c+c³a)(a³c+b³a+c³b)

(a+b+c)² .Can be strongened this inequality?

Mih´aly Bencze OQ. 3198. Determine all x_k>0 (k= 1,2, ..., n) for which from

Pn k=1

x_k> P

cyclic x1

x2 holds Pⁿ

k=1

x_k < P

cyclic x2

x1.

Mih´aly Bencze OQ. 3199. Ifx_k∈(0,1)∪(1,+∞) (k= 1,2, ..., n), then determine all a, b >0 such that P

cyclic x^a₁

(x2−1)^2b ≥1.

Mih´aly Bencze OQ. 3200. Ifx_k>0 (k= 1,2, ..., n) then

2 P

cyclic

px²₁−x₁x₂+x²₂ ≥ P

cyclic

px²₁+x₁x₂+x²₂.

Mih´aly Bencze OQ. 3201. Ifx_k∈R (k= 1,2, ..., n) and Pⁿ

k=1

x_k=a, Pⁿ

k=1

x²_k=b, then determine the best m_r, M_r∈R (r = 1,2) such that m₁ ≤ P

cyclic

x²₁x₂ ≤M₁ and m₂ ≤ P

cyclic

x₁x²₂ ≤M₂.

Mih´aly Bencze

OQ. 3202. Ifxk >0 (k= 1,2, ..., n) then determine the best m, M >0 such thatm≤ P

cyclic

x1

√r

x^r₁+(a^r−1)x2x3...xrxr+1 ≤M where r∈ {2,3, ..., n−1} and a≥2.

Mih´aly Bencze OQ. 3203. Ifx_k>0 (k= 1,2, ..., n) and

Pn k=1

x_k= 1,then determine max P

cyclic

(x1x2...xp)^α

1−(x2x3...xp+1)^β,where α, β >0 and p∈ {2,3, ..., n−1}.

Mih´aly Bencze OQ. 3204. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic

x₁x₂...x_p

! P

cyclic x1

x^p₂+x3

!

≥ _n^p−1ⁿ+1,for all p∈ {2,3, ..., n−1}.

Mih´aly Bencze OQ. 3205. LetABC be a triangle, then determine

max P

cyclic

(sinA)^b(sinB)^c(sinC)^a.

Mih´aly Bencze OQ. 3206. Ifx_k>0 (k= 1,2, ..., n), Pⁿ

k=1

x_k = 1,then determine the maximal constantα >0 such that P

cyclic

q

x₁+α(x₂−x₃)²+ 2 Pn k=1

√x_k≤3n.

Mih´aly Bencze OQ. 3207. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x²_k= 1,then 1≤ P

cyclic x1

1+x2x3...xn ≤ (^√n)ⁿ (^√n)ⁿ⁻¹⁺¹.

Mih´aly Bencze

OQ. 3208. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k= 1,then Qn

k=1

x_k+ ^{P α}

cyclic

x1x2...xp ≥1 +^α_n for all α >0 and allp∈ {2,3, ..., n−1}. Mih´aly Bencze OQ. 3209. Ifx_k>0 (k= 1,2, ..., n) andS =

Pn k=1

x_k, then Pn k=1

x^S_k^−xk ≥1.

Mih´aly Bencze OQ. 3210. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic x1

x2 ≥ _n+1^{n2 3} vu ut

Pn k=1

x³_k P

cyclic

x1x2x3 +^3n2+3n+1_n3 .

Mih´aly Bencze OQ. 3211. Iff :I →R (I ⊆R) is a convex function,x_k ∈I (k= 1,2, ..., n) such thatx₁≤x₂ ≤...≤x_n, then determine all y_k∈R (k= 1,2, ..., n)

Pn k=1

yk= 1 such thatf _n

P

k=1

xkyk

≤ Pⁿ

k=1

ykf(xk).

Mih´aly Bencze OQ. 3212. Ifx_k>0 (k= 1,2, ..., n) and

Qn k=1

x_k=Pⁿ,then P

cyclic 1+x1x2

1+x1 ≥ ^n(P_Pⁿ₊₁⁺¹⁾.

Mih´aly Bencze OQ. 3213. LetABC be a triangle. Determine all x, y, z >0 andn∈N such thatQ

(xaⁿ+ybⁿ+zcⁿ)≥((y^x+z^x)sr)ⁿ⁻¹.

Mih´aly Bencze OQ. 3214. Iff : [0,1]→(0,+∞) is a concave function, then determine all s, r∈R such that

(s+ 1)

R1 0

f^s(x)dx ^r

≤

(r+ 1) R1 0

f^r(x)dx ^s

.

OQ. 3215. Ifxk>0 (k= 1,2, ..., n) and P

cyclic

x1x2...xn−1 =n, then P

cyclic

x1

nxⁿ₁+x2x3...xn ≥ Qⁿ

k=1

x_k.

Mih´aly Bencze OQ. 3216. Ifx_k>0 (k= 1,2, ..., n) then

P

cyclic

x₁+^x

n−1 2

x3

n−1

≥

n(n−1)² Pⁿ

k=1

xⁿ_k Pn k=1

xk

.

Mih´aly Bencze and Zhao Changjian OQ. 3217. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x^a_k= 1,where a∈R, then find the minimum value of

Pn k=1

x^b_k

1−x^c_k,when b, c∈R.

Mih´aly Bencze OQ. 3218. If 0< xk<1 (k= 1,2, ..., n), then determine allf :Rⁿ→R for which Pⁿ

k=1 1

1−xk ≥ _1−f(x1,xⁿ_2,...,xn) ≥ ⁿ

1−_n¹ Pⁿ

k=1

xk

.

Mih´aly Bencze OQ. 3219. Ifx_k>0 (k= 1,2, ..., n) then

n−1

„ _n P

k=1

xk

« P

cyclic

x1x2

! ≥ _n ^P1

cyclic

x²₁x2 + _n ^P1

cyclic

x1x²₂.

Mih´aly Bencze OQ. 3220. Ify, x_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x^α_k =y^α,whereα≥1, then Pn

k=1

x^α_k⁻¹−y^α−1 ≥n(α−1) Qⁿ

k=1

(y−x_k).

Mih´aly Bencze

OQ. 3221. Ifx_k, y_k>0 (k= 1,2, ..., n) andS = Pⁿ

k=1

y_k,then Pn

k=1

(S−y_k)x_k ≥(n−1) vu ut P

cyclic

x1x2

! P

cyclic

y1y2

! .

Mih´aly Bencze OQ. 3222. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k= 1 and α≥1, then

n≤ P

cyclic x^α₁+1

x^α₂+1 ≤n+_n−¹₁.

Mih´aly Bencze OQ. 3223. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k= 1 then find the minimum and the maximum value of the expression P

cyclic

(x₁x₂...x_n−1)^α when α≥1 is given.

Mih´aly Bencze OQ. 3224. LetA₁A₂...A_n be a convex polygon inscribed in the unit circle.

IfM ∈Int(A₁A₂...A_n),then Qⁿ

k=1

M A_k≤1 +_n¹₂ + _n²n.

Mih´aly Bencze OQ. 3225. IfP0(x) = 0, Pn+1(x) =Pn(x) +^x−P_k^nk(x) for all n∈N^∗, where k∈N, k≥2 is given. Prove that, for alln∈N holds the inequalities 0≤ √^k

x−P_n(x)≤ _n+1^k ,whenx∈[0,1].

Mih´aly Bencze OQ. 3226. Ifx_k>0 (k= 1,2, ..., n) andP = Qⁿ

k=1

x_k, Pⁿ

k=1

x_k= 1 then Pn

k=1 xk

x²_k+P+1 ≤ _nⁿ_+nⁿⁿⁿ−2+1.

Mih´aly Bencze

OQ. 3227. Ifx_k>0 (k= 1,2, ..., n) and Pⁿ

k=1

x_k=nthen determine all p, q∈N^∗ such that Pⁿ

k=1 1 x^p_k ≥ Pⁿ

k=1

x^q_k.

Mih´aly Bencze OQ. 3228. Ifx_k, y_k>0 (k= 1,2, ..., n) then Pn¹

k=1 1 xk

+ Pn¹ k=1 1 yk

≤ _P^{n 1}

k=1 1 xk+

Pn k=1

1 yk

and more general, if x_ij >0 (i= 1,2, ..., n;j= 1,2, ..., m),then Pm

j=1

n1 P

i=1 1 xij

≤ _P^{m 1}

j=1 n1 P i=1xij

.

Mih´aly Bencze and Zhao Changjian.

OQ. 3229. Let be a1 = 1,16an+1 = 1 + 4an+√

1 + 24an for all n≥1.

Determine all n∈N^∗ for which a_n is prime.

Mih´aly Bencze OQ. 3230. The equationx²+ 4xy+y²= 1 have infinitely many solution in Z, because the sequencesx_n+1 =x²_n−y_n², y_n+1 = 2x_ny_n+ 4y²_n,

x1 = 1, y1=−4 offer infinitely many solution in Z.

1). Determine all solution in Z 2). Determine all solutution inQ Mih´aly Bencze OQ. 3231. Let be f(x) = Qⁿ

k=1

ln(akx+bk)

ln(ckx+dk),wherex >0

1). Determine all a_k, b_k, c_k, d_k>0 (k= 1,2, ..., n) for which f is increasing (decreasing)

2). Determine all a_k, b_k, c_k, d_k>0 (k= 1,2, ..., n) for which f is convex (concav)

Mih´aly Bencze

OQ. 3232. Let be f(x) = Qⁿ

k=1

sin (a_kx+b_k)

1). Determine all a_k, b_k∈R (k= 1,2, ..., n) and x∈R for which f is increasing (decreasing)

2). Determine all a_k, b_k∈R (k= 1,2, ..., n) and x∈R for which f is convex (concav)

Mih´aly Bencze OQ. 3233. 1). If Hn= 1 + ¹₂+...+_n¹,then 1^Hn+ 2^Hn+...+k^Hn ≤ ^n+k_n+1 for all k∈N^∗

2). Determine the best constants 0< a < b≤1 such that a ^n+k_n+1

≤1^Hn+ 2^Hn+...+k^Hn ≤b ^n+k_n+1

3). Determine the assymptotical expansion of the sum 1^Hn+ 2^Hn+...+k^Hn Mih´aly Bencze OQ. 3234. LetABC be a triangle. Determine all x, y, z >0 such that sin^A_x sin^B_y sin^C_z ≤

3 x+y+z

^x+y+z₂ .

Mih´aly Bencze OQ. 3235. LetABC be a triangle. Determine all x_k, y_k, z_k>0 (k= 1,2,3) such thatx1sin_y^A

1 +x2sin_y^B

2 +x3sin_y^C

3 ≤ ³(^x1y²₁+x2y²₂+x3y²₃)

4(y1+y2+y3) .

Mih´aly Bencze OQ. 3236. 1). Prove that

1

2ln 3 (n+ 2) lnⁿ⁺²₃ <ⁿ⁺¹P

k=3 lnk

k < ¹₂ln 3 (n+ 1) lnⁿ⁺¹₃ +¹₃ln 3 2). Determine the best constants a, b, c, d >0 such that

Pn k=1

lnx

k =alnbnlncn+d+O(n).

Mih´aly Bencze OQ. 3237. 1). If x≥y >0 then (x+ 1)^x−x1 y^y ≥(y+ 1)^y−1yx^x

2). Determine all a_k, b_k, c_k>0 and d_k∈R (k= 1,2) for which (a₁x+b₁)^c1x+dx1 y^y ≥(a₂y+b₂)^c2y+dy2 x^x for all x≥y >0.

OQ. 3238. 1). Ifx≥y >0, then 1 +_x¹x

+ (1 +y)^1y ≥

1 +¹_yy

+ (1 +x)^1x 2). Determine all a_k, b_k, c_k, d_k>0 (k= 1,2) such that

a1+^b_x¹x

+ (c1+d1y)^y1 ≥

a2+^b_y²y

+ (c2+d2x)^x1 for allx≥y >0.

Mih´aly Bencze OQ. 3239. Suppose that A₁, A₂, ..., A_n are the vertices of a simplex S. On the faces opposite to A₁, A₂, ..., A_n−1,construct simplex outsideS with apexes B1, B2, ..., Bn−1 and volumesV1, V2, ..., Vn−1,respectively.

LetB_n be the point such thatA₁B_n=BA_n, where B is the point of intersection of the planes through B_i parallel to the respective bases (i= 1,2, ..., n−1).LetVn be the volume of the simplexA1A2...An−1Bn. Prove that V_n=V₁+V₂+...+V_n−1.

Mih´aly Bencze OQ. 3240. IfM, N, K are the mid-points of sides BC, CA, ABin triangle ABC, then 1≥Q

cos^A−₂^B ≥sin (AM B) sin (BN C) sin (CKA)≥ ^2r_R3

,a refinement of Euler’s inequality.

Determine all M ∈BC, N ∈CA, K ∈AB such that 1≥Q

cos^A−₂^B ≥sin (AM B) sin (BN C) sin (CKA)≥ ^2r_R3

.

Mih´aly Bencze OQ. 3241. LetABC be a triangle,A₁∈(BC), B₁ ∈(CA), C₁∈(AB) such thatAA₁∩BB₁∩CC₁={M}

1). Determine all pointsM for which P ₁

√M A+√ M B−√

M C ≥ ^3PP√_{M A}^{M A}. I have obtainedM ≡G.

2). Determine all pointsM for which P ₁

M A^λ+M B^λ−M C^λ ≥ _a^λ_+b^9λ_+c^λ,where λ∈[0,1].

Mih´aly Bencze OQ. 3242. LetABC be a triangle and M ∈Int(ABC),such that

M AB∡+M BC∡+M CA∡= 90^◦.Determine allM for which the triangle is isoscele.

Mih´aly Bencze

OQ. 3243. Determine all a, b, c, d, e∈Z such that Pn

j=0

Pn k=1

(−1)^{j+k an}_{bj cn}

dk+e

= 0

Mih´aly Bencze OQ. 3244. LetABC be a triangle. Determine all x_k, y_k, z_k>0 (k= 1,2,3) such thatx₁cos_y^A

1 +x₂cos_y^A

2 +x₃cos_y^A

3 ≤ ¹₂+ _x ^x1+x2+x3

1y1+x2y2+x3y3.

Mih´aly Bencze OQ. 3245. LetABC be a triangle. Determine all x_k, y_k, z_k>0 (k= 1,2,3) such thatx₁cos_y^A

1 +x₂cos_y^A

2 +x₃cos_y^A

3 ≤ _y1+y³_2+y3 ^x1y1+x2₂^y2+x3y33₂ . Mih´aly Bencze OQ. 3246. Solve in Z the following equation

x₁+x²₂+x³₃+...+xⁿ_n

x₂+x²₃+x³₄+...+xⁿ₁

... x_n+x²₁+x³₂+...+xⁿ_n−1

= (x₁+x₂+...+x_n)^n(n+1)2 .

Mih´aly Bencze OQ. 3247. LetABC be a triangle, and denoteA the areea of the triangle.

Determine the best constants 1≤x < y≤3, such that xP

(a−b)² ≤P

a²−4A√

3≤yP

(a−b)².

Mih´aly Bencze OQ. 3248. Determine all n∈N for which Φ (n) divides σ(n) + Ψ (n).

Mih´aly Bencze

OQ. 3249. Solve in Z the equation Pn k=1

a_k b_k c_k d_k

−b_k a_k −d_k c_k

−c_k d_k a_k −b_k

−d_k −c_k b_k a_k

=xⁿ, wheren∈N, n≥2.

Mih´aly Bencze

OQ. 3250. LetA1A2...An+1 be a concyclic (n+ 1)−gon, denote Ωk the anticentres of A₁A₂...A_k−1A_k+1...A_n+1(k= 1,2, ..., n+ 1).Prove that Ω₁Ω₂...Ω_n+1 is concyclic.

Mih´aly Bencze OQ. 3251. Determine all functionsf :R →R for which from

Pn k=1

f(α_kx)≥ Pⁿ

k=1

f(β_kx) when α_k, β_k∈R (k= 1,2, ..., n) for x∈(−ε, ε), ε >0,implies Pⁿ

k=1

α_k≥ Pⁿ

k=1

β_k.

Mih´aly Bencze OQ. 3252. DenoteQ3 the set of important points of a triangle

(H, G, O, I etc)

1). Let ABCD be a concyclic quadrilateral and denote M_A, M_B, M_C, M_D ∈Q₃ the important points of triangles

BCD, CDA, DAB, ABC.Determine all M_A, M_B, M_C, M_D ∈Q₃ for which the quadrilaterals MAMBMCMD are concyclic

2). Prove thatcardQ₃ ≥2.We can show thatH, G∈Q₃ satisfys the point 1). From Sylvester’s theorem we have OHA=OB+OC+OD and his permutations. From others we have OA+OB+OC+OD=OH_A+OA=

=OH_B+OB+OH_C +OC=OH_D+OD=QT .FromOH_A+OA=OT we have T H_A=AO,O is the circumcenter ofABCD, etc.

We have T H_A=OA=T H_B=OB=T H_C =OC=T H_D =OD,which means that H_AH_BH_CH_D is concyclic. FromH_AG_A= 2G_AO holds that G_AG_BG_CG_D is concyclic. FinallycardQ₃ ≥2.

3). DenoteQ_n the set of important points of the convex n-gon.

LetA₁A₂...A_n+1 be a concyclic convex (n+ 1)−gon. Denote M_k∈Q_n (k= 1,2, ..., n+ 1) the important points of A₁A₂...A_k−1A_k+1...A_n+1 (k= 1,2, ..., n+ 1).Determine all M_k∈Q_n (k= 1,2, ..., n) for which A₁A₂...A_k−1A_k+1...A_n+1 (k= 1,2, ..., n) are concyclic.

Mih´aly Bencze OQ. 3253. DenoteA_2k the denominator of Bernoullli’s numberB_2k.

1). Compute P^∞

k=1 1 A_2k

2). Compute P^∞ ₁

A²

3). More general determine P^∞

k=1 1 A^α_2k

4). How many prime exist betweenA_2k and A_2k+2? 5). Determine all k∈N for which A_2k is prime 6). Determine all n∈N for which Pⁿ

k=1

A_2k is prime

Mih´aly Bencze OQ. 3254. Let be A₁A₂...A_n a convex polygon with sides a_k

(k= 1,2, ..., n).Prov that 1). _min(a ^a1

2,a3,...,an)+_min(a ^a2

1,a3,...,an)+...+_min(a ^an

1,a2,...,an−1) ≥n 2). _max(a^a1

2,a3,...,an) +_max(a ^a2

1,a3,...,an)+...+_max(a ^an

1,a2,...,an−1) ≥n

Mih´aly Bencze OQ. 3255. DenoteBnthe n−th Bernoulli’s number. Determine alln for which k+P^k

i=1

n_iB_ni−1≡0 (modn) if and only ifnis prime, when n= P^k

i=1

n_i. Mih´aly Bencze OQ. 3256. Determine allnfor which n!B₁B₂...B_n−1+ (−1)ⁿ≡0 (modn) if and only if nis prime.

Mih´aly Bencze OQ. 3257. Ifx, y, z >0 andλ∈[1,2],then P

cyclic (y+z)²

x²+λyz ≤ _λ+1¹² .

Mih´aly Bencze OQ. 3258. Solve in Z the following equation

x₁a³+x₂b³+x₃c³− xa²+yb²+zc²

(xa+yb+zc) =

=y₁(a+b) (a−b)²+y₂(b+c) (b−c)²+y₃(c+a) (c−a)².

Mih´aly Bencze OQ. 3259. Compute the following sums:

1). F = _F¹

1 + _F¹

1F2 +...+_F ¹

1F2...Fn +..., whenF_k denote the k-th Fibonacci number

2). P = _p¹

1 +_p¹

1p2 +...+_p ¹

1p2...pn +..., when p_k denote thek-th prime number 3). If e = 1 + ¹n

, then computeE = ¹ + ¹ +...+ ¹ +...