OQ4867. Let ABC be a triangle and AA 1 BB 1 CC 1 = {M} where A 1 BC, B 1 CA, C 1 AB. Determine all points M for which ana 1...

Transcription

1 764 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 Open questions OQ4867. Let ABC be a triangle and AA BB CC = {M} where A BC, B CA, C AB. Determine all points M for which 4 s 2 3r 2 2Rr AA 2 +BB2 +CC2 AA BB BB CC CC AA. OQ4868. Solve in Z the equation n x i x i2...x ik (n + 2) ks x k p =. i <...<i k n p= OQ4869. Let a, a 2,..., a n be digits in the base x. Determine all n N and all x N for which ana...a n a n...a a n n. OQ4870. Compute a a 2...a n a na n...a + a 2a 3...a na a a n...a 2 k= (+d(k))(2+d(k+))...(n+d(k+n )). and Stanciu Nicolae OQ487. Compute k= (+σ(k))(+σ(k+))...(+σ(k+n)). OQ4872. If x (0, ) then compute λ p = lim when p N. OQ4873. Find the sum S k = integer part. We have S = d (n). i <i 2 <...<i k n x ip +ip ip k n i <i 2 <...<i k n h i n k i i 2...i k when [ ] denote the

2 OQ4874. If a i = n n j= k= s nq m a ij i= Open Questions 765 n a ij where a ij > (i =, 2,..., m; j =, 2,..., n), then j= s n mq m a i i= OQ4875. If a k > 0 (k =, 2,..., n), then n n a n k n n Q a n +a 2 a 3...a n Q 2 a 2 +a a n n cyclic. k= a k for all n 3. OQ4876. Let A A 2...A 2n, B B 2...B 2n, C C 2...C 2n convex polygon inscribed in same circle. Diagonals A A n+, A 2 A n+2,..., A n A 2n, B B n+, B 2 B n+2,..., B n B 2n, C C n+, C 2 C n+2,..., C n C 2n intersect at points, Q, R. Determine all polygons such that: ). The triangle QR is equilateral 2). The triangle QR is isoscelle 3). The triangle QR is rightangled OQ4877. Solve in Q the equation x 2 y 3 z 4 = 2x 2 + 3y 3 + 4z 4. OQ4878. Solve in Z the following system: x a xa 2 x a 3 2 xa 4 2 = b 3 = b 2 x a 2n n x a 2n = b n OQ4879. Solve in N the inequation σ 2 (n) ϕ (n) 207 n. OQ4880. Solve in N the inequation σ 3 (n) ϕ 2 (n) 206 n..

3 766 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ488. Consider two n-convex gons that intersect exactly in 2n points. rove that the common area of these n-gons is greather than or equal to half of the area of one of them. OQ4882. If x k R (k =, 2,..., n) and n x 2 k k= = then find all x R (k =, 2,..., n) which maximizes min ( x x 2, x 2 x 3,..., x n x ). OQ4883. Let A A 2...A n be a regular polygon and let M Int (A A 2...A n ). rove that sin (A MA 2 ) > 2π n n 2 + cos π n. OQ4884. Compute n k=0 ( ) k d (k) σ n(n+)(n+2) 3 k(k+)(k+2) 3. OQ4885. If a ij > 0 (i =, 2,..., n; j =, 2,..., m), λ R and n M λ (x, x 2,..., x n ) = n x λ λ i then determine all λ R for which i=! m a j, m a 2j,..., m a nj m M λ (a j, a 2j,..., a nj ). M λ j= j= j= OQ4886. If a ij > 0 (i =, 2,..., m; j =, 2,..., n), then study the monotonicity and the convexity of the function f (x) = n m! x! x x m n a ij. j= a x ij i= i= j= j= OQ4887. If a ij > 0 (i =, 2,..., n; j =, 2,..., m), p i > 0 (i =, 2,..., n) and H (x, x 2,..., x n ) = n p i i= n p i x i= i then

4 H m j= Open Questions 767! a j, m a 2j,..., m a nj m H (a j, a 2j,..., a nj ). j= j= j= OQ4888. If a ij > 0 (i =, 2,..., n; j =, 2,..., m) p i > 0 (i =, 2,..., n) and nq n A (x, x 2,..., x n ) = x p p i i i= i. rove that i=! G m a j, m a 2j,..., m a nj m G (a j, a 2j,..., a nj ). j= j= j= j= OQ4889. If a ij > 0 (i =, 2,..., n; j =, 2,..., m) p i > 0 (i =, 2,..., n) and A (x, x 2,..., x n ) = A m j= n p i x i i= n p i i=,then! a j, m a 2j,..., m a nj m A (a j, a 2j,..., a nj ). j= j= j= OQ4890. If a ij > 0 (i =, 2,..., n; j =, 2,..., m) p i > 0 (i =, 2,..., n) and A (x, x 2,..., x n ) = A n Q j= a j, nq a 2j,..., j= n p i x i i= n p i i=,then! nq Q a nj m A (a j, a 2j,..., a nj ). j= j= OQ489. If a ij > 0 (i =, 2,..., n; j =, 2,..., m) p i > 0 (i =, 2,..., n) and nq n G (x, x 2,..., x n ) = x p p i i i= i, then i=! Q G n nq nq Q a j, a 2j,..., a nj m G (a j, a 2j,..., a nj ). j= j= j= j=

5 768 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4892. If a ij > 0 (i =, 2,..., n; j =, 2,..., m) p i > 0 (i =, 2,..., n) and H (x, x 2,..., x n ) = H n Q j= a j, n p i i= n i= nq a 2j,..., j=,then p i x i! nq a nj j= Q m H (a j, a 2j,..., a nj ). j= OQ4893. If a k > 0 (k =, 2,..., n), A n = n v u 0 then v 2 +A n u 2 +A n n nq k= v 2 +a k u 2 +a k n a k, H n = k= v 2 +H n u 2 +H n n. n n a k= k and OQ4894. Solve in Z the equation n x x +x x k k = n (x + x x k ) x k. k= k= OQ4895. If A n (x, x 2,..., x n ) = n n n s n nq x k, G n (x, x 2,..., x n ) = n x k, k= H n (x, x 2,..., x n ) =, K n (x, x 2,..., x n ) = x 2 k then x k= k= k A n (x, x 2,..., x n ) + G n (x, x 2,..., x n ) H n (x, x 2,..., x n ) + K (x, x 2,..., x n ). OQ4896. If a k > 0 (k =, 2,..., n) then n a 2 k 2 a a 2 ln a ln a 2 a a 2. k= cyclic cyclic s n n k= OQ4897. Let ABC be a triangle and AA, BB, CC his cevians, A (BC), B (CA), C (AB) and AA BB CC = {M}. Denote ω the Brocard angle of the triangle ABC.

6 Open Questions 769 Determine all cevians AA, BB, CC for which 3 2 sin ω AA BB + BB CC + CC AA OQ4898. Determine all a k {0,,..., 9} (k =, 2,..., n) for which a a 2...a n a n a n...a 2 a = a 2n + a2n a2n n. OQ4899. Determine all n, k N such that all divisors a of n and all divisors b of k holds that a + b is divisor of n + k. OQ4900. Determine all a k {0,,..., 9} (k =, 2,..., n) for which a a 2...a n = a n + an an n. OQ490. Determine all a k, b k {0,,..., 9} (k =, 2,..., n) for which a a 2...a n b b 2...b n = (a n + an an n) (b n + bn bn n). OQ4902. Determine all convex polygons A A 2...A 3n for which A A A 3A A 3n A 2 = A 2A A 4A A 3nA 2 2. OQ4903. Determine all numbers for which (a a 2...a n ) k = b b 2...b kn and a a 2...a n = b b 2...b n + b n+...b 2n b kn k+..b kn. OQ4904. Determine all p, p 2, p 3,..., p k prime numbers which dividet the number n N and for which p p 2...p k + dividet n k +. OQ4905. Denote p k the k th prime number. Solve in N the equation nq p r k k = m r. k=

7 770 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4906. ). Solve in N the equation (n!) k + = m p. If k =, p = 2 then (n, m) are the Brown numbers. 2). Solve in N the equation ((n!)!...)! + = m p {z } k time OQ4907. Determine all polindrome numbers a a 2...a 2n for which a a 2...a n and a n+ a n+2...a 2n are polindrome numbers too. OQ4908. Let ABC be a triangle. Determine the best constants a k, b k, c k R (k =, 2, 3, 4) for which a R 2 + b Rr + c r 2 a 2 R + b 2 r R + r 3 c 2 R 2 r p q 3 (a3 R 2 + b 3 Rr + c 3 r 2 ) r 3 (a 4 R + b 4 r) 2 r 3 c 4 s 2 r 3r. OQ4909. A convex quadrilateral ABCD is extremal if and only if a+b 2 = 2bc b+c = cd. rove that ). a + ab + b c + cd + d 2). (a + b c) (a + b) = d (b + c) + bc (d b) 3). Determine all properties of the extremal quadrilateral OQ490. Let be A, B M n n (C), A = (A A) 2, the singular values of A, i.e., the eigenvalues of the operator A, enumerated in decreasing order, will be denoted by S j (A), j =, 2,..., n, then ). (x + y) S j (AB ) S j (xa A + yb B) for all (j =, 2,..., n) and x, y > 0 A K 2). If C M n n (C) such that K 0, then B xa K (x + y) S j (C) S j K for all j =, 2,..., n and x, y > 0 yb m 3). If A i M n (C) (i =, 2,..., m) x i > 0 (i =, 2,..., m), x i =, then i= S j A x Ax Axm m + A x 2 Ax Ax m A xm Ax 2...Ax m m S j (A + A A m ) for all j =, 2,..., n.

8 Open Questions 77 OQ49. Denote F k and L k the k th Fibonacci respective Lucas numbers. Compute: ). A n (r, s, t) = n h F r i k+2 +L s k 2). B n (r, s, t) = n 3). C n (r, s, t) = n F t k= k+ h L r k+2 +Fk s i L t k= k+ h F r k+2 +L s k k= 4). D n (r, s, t) = n k= L r k+2 +F s k i t F s Lk L r F k t, when [ ] denote the integer part. OQ492. Denote F k and L k the k th Fibonacci respective Lucas numbers. Compute: ). A n (r, s, t) = n h L s i k L r k+2 2). B n (r, s, t) = n 3). C n (r, s, t) = n 4). D n (r, s, t) = n F t k= k+ h F s k Fk+2 r L t k= k+ h L s k Fk+2 r F t k= k+ h F s k L r k+2 k= L t k+ i i i, when [ ] denote the integer part. OQ493. Determine all m N for which m is prime, where Fk 2 n= k=n F k denote the k th Fibonacci number, and [ ] the integer part. " # OQ494. If a t = k, then determine all b t i Z (i =, 2,..., m) for which m b t a t = 0. t= k=n

9 772 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ495. If f (k) = " k m + a k m a m k + a m when a, a 2,..., a m N # then compute D t = when [ ] denote the integer part. k=n f t (k) " # OQ496. Denote k the k th ell number. Compute C t = p t k when [ ] denote the integer part. We have n C = n 2 if n is even n + n 2 if n is odd. " # OQ497. Denote L k the k th Lucas number. Compute B t = L t k when [ ] denote the integer part. OQ498. " Denote F k the k th Fibonacci number. Compute # A t = when [ ] denote the integer part. We have: F t k=n k Fn 2 if n is even ). A = F n 2 if n is odd Fn F 2). A 2 = n if n is even F n F n if n is odd OQ499. rove that S t = " k=n # k t = t i=0 k=n k=n a i n i and determine all a 0, a,..., a t where [ ] denote the integer part. We have the followings: ). S 2 = n 2). S 3 = 2n (n ) i 3). S 4 = 3n 2 5n 2 + 4n + 4). S 5 = 4n 4 8n 3 + 9n 2 5n + h (2n+)(n ) 4 h (n+)(n 2) 3 i

10 Open Questions 773 OQ4920. Solve in Z the equations: x k + xk 2 + xk 3 = yk + yk 2 + yk 3, k {2, 4}. If x = a + 3c, x 2 = b + 3c, x 3 = a b + 2c, y = c + d, y 2 = c + e, y 3 = 2c d e, a, b, c, d, e Z, then we obtain a solution. OQ492. Solve in Z the equations: x k + xk 2 + xk 3 + xk 4 + xk 5 = yk + yk 2 + yk 3 + yk 4 + y5 k, k {, 3, 9, 7}. If x = a, x 2 = a + 2c, x 3 = b, x 4 = b + 2c, x 5 = c + d + e, y = a + b c, y 2 = a + b + c, y 3 = d, y 4 = e, y 5 = 3c when a, b, c, d, e Z, then we obtain a solution. OQ4922. Solve in Z the equation 3 64 x 6 k yk 3 6 z 0 k 3 t0 k = 45 u 8 k vk 8 2. If k= k= k= x = a + b + c, x 2 = b + c + d, x 3 = a d, y = c + d + a, y 2 = d + a + b, y 3 = b c, u k = z k = xk, v k = t k = y k (k =, 2, 3) ; a, b, c, d Z, ad = bc, then we obtain a solution. OQ4923. Solve in Z the equation 25 x 3 x3 2 x3 3 + x3 4 x3 5 + x3 6 x 7 x 7 2 x7 3 + x7 4 x7 5 + x7 6 = 2 x 5 x5 2 x5 3 + x5 4 x5 5 + x If x = a + b + c, x 2 = b + c + d, x 3 = a d, x 4 = c + d + a, x 5 = d + a + b, x 6 = b c, a, b, c, d Z, ad = bc, then we obtain a solution. OQ4924. Let be f (n) = ). Compute k= 2). Compute k= 2). Compute n k= f(k) f 2 (k) r +f(k)+f 2 (k) n + q n + p n n +...

11 774 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4925. Denote G k (n) = qn k + kp n k n +... Compute ). 2). 3). n= k= k= G k (n) G 2 k (n) +G k (n)+g 2 k (n) OQ4926. Solve in Z the equation x 3 + x3 2 + x3 3 y 3 + y y3 3 = z 2. If x = 3a 3 +, x 2 = 3a 3 + 2, x 3 = 3a, y = a 3 +, y 2 = a 3 + 2, y 3 = 3a, z = 9a a 3 + 3, a Z, then we obtain a solution. OQ4927. Solve in Z the equation x y 3 + x 2y 3 2 = x 3y 2 3. If x = a + 2b, y = a, x 2 = b, y 2 = 2a + 3b, x 3 = a + b, y 3 = a + 3b, a, b Z, then we obtain a solution. OQ4928. Solve in Z the equation x 3 + y 3 + z 3 = 3t 3. If x = 2a 3 + 6a 8, y = 2a 3 6a 8, z = 6a 2 + 0, t = 2a 2 2, a Z, then we obtain a solution. OQ4929. Solve in Z the equation x 3 + y 3 + z 3 = 6t 3. If x = a 3 + 9a a + 9, y = a 3 9a 2 39a 35, z = 6a a + 70, t = 4a a + 56, a Z, then we obtain a solution. OQ4930. Solve in Z the equation x 4 + 4y 4 = z 4 + 3t 4. If x = 2a 8 6a 4, y = 2a 7, z = 2a 8 + 6a 4, t = 2a, a Z, then we obtain a solution. h i h i OQ493. Denote F (n) = e π n and G (n) = π e n, when [ ] denote the integer part. Compute ). k= F (k) 2). k= G(k) 3). k= F (k)+g(k)

12 Open Questions 775 4). k= ( ) k+ F (k)g(k) 5). k= ( ) k+ F 2 (k)+g 2 (k) OQ4932. Solve in Z the equation x y 2 = 2 n. OQ4933. Solve in Z the equations x 2 207y 2 = 2 n, and z 3 207t 3 = 2 m. OQ4934. Solve in Z the equation 2x 3 + y 3 7z 3 = 3 n. OQ4935. Solve in Z the equation x 4 + 2y 4 = z 4 + 6t 4. If x = 48a 2 2a 4, y = 48a 7, z = 48a 8 + 2a 4, t = 2a, a Z, then we obtain a solution. OQ4936. Solve in Z the equation 2x 3 + y 3 + 7z 3 = t 3. OQ4937. Solve in Z the equation x 6 + y6 + 2z t 6 = x6 2 + y z 2 t 6 2. If x = au 6 + bv 6, y = bu 6 v 6, z = ab a 4 b 4, t = uv 5, x 2 = au 6 bv 6, y 2 = au 6 + bv 6, z 2 = au a 4 b 4, t 2 = u 5 v, when a, b, u, v Z, then we obtain a solution. OQ4938. Solve in Z the equation x 3 + y 3 + z 3 = ut 3. If x = 27a 3 b 9, y = 27a 3 + 9ab 6 + b 9, z = 27a 2 b 3 + 9ab 6, u = a, t = 27a 2 b 2 + 9ab 5 + 3b 8, a, b Z then we obtain a solution. OQ4939. Solve in Z the equation x y z 24 + t 7 = u 7. If x = a, y = a, z = a, t = a , u = a , a Z, then we obtain a solution.

13 776 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4940. Solve in Z the equation 2x 7 + y z 7 = 207 t 7. OQ494. Solve in Z the equation 2x 5 + y z 3 = 207t 3. OQ4942. Solve in Z the equation x n = yz m tu r, when n, m, r N. OQ4943. Solve in Z the equation n k= a k b n k =. OQ4944. Solve in Z the equation x 3 + y 4 + z 5 + t 7 + u 8 = v 8. If x = a 4, y = a, z = a 4, t = a 4, u = a , v = a a Z then we obtain a solution. OQ4945. Solve in Z the equation x 5 = yz 2 tu 2. If x = a b, y = a, z = a 2 + 0ab + 5b 2, t = b, u = 5a 2 + 0ab + b 2, a, b Z then we obtain a solution. OQ4946. Solve in Z the equation x 7 = yz 2 tu 2. If x = a b, y = a, z = a 3 + 2a 2 b + 35ab 2 + 7b 3, t = b, u = 7a a 2 b + 2ab 2 + b 3, a, b Z then we obtain a solution. OQ4947. Solve in N the equation n Q k=0 (m + k) + x p = y r. Remark. If m =, n n, x =, r = 2, then we get n! + = m 2 which have the solutions (n, m) {(4, 5), (5, ), (7, 7)} and Erdős conjectured thad these are the only ones solutions.

14 Open Questions 777 OQ4948. Solve in Z the equation x 3 = yz 2 tu 2. If x = a b, y = a, z = a + 3b, t = b, u = 3a + b, a, b Z then we obtain a solution. OQ4949. Solve in Z the equation x k + y k + 2z k = x k 2 + yk 2 + 2zk 2, when k {, 5}. If x = 8a 6 + 2ab 5, y = 8a 5 b + b 6, z = 8b 6, x 2 = 8a 6 + 2ab 5, y 2 = 8a 5 b b 6, z 2 = a 6, a, b Z then we obtain a solution. OQ4950. Solve in Z the equation x 5 + x5 2 + x5 3 = y5 + y5 2 + y5 3. If x = a a 3 2a 5 + a 9, x 2 = + a 2 2a 6 + 2a 7 + a 8, x 3 = 2a 3 + 2a 4 2a 7, y = a + 3a 3 2a 5 + a 9, y 2 = + a 2 2a 6 2a 7 + a 8, y 3 = 2a 3 + 2a 4 + 2a 7, a Z then we obtain a solution. OQ495. Solve in Z the equation x y 3 + x 2y 3 2 = z t 3 + z 2t 3 2. If x = + 6a 6a 2 + 5a 4 + 7a 6, y = a + 3a 2 2a 3 + a 5 + a 7, x 2 = 2a 8a 3 5a 4 + 6a 5, y 2 = + a 2 2a 4 3a 5 + a 6 z = 6a 6a 2 + 5a 4 + 7a 6 t = a 3a 2 2a 3 + a 5 + a 7 z 2 = 2a 8a 3 + 5a 4 + 6a 5 t 2 = + a 2 2a 4 + 3a 5 + a 6, a Z then we obtain a solution. OQ4952. Solve in Z the equation x 4 + y 4 = z 4 + 2t 4. If x = 92n 7, y = 92n 8 24n 4, z = 92n n 4, t = 2n, n Z then we obtain a solution. OQ4953. Solve in Z the equation 4x 24 + y z 3 = tu 3 +. If x = a, y = 6a 4, z = 92a 8 24a 4, t = 6a 4 +, u = 92a a 4, a Z then we obtain a solution.

15 778 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4954. Solve in Z the equation x 2 y 2 εz 2 =, where ε {, }. If x = 2a 2 + ε, y = 2a 2, z = 2a, a Z then we obtain a solution. OQ4955. Solve in Z the equation x 3 y 3 εz 3 t 3 u 3 = ε, where ε {, }. If x = 3a 2 + ε, y = 3a 3, z = 3a 2, t = 2a, u = a, a Z then we obtain a solution. OQ4956. Solve in Z the equation x 4 y 4 εz 4 εt 4 u 4 v 4 + w 4 =, when ε {, }. If x = 4a 4 + ε, y = 4a 4, z = 4a 3, t = 2a, u = 3a 2, v = 2a 2, w = a 2, a Z then we obtain a solution. OQ4957. Solve in Z the equation x 3 y 4 = 8z u 2 + v 2. If x = a + b, y = a b, z = ab, u = a, v = b, a, b Z then we obtain a solution. OQ4958. Solve in Z the equation x y 3 + z 3 + t 3 6u = 3v s 2 + r 2 + p 2. If x = a + b + c, y = a, z = b, t = c, u = abc, v = a + b + c, s = a, r = b, p = c, a, b, c Z then we obtain a solution. OQ4959. Solve in Z the equation x 3 + 3y 3 + 4z 3 + t 3 = 239. If x = 5, y = 3, z = 2, t = then we obtain a solution. OQ4960. Solve in Z the equation x x 2 x 3 x 4 + y 4 = z 2. If x = a + b, x 2 = a + 2b, x 3 = a + 3b, x 4 = a + 4b, y = b, z = a 2 + 5ab + 5b 2, a, b Z then we obtain a solution. OQ496. Solve in Z the equation x x 2 x 3 x 4 + y 4 = z 2 + t 2.

16 Open Questions 779 OQ4962. Solve in Z the equation x 4 + x4 2 + x4 3 = y4 + y4 2 + y4 3. If x = a 4 b 2b, x 2 = ab 4 + 2a, x 3 = 2a 3 b, y = a 4 b + 2b, y 2 = ab 4 2a, y 3 = 2ab 3, a, b Z then we obtain a solution. OQ4963. Solve in Z the equation x 4 + y 4 + 4z 4 = t 4. If x = a 4 2b 4, y = 2a 3 b, z = 2ab 3, t = a 4 + 2b 4, a, b Z then we obtain a solution. OQ4964. Solve in Z the equation x x 2 x 3 x 4 + y y 2 y 3 y 4 + 2z 4 = u 2 + v 2. If x = a + b, x 2 = a + 2b, x 3 = a + 3b, x 4 = a + 4b, y = a + b, y 2 = 3a + 2b, y 3 = 5a + 3b, y 4 = 5a + b, z = b, u = a 2 + 5ab + 5b 2, v = 5a 2 + 9ab + 5b 2, a, b Z then we obtain a solution. OQ4965. Solve in Z the equation x x 2 x 3 x 4 x 5 x 6 + 9y 6 = z 2. If x = 2a + b, x 2 = 2a + 2b, x 3 = 3a + 3b, x 4 = 2a + 4b, x 5 = 2a + 5b, x 6 = 2a + 6b, y = b, z = 2a s 2 b + 72ab b 3, a, b Z then we obtain a solution. OQ4966. Solve in Z the equation x 2 + y 2 = z 2 + u + v 2 2. If x = a 2 + ab + b 2, y = c 2 + cd + d 2, z = a + b, u = (a + b) (c + d), v = c + d when a, b, c, d Z, then we obtain a solution. OQ4967. Solve in Z the equation x 2 + y 2 = z 2 + t 2 v. If x = a 2 + b 2 a + 2ab 2, y = a 2 + b 2 b + 2a 2 b, z = a, t = b, v = a 4 + 4a 2 b 2 + b 4 ; a, b Z then we obtain a solution. OQ4968. Solve in Z the equation x 2 + y 2 = z 3. If x = a 2 b 2 a 2ab 2, y = a 2 b 2 b + 2a 2 b, z = a 2 + b 2 ; a, b Z then we obtain a solution. OQ4969. Solve in Z the equation x 5 + x5 2 + x5 3 = y5 + y5 2 + y5 3. If x = a b 5, x 2 = a 5 25b 5, x 3 = 0a 3 b 2, y = a b 5, y 2 = a 5 75b 5,

17 780 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 y 3 = 50ab 4 ; a, b Z then we obtain a solution. OQ4970. Solve in Z the equation x 2 + y 2 = 2 + z 2. If x = 2a +, y = 2a 2 + 2a, z = 2a 2 + 2a, a Z then we obtain a solution. OQ497. Solve in Z the equation x 3 + y 3 + z 3 = 2. If x = 6a 3 +, y = 6a 3 +, z = 6a 2 ; a Z then we obtain a solution. OQ4972. Solve in Z the equation x 4 + y 4 + z 4 = 2 + t 4 + u 4. If x = 8a 5 2a, y = 8a 4 +, z = 8a 4, t = 8a 5 + 2a, u = 4a 2 ; a Z then we obtain a solution. OQ4973. Solve in Z the equation x 5 + y 5 + z 5 + t 5 + 2u 5 = 2. If x = 8a 6 + 2a, y = 8a 6 2a, z = 8a 5 +, t = 8a 5 +, u = 8a 6, a Z then we obtain a solution. OQ4974. Solve in Z the equation x k + y k + z k = t k + u k + v k, when k {2, 6}. If x = ad + b, y = c 2d, z = de + f, t = de f, u = c + 2d, v = ad b, a = 4s + r, b = r, c = 2 (s 3r) (s 2r) + r 2, d = s r, e = 2s 3r, f = 2s 5r, s 2 6r 2 = ; s, r Z then we obtain a solution. OQ4975. Let n be a composite number such L n (mod n), this is defined as Lucas pseudoprimes, when L n denote the Lucas number. Denote L the set of Lucas pseudoprimes. Compute p L p 2. OQ4976. Let F k and L k denote the k th Fibonacci respective Lucas numbers, and p be a prime, f a (n) = a n F k and g p k a (n) = a n where [ ] denote the integer part. k=0 k=0 L k p k

18 Open Questions 78 Determine all a, b > 0 and n N for which f a (n) and g a (n) are prime. OQ4977. Denote n the n th ell number. Determine all a, b, a i, b i N (i =, 2,..., r), for which an+b 2 k = an +b + a2n +b arn +b 3 k=0 when a = a + a a r and b = b + b b r. OQ4978. Denote F k the k th Fibonacci number. Determine all p N for p which = n= ( ) n+ F nf n+...f n+p. OQ4979. Determine a recurrence relation for the sequence x n = [π n ] and compute the general form, where [ ] denote the integer part. Same question for the sequence y n = [e n ]. OQ4980. Compute ). α (p) = k= 3). γ (p) = k= ( ) k+ F k F k+...f k+p ( ) k+ k k+... k+p 2). β (p) = k= ( ) k+ L k L k+...l k+p OQ498. Compute ). α (p) = k= 3). γ (n) = k= ( ) k+ F n +F n F n k ( ) k+ n + n n k OQ4982. Compute k= ( ) k+ Fn k +L n k +. k n 2). β (n) = k= ( ) k+ L n +Ln Ln k

19 782 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ4983. Compute OQ4984. Compute ). 2). 3). k= ( ) k+ F k +F 2 k= 2k+ +F 3k F p pn+p ( ) k+ L k +L 2 k= 2k+ +L3 3k Lp pn+p ( ) k+ k + 2 k= 2k+ + 3k p pn+p OQ4985. Compute ). 2). k= k= ( ) k+ (F k +L k ) p +(L k + k ) p +( k +F k ) p ( ) k+ F Lk +L k + Fk 3). ( ) k+ (F k L k ) n +(L k k ) n +( k F k ) n. k= F L k k ( ) k+ +L k k + F k k OQ4986. Compute ). 3). k= k= ( ) k+ F k L k F 2 k +L2 k ( ) k+ F k k F 2 k + 2 k 2). k= ( ) k+ L k k L 2 k + 2 k OQ4987. Compute ). 3). k= ( ) k+ F k L k k F 3 k +L3 k + 3 k ( ) k F k k= k +Lk k + k k 2). k= ( ) k+ F k+k 2 +L k+k 2+ k+k 2

20 Open Questions 783 OQ4988. Solve in Z the equation 2 x 2 + y 2 z 2 t 2 = u 4 v 4. If x = a, y b, z = a + b, t = a b, u = a + b, v = a b, a, b Z, then we obtain a solution. OQ4989. Solve in Z the equation x 3 + y 3 + z 3 + t 3 = 24u. If x = a + b + c, y = a + b c, z = a b + c, t = a b c, u = abc, then we obtain a solution. OQ4990. Solve in Z the equation 6 x 2 + y 2 + z 2 x 3 + x3 2 + x3 3 + x3 4 = 3 x x x5 7 + x5 8. If x = a, y = b, z = c, x = a + b + c, x 2 = a + b + c, x 3 = a b + c, x 4 = a b c, x 5 = a + b + c, x 6 = a + b + c, x 7 = a b + c, x 8 = a b c, a, b, c Z, then we obtain a solution. OQ499. Solve in Z the equation 5 x 2 + y 2 + z 2 + t 2 x 4 + x4 2 + x4 3 + x4 4 x4 5 x4 6 x4 7 8 x4 = y 6 + y6 2 + y6 3 + y6 4 z6 z6 2 z6 3 z6 4. If x = a, y = b, z = c, t = d, x = a + b + c d, x 2 = a + b c + d, x 3 = a + b + c + d, x 4 = a + b + c + d, x 5 = 2a, x 6 = 2b, x 7 = 2c, x 8 = 2d, y = a + b + c + d, y 2 = a + b c + d, y 3 = a + b + c + d, y 4 = a + b + c + d, z = 2a, z 2 = 2b, z 3 = 2c, z 4 = 2d, a, b, c, d Z then we obtain a solution. OQ4992. Solve in Z the equation x 2 + y 2 z 2 + t 2 = u 2 + v 2. If x = a, y = a 2, z = b, t = b 2, u = a b + a 2 b 2, v = a b 2 a 2 b, a, a 2, b, b 2 Z then we obtain a solution. OQ4993. Solve in Z the equation x 2 + x x2 3 y 2 + y2 2 + y3 3 = z 2 + z2 2 + z2 3 + z4 4. If x, x 2, x 3, y, y 2, y 3 Z and z = x y + x 2 y 2 + x 3 y 3, z 2 = x y 2 x 2 y, z 3 = x y 3 x 3 y, z 4 = x 2 y 3 x 3 y 2 then we obtain a solution.

21 784 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 n OQ4994. Solve in Z the equation a 2 k k= n b 2 k k= = n c 2 k + k= i<j n If c k = a k b k (k =, 2,..., n) and d ij = a i b j a j b i ( i, j n) then we obtain a solution. d 2 ij. OQ4995. Solve in Z the equation x 4 x4 2 x4 3 x4 4 + x4 5 + x4 6 + x4 7 x4 8 = 92y. If x = a + b + c + d, x 2 = a b + c + d, x 3 = a + b + c + d, x 4 = a + b + c d, x 5 = a b c + d, x 6 = a b + c + d, x 7 = a + b c d, x 8 = a b c d, y = abcd, a, b, c, d Z, then we obtain a solution. OQ4996. Solve in Z the equation 6 x 2 + y 2 3 = z 6 + t u 6 + v 6. If x = a, y = b, z = a + b, t = a b, u = a, v = b, a, b Z, then we obtain a solution. OQ4997. Solve in Z the equation 2 i= x 6 i = 6 y 2 + y2 2 + y2 3 + k y2 2 4 for k {2, 4}. If x = a + b, x 2 = a + c, x 3 = a + d, x 4 = b + c, x 5 = b + d, x 6 = c + d, x 7 = a b, x 8 = a c, x 9 = a b, x 0 = b c, x = b d, x 2 = c d, y = a, y 2 = b, y 3 = c, y 4 = d, a, b, c, d Z, then we obtain a solution. OQ4998. Solve in Z the equation x 3 + y 3 + z 3 = t 3. If x = a 4 2ab, y = a 3 b + b 4, z = 2a 3 b b 4, t = a 4 + ab 3, a, b Z, then we obtain a solution. OQ4999. Solve in Z the equation x3 +x3 2 +x3 3 +x3 4 = z 3. If x y 3 = ap q, +y3 2 +y3 3 +yy 4 x 2 = bp q, x 3 = cp + q, x 4 = dp + q, y = a, y 2 = b, y 3 = c, y 4 = d, z = a + b + c + d, {p, q} = a + b + c + d, a 2 + b 2 c 2 d 2, then we obtain a solution.

22 OQ5000. Solve in Z the equation Open Questions k= x 5 k = 2. If x = + ax 5, x 2 = ax 5, x 3 = + bx 5, x 4 = bx 5, x 5 = + cx 5, x 6 = cx 5, x 7 = dx 4, {a, b, c, d} = {270, 360, 450, 80}, x Z then we obtain a solution. OQ500. Solve in Z the equation x 3 + y 3 + z 3 + t 3 = 36u 2. If x = a, y = a, z = c +, t = c +, a 2 + 6b 2 = c 2, a, b, c Z, then we obtain a solution. OQ5002. Solve in Z the equation x 5 + x5 2 + x5 3 + x5 4 + x5 5 + x5 6 = 0. If x = a a 3 2a 5 + a 9, x 2 = + a 2 2a 6 + 2a 7 + a 8, x 3 = 2a 3 + 2a 4 2a 7, x 4 = a + 3a 3 2a 5 + a 9, x 5 = + a 2 2a 6 2a 7 + a 8, x 6 = 2a 3 2a 4 2a 7, a Z, then we obtain a solution. OQ5003. Solve in Z the equation x 4 + y 4 = z 4 + 2t 4. If x = 92a 8 24a 4, y = 92a 7, z = 92a a 4, t = 2a, a Z, then we obtain a solution. OQ5004. Solve in Z the equation x 8 + y 8 + z 4 = 2t 2. If x = a + b, y = a b, z = 4ab, t = a 4 + 4a 2 b 2 + b 4, a, b Z, then we obtain a solution. OQ5005. Soolve in Z the equation 2x 4 + 2y 4 + z 4 = t 4. If x = 4a 3, y = 2a, z = 4a 4, t = 4a 4 +, a Z, then we obtain a solution. OQ5006. Solve in Z the equation x 8 + 7y 8 + 7z 8 + t 8 + u 8 = v 8. If x = 2 4 a 7, y = 2 3 a 5, z = 2 2 a 3, t = 2a, u = 6a 8, v = 6a 8 +, a Z, then we obtain a solution.

23 786 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ5007. Solve in Z the equation x x x x x x x6 7 + x x6 9 = 8x6 If x = 2 8 a 5, x 2 = 2 7 a 3, x 3 = 2 6 a, x 4 = 2 5 a 9, x 5 = 2 4 a 7, x 6 = 2 3 a 5, x 7 = 2 2 a 3, x 8 = 2a, x 9 = 256a 6, x 0 = 256a 6 +, a Z, then we obtain a solution. 0. OQ5008. Solve in Z the equation x 8 + 7y 8 + z 8 + t 8 + u 8 = v 8. If x = 2 3n+2, y = 2 5n+3, z = 2 n+, t = 2 7n+4, u = 8 2n+4, v = 2 8n+4 +, n N, then we obtain a solution. OQ5009. Solve in Z the equation 96x y 3 = z 3. If x = a, y = 4a 4, z = 4a 4 +, a Z, then we obtain a solution. OQ500. Solve in Z the equation x y z t 7 = u 7. If x = a, y = a, z = a, t = 6a 8, u = 6a 8 +, a Z, then we obtain a solution. OQ50. Solve in Z the equation x y z t u v w r 5 = s 5. If x = a, y = a, z = a, t = a, u = a, v = a, w = a, r = 256a 6, s = 256a 6 +, a Z, then we obtain a solution. OQ502. Solve in Z the equation x3 y+ + y3 z+ + z3 x+ = 8. OQ503. Solve in Z the equation x 5 y y5 z z5 x+207 = OQ504. Solve in Z the equation x 4 + y 4 = z 4 + t 4. If x = a + 3a 2 2a 3 + a 5 + a 7, y = + a 2 2a 4 3a 5 + a 6, z = a 3a 2 2a 3 + a 5 + a 7, t = + a 2 2a 4 + 3a 5 + a 6, a Z,

24 Open Questions 787 then we obtain a solution. OQ505. If a k > 0 (k =, 2,..., n), then n a a 2 +a a n + a a 2...a n n + n2 n. k= a n k OQ506. Solve in Z the equation n Q k= + x k = n. OQ507. Determin all x k R (k =, 2,..., n), x i 6= x j (i 6= j) for which m m m x x 2 + x2 x xn x N for all m N. OQ508. If a k > 0 (k =, 2,..., n), then n a a +a 2 + a a 2 n a 2 k k= + n2 2. nq OQ509. Let a k 2p (k =, 2,..., n) be positive integers such that k= is divisible by Q a 2p a 2p 2 a a 2p 2. Determine all n, p N cyclic for which (a, a 2,..., a n ) =. OQ5020. Solve in Z the equation n a 2p s n nq x k = a + n x k, when a Z. k= s nq OQ502. Solve in Z the equation n x k = a + k= k= n n k=, when a Z. x k

25 788 Octogon Mathematical Magazine, Vol. 24, No.2, October 206 OQ5022. Determine all magic squares M = (a ij ) i,j n, when a ij = F i + L j, F i the i th Fibonacci number, L j the j th Lucas number. OQ5023. Determine all magic squares M = (a ij ) i,j n where a ij = 0 n sin 2 b ij ( i, j n) when [ ] denote the integer part and b ij = p i+j (p prime). OQ5024. Let be p ij ( i, j n) prime numbers and p ij = 0, a i a i2...a in... (i =, 2,...). Determine all magic squares M = (a ij ) i,j n. OQ5025. Determine all magic squares M = (a ij ) i,j n for which a ij = p k ij ( i, j n) where k N and p ij are prime numbers. OQ5026. rove that don t exist prime p and q such that tgp = q. OQ5027. rove that don t exist prime p and q such that ctgp = q. OQ5028. rove that don t exist prime p and q such that sin p = q. OQ5029. rove that don t exist prime p and q such that cos p = q.

26 OQ5030. If x k > 0 (k =, 2,..., n), then x x 2 x p x 2... x 2 x 2 2 x p +xp Open Questions 789 x x x p 2 +x p 2 x 3, when p N, p 2. OQ503. If a k > s 0 (k =, 2,..., n), then determine all x, y > 0 for which (x + y) n a k x 4 n r n a 4 k + y a ( n a 2. 2) k= k= cyclic OQ5032. Compute n=0 (n!) k (kn+)!(kn+2)!...(kn+k )!. OQ5033. Let ABC be a triangle, determine the best constants u, v, u 2, v 2 0 for which u R+v r m a + m b + m c u 2 R+v 2 r is the best possible. Remark. u = 4 9, v = 9, u 2 = 0, v 2 = is a solution. OQ5034. If x k > 0 (k =, 2,..., n), then n x 6 k k= x 3 x3 2 cyclic + x x 2 cyclic n x 2 k k= 2. OQ5035. Compute + f()+f(2) 2 + f()+f(2)+f(3) f()+f(2)+f(3)+f(4) for the cases: ). f (k) = k 2). f (k) = p k 3). f (k) = F k 4). f (k) = L k