Polynomials ProblemsAmir Hossein Parvardi March 20, 2011 ∗ 1. Find all polynomial P satisfying: P (x2 + 1) = P (x)2 + 1. 2. Find all functions f : R → R such that f (xn + 2f (y )) = (f (x))n + y + f (y ) ∀x, y ∈ R, 3. Find all functions f : R → R such that x2 y 2 (f (x + y ) − f (x) − f (y )) = 3(x + y )f (x)f (y ) 4. Find all polynomials P (x) with real coefficients such that P (x)P (x + 1) = P (x2 ) ∀x ∈ R. 5. Find all polynomials P (x) with real coefficient such that P (x)Q(x) = P (Q(x)) ∀x ∈ R. n ∈ Z≥2 . 6. Find all polynomials P (x) with real coefficients such that if P (a) is an integer, then so is a, where a is any real number. 7. Find all the polynomials f ∈ R[X ] such that sin f (x) = f (sin x), (∀)x ∈ R. 8. Find all polynomial f (x) ∈ R[x] such that f (x)f (2x2 ) = f (2x3 + x2 ) ∀x ∈ R. 9. Find all real polynomials f and g , such that: (x2 + x + 1) · f (x2 − x + 1) = (x2 − x + 1) · g (x2 + x + 1), for all x ∈ R. 10. Find all polynomials P (x) with integral coefficients such that P (P ′ (x)) = P ′ (P (x)) for all real numbers x. ∗ email: [email protected], blog: http://math-olympiad.blogsky.com 1 b.11. b. Prove that P (x) = ax + b for some rational a and b. 21. 2 . and k a natural number. Find all polynomials with integer coefficients f such that for all n > 2005 the number f (n) is a divisor of nn−1 − 1. Q. Find all polynomials p(x) ∈ R[x] such that the equation f (x) = n has at least one rational solution. a) If P. Prove that deg[P] ≤ 1 18. 16. such that |f (0)| is not a perfect square. c such that ab + bc + ca = 0 we have the following relations f (a − b) + f (b − c) + f (c − a) = 2f (a + b + c). R are non-zero polynomials that for each z ∈ C. Find all polynomials f with real coefficients such that for all reals a. c that aP (x) + bQ(x) + c = 0 15. P (x). Find all polynomials with complec coefficients f such that we have the equivalence: for all complex numbers z. 12. Prove that W has at least k + 1 nonzero coefficients. Q. 24. Suppose f is a polynomial in Z[X ] and m is integer . Q(x) ∈ R[x] and we know that for real r we have p(r) ∈ Q if and only if Q(r) ∈ Q I want some conditions between P and Q.Consider the sequence ai like this a1 = m and ai+1 = f (ai ) find all polynomials f and alll integers m that for each i: ai |ai+1 14. 23. P (z )Q(¯ z ) = R(z ). 13. Prove that any polynomial ∈ R[X ] can be written as a difference of two strictly increasing polynomials. P. z ∈ [−1. Prove that if the leading coefficient of f is 1 (the coefficient of the term having the highest degree in f ) then f (X 2 ) is also irreducible in the ring of integer polynomials. Let f ∈ Z[X ] be an irreducible polynomial over the ring of integer polynomials. 17. 1]. Q is a nonzero polynomial. 1] if and only if f (z ) ∈ [−1. prove that Q is constant polynomial. where a = 0. Consider the polynomial W (x) = (x − a)k Q(x). Let P be a polynomial such that P (x) is rational if and only if x is rational. Q. R ∈ R[x]. 22. 19.p(a) is integer then a is integer. for each positive integer n. R ∈ C[x]? 20. Show that the odd number n is a prime number if and only if the polynomial Tn (x)/x is irreducible over the integers. P is a real polynomail such that if α is irrational then P(α) is irrational.My conjecture is that there exist ratinal a. Find all polynomials p with real coefficients that if for a real a. b) Is the above statement correct for P. the equation P (x) = 2n has an integer root. Suppose that there exists r ≥ 1. Find all polynomials of two variables P (x. Prove that deg P ≤ 1. Let f be a polynomial with integer coefficients such that |f (x)| < 1 on an interval of length at least 4. Let p be a prime number and f an integer polynomial of degree d such that f (0) = 0. Let P be a polynomail with rational coefficients such that P −1 (Q) ⊆ Q. 26. Find r. c. ∀a. Find all polynomials P (x) with odd degree such that P (x2 − 2) = P 2 (x) − 2. 35. b)P (c.25. 30. y ≥ 0: p(x + y ) ≥ p(x) + p(y ). 36. ϕ ∈ R such that P (reiϕ ) = 0. 37. prove that xn − x − 1 is irreducible over Q for all n ≥ 2. d ∈ R. Prove that d ≥ p − 1. ad + bc). Find all polynomials P ∈ C[X ] such that P (X 2 ) = P (X )2 + 2P (X ). a2 ≤ a1 ≤ 1. . f (1) = 1 and f (n) is congruent to 0 or 1 modulo p for every integer n. . Find all real polynomials that p(x + p(x)) = p(x) + p(p(x)) 33. 31. d) = P (ac + bd. Find all polynomials of degree 3. 3 1 x + p2 1 x = p(x2 )p 1 x2 . Find all polynomials P (x) ∈ Z[x] such that for any n ∈ N. y ) which satisfy P (a. Prove that f = 0. 28. b. 29. Let P (x) := xn + n k=1 ak xn−k with 0 ≤ an ≤ an−1 ≤ . 32. Find all real polynomials p(x) such that p2 (x) + 2p(x)p For all non-zero real x. 34. such that for each x. 27. Find all real polynomials f (x) satisfying f (x2 ) = f (x)f (x − 1)∀x ∈ R. where S (n) = p + 1 in varable n . . . have the same leading coefficient. pn + p1 have a real root. Find all real polynomials f with x. a2 . . p1 p2 . . . Find all polynomials p with real coefficients such that for all reals a. Prove that P ′ (x) = nQ(x) 46. an such that the equation an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 has n roots setting an arithmetic progression. Find a rigurous proof n that S (n). . and xn f (x) + 1 is a reducible polynomial for every n ∈ N? 44. k=0 f (k ). Let f and g be polynomials such that f (Q) = g (Q) for all rationals Q. The polynomials P. Q are such that deg P = n. Show that there exists an integer A such that f (X ) = g (A + x) or f (x) = g (A − x). 40. 50. 49. Let F be the set of all polynomials Γ such that all the coefficients of Γ(x) are integers and Γ(x) = 1 has integer roots. 47. . Given distinct prime numbers p and q and a natural number n ≥ 3. . . Find all positive integers n ≥ 3 such that there exists an arithmetic progression a0 . 48. . p2 p3 . 39. . . c such that ab + bc + ca = 1 we have the relation p(a)2 + p(b)2 + p(c)2 = p(a + b + c)2 . Find all positive real numbers a1 . pn−2 pn +pn−1 . 42. 1 n + 41. Let f. Find all polynomials P (x) with integer coefficients such that the polynomial Q(x) = (x2 + 6x + 10) · P 2 (x) − 1 is the square of a polynomial with integer coefficients. . Prove that at least n−2 of polynomials p1 p2 . 43. for all real numbers X . . . Given non-constant linear functions p1 (x). find all a ∈ Z such that the polynomial f (x) = xn + axn−1 + pq can be factored into 2 integral polynomials of degree at least 1. Prove that there exist reals a and b such that f (X ) = g (aX + b). Does there exist a polynomial f ∈ Q[x] with rational coefficients such that f (1) = −1. y ∈ R such that 2yf (x + y ) + (x − y )(f (x) + f (y )) ≥ 0. find the smallest integer m(k ) > 1 such that there exist Γ ∈ F for which Γ(x) = m(k ) has exactly k distinct integer roots. Suppose that f is a polynomial of exact degree p. . . 4 . and P 2 (x) = (x2 − 1)Q2 (x) + 1. . a1 . p2 (x). . Given a positive intger k . . pn (x). ak such that the number a1 1 n · · · + ak is rational for all positive integers n. where k is a fixed positive integer.38. is a polynomial function of (exact) degree 45. pn−1 +pn . b.deg Q = m. g be real non-constant polynomials such that f (Z) = g (Z). .. 59. 54. Find all polynomials P (x) with real coefficients. Problem: Find all polynomials p(x) with real coefficients such that (x + 1)p(x − 1) + (x − 1)p(x + 1) = 2xp(x) for all real x. xn . such that P (x2 − 1) = P (x)P (−x). Find all couples of polynomials (P. Find all polynomials P (x) ∈ R[x]such that: P (x2 ) + x · (3P (x) + P (−x)) = (P (x))2 + 2x2 ∀x ∈ R 56.. Find all polynomials P (x) ∈ R[x] such that P (x2 + 1) = P (x)2 + 1 holds for all x ∈ R. Q) with real coefficients.. Find all polynomials p(x. x − y ) = 2p(x. such that for infinitely many x ∈ R the condition P (x) P (x + 1) 1 − = Q(x) Q(x + 1) x(x + 2) Holds. 58.. y ). .. Find all polynomials f. Find all polynomials such that P (x3 + 1) = P ((x + 1)3 ). for k = 1. x2 . 57. Find all polynomials P (x) that have only real roots. Find all polynomials Pn (x) = n!xn + an−1 xn−1 + .. y ∈ R we have p(x + y. 53. 52. 2. 5 .51. Find all polynomials P (x) with real coefficients such that there exists a polynomial Q(x) with real coefficients that satisfy P (x2 ) = Q(P (x)). y ] such that for each x. y ) ∈ R[x. g which are both monic and have the same degree and f (x)2 − f (x2 ) = g (x). such that P (P (x)) = P (x)k (k is a given positive integer) 61. such that k ≤ xk ≤ k + 1. n. + a1 x + (−1)n (n + 1)n with integers coefficients and with n real roots x1 . 60. 55.. The function f (n) satisfies f (0) = 0 and f (n) = n − f (f (n − 1)). n = 1. + c1 x + c0 where each of c0 . 64. Find all pairs of integers a.. for some polynomials Q(x). Find all reals α for which there is a nonzero polynomial P with real coefficients such that P (1) + P (3) + P (5) + · · · + P (2n − 1) = αP (n) ∀n ∈ N. There exists a polynomial P of degree 5 with the following property: if z is a complex number such that z 5 + 2004z = 1. y ) = P (x. c1 . R(x) with degree greater than 1. b for which there exists a polynomial P (x) ∈ Z[X ] such that product (x2 + ax + b) · P (x) is a polynomial of a form xn + cn−1 xn−1 + .. y ∈ R. then P (z 2 ) = 0. Find all polynomials g (x) with real coefficient such that f (n) = [g (n)]. Find all polynomials P (x.. y + 1) 67. Find all such polynomials P 65.. n and find all such polynomials for α = 2. 6 ∀x. y ) = P (x + 1. y + 1) = P (x + 1. Find all polynomials P (x) ∈ R[x] such that P (x2 − 2x) = (P (x) − 2)2 . Find all polynomials P (x) with real coefficients satisfying the equation (x + 1)3 P (x − 1) − (x − 1)3 P (x + 1) = 4(x2 − 1)P (x) for all real numbers x. Find all polynomials P (x) ∈ R[X ] satisfying (P (x))2 − (P (y ))2 = P (x + y ) · P (x − y ). 2 · · · Where [g (n)] denote the greatest integer that does not exceed g (n). 2. n = 0. . 71. Find all polynomials P (x) with reals coefficients such that (x − 8)P (2x) = 8(x − 1)P (x). 66. cn−1 is equal to 1 or −1. . 63. Find all n ∈ N such that polynomial P (x) = (x − 1)(x − 2) · · · (x − n) can be represented as Q(R(x)). 1. 68. 3 · · · . 70. y ) with real coefficients such that: P (x.62. 69.. 7 . Find all integers k such that for infinitely many integers n ≥ 3 the polynomial P (x) = xn+1 + kxn − 870x2 + 1945x + 1995 can be reduced into two polynomials with integer coefficients. 81. 76. √ 79. Find all polynomials P (x) = a0 + a1 x + · · · + an xn having exactly n roots not greater than −1 and satisfying 2 a2 0 + a1 an = an + a0 an−1 . Does there exist a polynomial q (x) with integer coefficients such that q (k + k 2 ) = 3 + k ? 80. Find all values of the positive integer m such that there exists polynomials P (x). Let n ≥ 2 be a positive integer. b which satisfying am − b2 = 0. b)) = b. 77. 83. y . for all real numbers x. Find all polynomials P (x). y ) with real coefficient satisfying the condition: For every real numbers a. then p (a) + p (b) is also a perfect square. Find all polynomials p(x) ∈ R[x] such that p(x2008 + y 2008 ) = (p(x))2008 + (p(y ))2008 .72. Let k = 3 3. Find all polynomials W (x) ∈ R[x] such that W (x2 )W (x3 ) = W (x)5 ∀x ∈ R. Find all non-constant real polynomials f (x) such that for any real x the following equality holds f (sin x + cos x) = f (sin x) + f (cos x). Q(x). Q(x) such that P (Q(X )) = Q(P (x))∀x ∈ R. Find a polynomial p(x) with rational coefficients and degree as small as possible such that p(k + k 2 ) = 3 + k . R(x) with real coefficients such that P (x) − Q(x) = R(x) ∀x ∈ R. Q(x). R(x. 74. Find all polynomials P (x). Find all polynomials P (x) ∈ Q[x] such that √ −x + 3 − 3x2 P (x) = P for all 2 |x| ≤ 1. Find all polynomials p of one variable with integer coefficients such that if a and b are natural numbers such that a + b is a perfect square. 82. 84. Find all Polynomials P (x) satisfying P (x)2 − P (x2 ) = 2x4 . Find all the polynomials f (x) with integer coefficients such that f (p) is prime for every prime p. b)) = a and Q(R(a. 73. 78. 75. we always have that P (R(a. 93. y . with f (a) = 0. fn (x). b) = 0. 2 2 . Find all polynomials P (x) satisfying the equation (x + 1)P (x) = (x − 2010)P (x + 1). c. 94. Find all polynomials f with real coefficients such that for all reals a. y ) such that for all reals x. 92. 89. Find all polynomials p(x) that satisfy (p(x))2 − 1 = 4p(x2 − 4X + 1) ∀x ∈ R. Determine the polynomials P of two variables so that: a. . having the property that f (n) divides f 2k · a . Let f (x) = x4 − x3 + 8ax2 − ax + a2 . Let n and k be two positive integers. 96. f2 (x) . .) for any real numbers a. Determine all monic polynomials f ∈ Z[X ]. 87. Find all nonconstant polynomials with real coefficients f1 (x) . 86. Find all real number a such that f (x) = 0 has four different positive solutions. c such that ab + bc + ca = 0 we have the following relations f (a − b) + f (b − c) + f (c − a) = 2f (a + b + c). Find all polynomials P (x) such that P (x2 − y 2 ) = P (x + y )P (x − y ). b. 91. b. the same for all t. y we have P (x2 . 8 . 90. for every real x (with fn+1 (x) ≡ f1 (x) and fn+2 (x) ≡ f2 (x)). ty ) = tn P (x. a)+ P (c + a. y 2 ) = P (x + y )2 (x − y )2 . for which fk (x) fk+1 (x) = fk+1 (fk+2 (x)) . forall a ∈ Z. Let n ≥ 3 be a natural number. .) P (1. Find all integers n such that the polynomial p(x) = x5 − nx − n − 2 can be written as product of two non-constant polynomials with integral coefficients. Find all polynomials p(x) that satisfy (p(x))2 − 2 = 2p(2x2 − 1) ∀x ∈ R. c)+ P (b + c. y ) where n is a positive integer.85. x.) for any real numbers t. Find All Polynomials P (x. b. of degree n. y we have P (tx. 88. x. c we have P (a + b. 95. . 1 ≤ k ≤ n. Find all polynomial P ∈ R[x] such that: P (x2 + 2x + 1) = (P (x))2 + 1. 0) = 1. c ∈ R: p(ab. c ∈ R.97. 98. y ) such that for each a. Find all polynomials P (x) with real coefficients such that P (x2 − 2x) = (P (x − 2)) 2 100. b2 + 1) = 0. 99. 9 . Find all polynomials of degree 3 such that for all non-negative reals x and y we have p(x + y ) ≤ p(x) + p(y ). b. Find all two-variable polynomials p(x. Find all polynomials p(x) with real coefficients such that p(a + b − 2c) + p(b + c − 2a) + p(c + a − 2b) = 3p(a − b) + 3p(b − c) + 3p(c − a) for all a. c2 + 1) + p(bc. a2 + 1) + p(ca. b. http://www. 10 . http://www.artofproblemsolving.php?t=392444. http://www.com/Forum/viewtopic.artofproblemsolving.php?t=53450. 19.php?t=6122.php?t=385331.com/Forum/viewtopic.com/Forum/viewtopic.artofproblemsolving. 13.com/Forum/viewtopic.Solutions 1.artofproblemsolving.php?t=78454.php?t=382979. http://www. http://www. 12.com/Forum/viewtopic. 25.php?t=391333.artofproblemsolving. 3.artofproblemsolving. http://www.artofproblemsolving.artofproblemsolving. 26. 15.php?t=111404.php?t=49530.artofproblemsolving. 17.com/Forum/viewtopic. 2.com/Forum/viewtopic.php?t=53271. http://www.php?t=395325. http://www. 6. http://www.com/Forum/viewtopic.artofproblemsolving.artofproblemsolving.com/Forum/viewtopic. http://www.php?t=66713.artofproblemsolving.com/Forum/viewtopic. 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