Algebra Rmo
Algebra
Archit Pal Singh Sachdeva
1. Consider the sequence of polynomials defined by P1 (x) = x2 − 2 and Pj (x) = P1 (Pj−1 (x)) for j = 2, 3, . . .. Show that for any positive integer n the roots of equation Pn (x) = x are all real and distinct. 2. Prove that every polynomial over integers has a nonzero polynomial multiple whose exponents are all divisible by 2012. 3. Let fn (x) denote the Fibonacci polynomial, which is defined by f1 = 1, f2 = x, fn = xfn−1 + fn−2 . Prove that the inequality
2 fn ≤ (x2 + 1)2 (x2 + 2)n−3
holds for every real x and n ≥ 3. 4. Find all polynomials f with with real coefficients satisfying, for any real number x, the relation f (x)f (2x2 ) = f (2x3 + x). 5. Consider the equation with real coefficients x6 + ax5 + bx4 + cx3 + bx2 + ax + 1 = 0, and denote by x1 , x2 , . . . , x6 the roots of the equation. Prove that
6
k=1
(x2 + 1) = (2a − c)2 . k
6. Let a, b, c, d be real numbers such that (a2 + 1)(b2 + 1)(c2 + 1)(d2 + 1) = 16. Prove that −3 ≤ ab + bc + cd + da + ac + bd − abcd ≤ 5. 7. Solve the equation x3 − 3x = 1 √ x + 2.
8. For positive numbers a, b, c prove the inequality a2 − ab + b2 + b2 − bc + c2 ≥ a2 + ac + c2 .
9. Let a, b, c be real numbers. Show that a ≥ 0, b ≥ 0, and c ≥ 0 if and only if a + b + c ≥ 0, ab + bc + ca ≥ 0, and abc ≥ 0. 10. Find the minimum possible value of a2 + b2 if a and b are real numbers such that x4 + ax3 + bx2 + ax + 1 = 0 has at least one real root. 11. If x and y are positive real numbers such that (x + x2 + 1)(y + y 2 + 1) = 2011, find the minimum possible value of x + y. 12. Let p ≥ 5 be a prime number. Prove that 43 divides 7p − 6p − 1.
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Archit Pal Singh Sachdeva
1. Consider the sequence of polynomials defined by P1 (x) = x2 − 2 and Pj (x) = P1 (Pj−1 (x)) for j = 2, 3, . . .. Show that for any positive integer n the roots of equation Pn (x) = x are all real and distinct. 2. Prove that every polynomial over integers has a nonzero polynomial multiple whose exponents are all divisible by 2012. 3. Let fn (x) denote the Fibonacci polynomial, which is defined by f1 = 1, f2 = x, fn = xfn−1 + fn−2 . Prove that the inequality
2 fn ≤ (x2 + 1)2 (x2 + 2)n−3
holds for every real x and n ≥ 3. 4. Find all polynomials f with with real coefficients satisfying, for any real number x, the relation f (x)f (2x2 ) = f (2x3 + x). 5. Consider the equation with real coefficients x6 + ax5 + bx4 + cx3 + bx2 + ax + 1 = 0, and denote by x1 , x2 , . . . , x6 the roots of the equation. Prove that
6
k=1
(x2 + 1) = (2a − c)2 . k
6. Let a, b, c, d be real numbers such that (a2 + 1)(b2 + 1)(c2 + 1)(d2 + 1) = 16. Prove that −3 ≤ ab + bc + cd + da + ac + bd − abcd ≤ 5. 7. Solve the equation x3 − 3x = 1 √ x + 2.
8. For positive numbers a, b, c prove the inequality a2 − ab + b2 + b2 − bc + c2 ≥ a2 + ac + c2 .
9. Let a, b, c be real numbers. Show that a ≥ 0, b ≥ 0, and c ≥ 0 if and only if a + b + c ≥ 0, ab + bc + ca ≥ 0, and abc ≥ 0. 10. Find the minimum possible value of a2 + b2 if a and b are real numbers such that x4 + ax3 + bx2 + ax + 1 = 0 has at least one real root. 11. If x and y are positive real numbers such that (x + x2 + 1)(y + y 2 + 1) = 2011, find the minimum possible value of x + y. 12. Let p ≥ 5 be a prime number. Prove that 43 divides 7p − 6p − 1.
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