RMO Question..

AIEEE Online › Olympiad Stuff questions › RMO Question..

Prove that x⁹⁹⁹⁹+x⁸⁸⁸⁸+x⁷⁷⁷⁷+.....+x¹¹¹¹+1 is divisible by x+1

#Mathematics #Olympiad Stuff

UP 0 DOWN 0 0 6
Post on FacebookPost anonymously?

6 Answers

1

yes no ·2009-07-18 05:57:43

f(-1) = 0 ??

UP 0 DOWN 0
Post on FacebookPost anonymously?

1

Samarth Kashyap ·2009-07-18 05:59:32

putting x=-1
(-1)⁹+(-1)⁸+.......+1=-1+1-1+1-1+1-1+1-1+1=0
since x=-1 is a root of x⁹⁹⁹⁹+x⁸⁸⁸⁸+x⁷⁷⁷⁷+.....+x¹¹¹¹+1=0
(x+1) is a factor.

UP 0 DOWN 0
Post on FacebookPost anonymously?

9

Celestine preetham ·2009-07-18 06:30:03

are u joking that this is a RMO question ?

UP 0 DOWN 0
Post on FacebookPost anonymously?

1

rudra ·2009-07-19 04:32:37

actually yes.. :D RMO 1992(the book says so) .. there was a reason behind the question :P ... having fun

UP 0 DOWN 0
Post on FacebookPost anonymously?

1

rudra ·2009-07-19 04:36:24

here is another one..

show that (x-1)² is a factor of xⁿ⁺¹-xⁿ-x+1

UP 0 DOWN 0
Post on FacebookPost anonymously?

106

Asish Mahapatra ·2009-07-19 04:39:32

xⁿ(x-1)-(x-1)
=(x-1)(xⁿ-1)
=(x-1)²(1+x+x²+...+x^n-1)

So (x-1)² is a factor of xⁿ⁺¹-xⁿ-x+1

UP 0 DOWN 0
Post on FacebookPost anonymously?

Your Answer

Basic Integral Calculus Operators Symbols Matrix Cases

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω