MathDB

2010 IMC

Part of IMC

Subcontests

(5)
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IMC 2010 - Problem 3

Define the sequence x1,x2,...x_1, x_2, ... inductively by x1=5x_1 = \sqrt{5} and xn+1=xn2−2x_{n+1} = x_n^2 - 2 for each n≥1n \geq 1. Compute lim⁡n→∞x1⋅x2⋅x3⋅...⋅xnxn+1\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}.

IMC 2010 Problem 3, Day 2

Denote by SnS_n the group of permutations of the sequence (1,2,…,n).(1,2,\dots,n). Suppose that GG is a subgroup of Sn,S_n, such that for every π∈G∖{e}\pi\in G\setminus\{e\} there exists a unique k∈{1,2,…,n}k\in \{1,2,\dots,n\} for which π(k)=k.\pi(k)=k. (Here ee is the unit element of the group Sn.S_n.) Show that this kk is the same for all π∈G∖{e}.\pi \in G\setminus \{e\}.
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IMC 2010 - Problem 2

Compute the sum of the series ∑k=0∞1(4k+1)(4k+2)(4k+3)(4k+4)=11⋅2⋅3⋅4+15⋅6⋅7⋅8+...\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...

IMC 2010, Problem 2, Day 2

Let a0,a1,…,ana_0,a_1,\dots,a_n be positive real numbers such that ak+1−ak≥1a_{k+1}-a_k \geq 1 for all k=0,1,…,n−1.k=0,1,\dots,n-1. Prove that 1+1a0(1+1a1−a0)⋯(1+1an−a0)≤(1+1a0)(1+1a1)⋯(1+1an).1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).
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IMC 2010 Problem 1, Day 2

(a)(a) A sequence x1,x2,…x_1,x_2,\dots of real numbers satisfies xn+1=xncos⁡xn for all n≥1.x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1. Does it follows that this sequence converges for all initial values x1?x_1? (5 points)
(b)(b) A sequence y1,y2,…y_1,y_2,\dots of real numbers satisfies yn+1=ynsin⁡yn for all n≥1.y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1. Does it follows that this sequence converges for all initial values y1?y_1? (5 points)

IMC 2010 - Problem 1

Let 0<a<b0 < a < b. Prove that ∫ab(x2+1)e−x2dx≥e−a2−e−b2\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}.