MathDB

2004 IMC

Part of IMC

Subcontests

(6)
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Problem 6 IMC 2004 Macedonia

For every complex number zz different from 0 and 1 we define the following function f(z):=∑1log⁡4z f(z) := \sum \frac 1 { \log^4 z } where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials PP and QQ such that f(z)=P(z)Q(z)f(z) = \displaystyle \frac {P(z)}{Q(z)} for all z∈C−{0,1}z\in\mathbb{C}-\{0,1\}. b) Prove that for all z∈C−{0,1}z\in \mathbb{C}-\{0,1\} we have f(z)=z3+4z2+z6(z−1)4. f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}.

Problem 12 IMC 2004 Macedonia

For n≥0 n\geq 0 define the matrices An A_n and Bn B_n as follows: A_0 \equal{} B_0 \equal{} (1), and for every n>0 n>0 let A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). Denote by S(M) S(M) the sum of all the elements of a matrix M M. Prove that S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1}), for all n,k≥2 n,k\geq 2.
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Problem 4 IMC 2004 Macedonia

Suppose n≥4n\geq 4 and let SS be a finite set of points in the space (R3\mathbb{R}^3), no four of which lie in a plane. Assume that the points in SS can be colored with red and blue such that any sphere which intersects SS in at least 4 points has the property that exactly half of the points in the intersection of SS and the sphere are blue. Prove that all the points of SS lie on a sphere.

Problem 10 IMC 2004 Macedonia

For n≥1n\geq 1 let MM be an n×nn\times n complex array with distinct eigenvalues λ1,λ2,…,λk\lambda_1,\lambda_2,\ldots,\lambda_k, with multiplicities m1,m2,…,mkm_1,m_2,\ldots,m_k respectively. Consider the linear operator LML_M defined by LMX=MX+XMTL_MX=MX+XM^T, for any complex n×nn\times n array XX. Find its eigenvalues and their multiplicities. (MTM^T denotes the transpose matrix of MM).
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Problem 2 IMC 2004 Macedonia

Let f1(x)=x2−1f_1(x)=x^2-1, and for each positive integer n≥2n \geq 2 define fn(x)=fn−1(f1(x))f_n(x) = f_{n-1}(f_1(x)). How many distinct real roots does the polynomial f2004f_{2004} have?

Problem 8 IMC 2004 Macedonia

Let f,g:[a,b]→[0,∞)f,g:[a,b]\to [0,\infty) be two continuous and non-decreasing functions such that each x∈[a,b]x\in [a,b] we have ∫axf(t) dt≤∫axg(t) dt  and ∫abf(t) dt=∫abg(t) dt. \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. Prove that ∫ab1+f(t) dt≥∫ab1+g(t) dt. \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt.
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