MathDB

2009 IMC

Part of IMC

Subcontests

(5)
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IMC 2009 Day 1 P5

Let nn be a positive integer. An n-\emph{simplex} in Rn\mathbb{R}^n is given by n+1n+1 points P0,P1,⋯ ,PnP_0, P_1,\cdots , P_n, called its vertices, which do not all belong to the same hyperplane. For every nn-simplex S\mathcal{S} we denote by v(S)v(\mathcal{S}) the volume of S\mathcal{S}, and we write C(S)C(\mathcal{S}) for the center of the unique sphere containing all the vertices of S\mathcal{S}. Suppose that PP is a point inside an nn-simplex S\mathcal{S}. Let Si\mathcal{S}_i be the nn-simplex obtained from S\mathcal{S} by replacing its ithi^{\text{th}} vertex by PP. Prove that : ∑j=0nv(Sj)C(Sj)=v(S)C(S) \sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S})

IMC 2009 Day 2 P5

Let M\mathbb{M} be the vector space of m×pm \times p real matrices. For a vector subspace S⊆MS\subseteq \mathbb{M}, denote by δ(S)\delta(S) the dimension of the vector space generated by all columns of all matrices in SS. Say that a vector subspace T⊆MT\subseteq \mathbb{M} is a \emph{covering matrix space} if ⋃A∈T,A≠0ker⁡A=Rp \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p Such a TT is minimal if it doesn't contain a proper vector subspace S⊂TS\subset T such that SS is also a covering matrix space.
(a) (8 points) Let TT be a minimal covering matrix space and let n=dim⁡(T)n=\dim (T) Prove that δ(T)≤(n2) \delta(T)\le \dbinom{n}{2} (b) (2 points) Prove that for every integer nn we can find mm and pp, and a minimal covering matrix space TT as above such that dim⁡T=n\dim T=n and δ(T)=(n2)\delta(T)=\dbinom{n}{2}
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IMC 2009 Day 1 P4

Let p(z)=a0+a1z+a2z2+⋯+anznp(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n be a complex polynomial. Suppose that 1=c0≥c1≥⋯≥cn≥01=c_0\ge c_1\ge \cdots \ge c_n\ge 0 is a sequence of real numbers which form a convex sequence. (That is 2ck≤ck−1+ck+12c_k\le c_{k-1}+c_{k+1} for every k=1,2,⋯ ,n−1k=1,2,\cdots ,n-1 ) and consider the polynomial q(z)=c0a0+c1a1z+c2a2z2+⋯+cnanzn q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n Prove that : max⁡∣z∣≤1q(z)≤max⁡∣z∣≤1p(z) \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z)

IMC 2009 Day 2 P4

Let pp be a prime number and W⊆Fp[x]\mathbf{W}\subseteq \mathbb{F}_p[x] be the smallest set satisfying the following :
(a) x+1∈Wx+1\in \mathbf{W} and xp−2+xp−3+⋯+x2+2x+1∈Wx^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W} (b) For γ1,γ2\gamma_1,\gamma_2 in W\mathbf{W}, we also have γ(x)∈W\gamma(x)\in \mathbf{W}, where γ(x)\gamma(x) is the remainder (γ1∘γ2)(x)(modxp−x)(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}. How many polynomials are in W?\mathbf{W}?
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IMC 2009 Day 1 P3

In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 11 to nn and let aia_i be the number of friends of the ithi^{\text{th}} resident. Suppose that ∑i=1nai2=n2−n \sum_{i=1}^{n}a_i^2=n^2-n Let kk be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of k.k.

IMC 2009 Day 2 P3

Let A,B∈Mn(C)A,B\in \mathcal{M}_n(\mathbb{C}) be two n×nn \times n matrices such that A2B+BA2=2ABA A^2B+BA^2=2ABA Prove there exists k∈Nk\in \mathbb{N} such that (AB−BA)k=0n (AB-BA)^k=\mathbf{0}_n Here 0n\mathbf{0}_n is the null matrix of order nn.
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