MathDB

1994 IMC

Part of IMC

Subcontests

(6)
6
2

IMC 1994 D1 P6

Let f∈C2[0,N]f\in C^2[0,N] and ∣f′(x)∣<1|f'(x)|<1, f′′(x)>0f''(x)>0 for every x∈[0,N]x\in [0, N]. Let 0≤m0 <m1<⋯<mk≤N0\leq m_0\ <m_1 < \cdots < m_k\leq N be integers such that ni=f(mi)n_i=f(m_i) are also integers for i=0,1,…,ki=0,1,\ldots, k. Denote bi=ni−ni−1b_i=n_i-n_{i-1} and ai=mi−mi−1a_i=m_i-m_{i-1} for i=1,2,…,ki=1,2,\ldots, k.
a) Prove that −1<b1a1<b2a2<⋯<bkak<1-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1
b) Prove that for every choice of A>1A>1 there are no more than N/AN / A indices jj such that aj>Aa_j>A.
c) Prove that k≤3N2/3k\leq 3N^{2/3} (i.e. there are no more than 3N2/33N^{2/3} integer points on the curve y=f(x)y=f(x), x∈[0,N]x\in [0,N]).

IMC 1994 D2 P6

Find lim⁡N→∞ln⁡2NN∑k=2N−21ln⁡k⋅ln⁡(N−k)\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}
4
2

IMC 1994 D1 P4

Let α∈R\{0}\alpha\in\mathbb R\backslash \{ 0 \} and suppose that FF and GG are linear maps (operators) from Rn\mathbb R^n into Rn\mathbb R^n satisfying F∘G−G∘F=αFF\circ G - G\circ F=\alpha F.
a) Show that for all k∈Nk\in\mathbb N one has Fk∘G−G∘Fk=αkFkF^k\circ G-G\circ F^k=\alpha kF^k.
b) Show that there exists k≥1k\geq 1 such that Fk=0F^k=0.

IMC 1994 D2 P4

Let AA be a n×nn\times n diagonal matrix with characteristic polynomial (x−c1)d1(x−c2)d2…(x−ck)dk(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k} where c1,c2,…,ckc_1, c_2, \ldots, c_k are distinct (which means that c1c_1 appears d1d_1 times on the diagonal, c2c_2 appears d2d_2 times on the diagonal, etc. and d1+d2+…+dk=nd_1+d_2+\ldots + d_k=n).
Let VV be the space of all n×nn\times n matrices BB such that AB=BAAB=BA. Prove that the dimension of VV is d12+d22+⋯+dk2d_1^2+d_2^2+\cdots + d_k^2
3
2

IMC 1994 D1 P3

Given a set SS of 2n−12n-1, n∈Nn\in \mathbb N, different irrational numbers. Prove that there are nn different elements x1,x2,…,xn∈Sx_1, x_2, \ldots, x_n\in S such that for all non-negative rational numbers a1,a2,…,ana_1, a_2, \ldots, a_n with a1+a2+…+an>0a_1+a_2+\ldots + a_n>0 we have that a1x1+a2x2+⋯+anxna_1x_1+a_2x_2+\cdots +a_nx_n is an irrational number.

IMC 1994 D2 P3

Let ff be a real-valued function with n+1n+1 derivatives at each point of R\mathbb R. Show that for each pair of real numbers aa, bb, a<ba<b, such that ln⁡(f(b)+f′(b)+⋯+f(n)(b)f(a)+f′(a)+⋯+f(n)(a))=b−a\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a there is a number cc in the open interval (a,b)(a,b) for which f(n+1)(c)=f(c)f^{(n+1)}(c)=f(c)
2
2

IMC 1994 D1 P2

Let f∈C1(a,b)f\in C^1(a,b), lim⁡x→a+f(x)=∞\lim_{x\to a^+}f(x)=\infty, lim⁡x→b−f(x)=−∞\lim_{x\to b^-}f(x)=-\infty and f′(x)+f2(x)≥−1f'(x)+f^2(x)\geq -1 for x∈(a,b)x\in (a,b). Prove that b−a≥πb-a\geq\pi and give an example where b−a=πb-a=\pi.

IMC 1994 D2 P2

Let f ⁣:R2→Rf\colon \mathbb R ^2 \rightarrow \mathbb R be given by f(x,y)=(x2−y2)e−x2−y2f(x,y)=(x^2-y^2)e^{-x^2-y^2}.
a) Prove that ff attains its minimum and its maximum.
b) Determine all points (x,y)(x,y) such that ∂f∂x(x,y)=∂f∂y(x,y)=0\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0 and determine for which of them ff has global or local minimum or maximum.
1
2

IMC 1994 D1 P1

a) Let AA be a n×nn\times n, n≥2n\geq 2, symmetric, invertible matrix with real positive elements. Show that zn≤n2−2nz_n\leq n^2-2n, where znz_n is the number of zero elements in A−1A^{-1}.
b) How many zero elements are there in the inverse of the n×nn\times n matrix A=(1111…11222…21211…11212…2⋮⋮⋮⋮⋱⋮1212…⋱)A=\begin{pmatrix} 1&1&1&1&\ldots&1\\ 1&2&2&2&\ldots&2\\ 1&2&1&1&\ldots&1\\ 1&2&1&2&\ldots&2\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&2&1&2&\ldots&\ddots \end{pmatrix}

IMC 1994 D2 P1

Let f∈C1[a,b]f\in C^1[a,b], f(a)=0f(a)=0 and suppose that λ∈R\lambda\in\mathbb R, λ>0\lambda >0, is such that ∣f′(x)∣≤λ∣f(x)∣|f'(x)|\leq \lambda |f(x)| for all x∈[a,b]x\in [a,b]. Is it true that f(x)=0f(x)=0 for all x∈[a,b]x\in [a,b]?
5
2

IMC 1994 D1 P5

a) Let f∈C[0,b]f\in C[0,b], g∈C(R)g\in C(\mathbb R) and let gg be periodic with period bb. Prove that ∫0bf(x)g(nx) dx\int_0^b f(x) g(nx)\,\mathrm dx has a limit as n→∞n\to\infty and lim⁡n→∞∫0bf(x)g(nx) dx=1b∫0bf(x) dx⋅∫0bg(x) dx\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx
b) Find lim⁡n→∞∫0πsin⁡x1+3cos⁡2nx dx\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx

IMC problem 5

problem 5. Let x1,x2,…,xkx_1, x_2,\ldots, x_k be vectors of mm-dimensional Euclidean space, such that x1+x2+…+xk=0x_1+x_2+\ldots + x_k=0. Show that there exists a permutation π\pi of the integers {1,2,…,k}\{ 1, 2, \ldots, k \} such that: ∥∑i=1nxπ(i)∥≤(∑i=1k∥xi∥2)1/2\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}for each n=1,2,…,kn=1, 2, \ldots, k. Note that ∥⋅∥\lVert \cdot \rVert denotes the Euclidean norm. (18 points).