MathDB

2001 IMC

Part of IMC

Subcontests

(6)
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IMC 2001 Problem 6

Suppose that the differentiable functions a,b,f,g:R→Ra, b, f, g:\mathbb{R} \rightarrow \mathbb{R} satisfy f(x)≥0,f′(x)≥0,g(x)≥0,g′(x)≥0 for all x∈R, f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, lim⁡x→∞a(x)=A≥0,lim⁡x→∞b(x)=B≥0,lim⁡x→∞f(x)=lim⁡x→∞g(x)=∞,\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty, and f′(x)g′(x)+a(x)f(x)g(x)=b(x).\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x). Prove that lim⁡x→∞f(x)g(x)=BA+1\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}.

IMC 2001 Problem 12

For each positive integer nn, let fn(ϑ)=sin⁡(ϑ)⋅sin⁡(2ϑ)⋅sin⁡(4ϑ)⋯sin⁡(2nϑ)f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta). For each real ϑ\vartheta and all nn, prove that ∣fn(ϑ)∣≤23∣fn(π3)∣|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})|
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IMC 2001 Problem 4

p(x)p(x) is a polynomial of degree nn with every coefficient 00 or ±1\pm1, and p(x)p(x) is divisible by (x−1)k (x - 1)^k for some integer k>0 k > 0. qq is a prime such that qln⁡q<kln⁡n+1\frac{q}{\ln q}< \frac{k}{\ln n+1}. Show that the complex qq-th roots of unity must be roots of p(x). p(x).

IMC 2001 Problem 10

Let A=(ak,l)k,l=1,...,nA=(a_{k,l})_{k,l=1,...,n} be a complex n×nn \times n matrix such that for each m∈{1,2,...,n}m \in \{1,2,...,n\} and 1≤j1<...<jm1 \leq j_{1} <...<j_{m} the determinant of the matrix (ajk,jl)k,l=1,...,n(a_{j_{k},j_{l}})_{k,l=1,...,n} is zero. Prove that An=0A^{n}=0 and that there exists a permutation σ∈Sn\sigma \in S_{n} such that the matrix (aσ(k),σ(l))k,l=1,...,n(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n} has all of its nonzero elements above the diagonal.
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easy group theory

Let r,s,tr,s,t positive integers which are relatively prime and a,b∈Ga,b \in G, GG a commutative multiplicative group with unit element ee, and ar=bs=(ab)t=ea^r=b^s=(ab)^t=e. (a) Prove that a=b=ea=b=e. (b) Does the same hold for a non-commutative group GG?

IMC 2001 Problem 8

Let a0=2,b0=2,an+1=2−4−an2,bn+1=2bn2+4+bn2a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}. a) Prove that the sequences (an)(a_{n}) and (bn)(b_{n}) are decreasing and converge to 00. b) Prove that the sequence (2nan)(2^{n}a_{n}) is increasing, the sequence (2nbn)(2^{n}b_{n}) is decreasing and both converge to the same limit. c) Prove that there exists a positive constant CC such that for all nn the following inequality holds: 0<bn−an<C8n0 <b_{n}-a_{n} <\frac{C}{8^{n}}.
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integer matrix

Let n n be a positive integer. Consider an n×n n\times n matrix with entries 1,2,...,n2 1,2,...,n^2 written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for n \equal{} 3 we get [123456789] \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]) We choose n n entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

IMC 2001 Problem 7

Let r,s≥1r, s \geq 1 be integers and a0,a1,...,ar−1,b0,b1,...,bs−1a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} be real non-negative numbers such that (a0+a1x+a2x2+...+ar−1xr−1+xr)(b0+b1x+b2x2+...+bs−1xs−1+xs)=1+x+x2+...+xr+s−1+xr+s(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}. Prove that each aia_i and each bjb_j equals either 00 or 11.