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IMC 2017 – Day 2 – Problems

August 3, 2017 Leave a comment

Problem 6. Let {f: [0,\infty) \rightarrow \Bbb{R}} be a continuous function such that {\lim_{x \rightarrow \infty}f(x) = L} exists (finite or infinite).

Prove that

\displaystyle \lim_{n \rightarrow \infty} \int_0^1 f(nx) dx = L.

Problem 7. Let {p(x)} be a nonconstant polynomial with real coefficients. For every positive integer {n} let

\displaystyle q_n(x) = (x+1)^n p(x)+x^n p(x+1).

Prove that there are only finitely many numbers {n} such that all roots of {q_n(x)} are real.

Problem 8. Define the sequence {A_1,A_2,...} of matrices by the following recurrence

\displaystyle A_1 = \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix}, \ A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \end{pmatrix} \ \ (n=1,2,...)

where {I_m} is the {m\times m} identity matrix.

Prove that {A_n} has {n+1} distinct integer eigenvalues {\lambda_0<\lambda_1<...<\lambda_n} with multiplicities {{n \choose 0},\ {n\choose 1},...,{n \choose n}}, respectively.

Problem 9. Define the sequence {f_1,f_2,... : [0,1) \rightarrow \Bbb{R}} of continuously differentiable functions by the following recurrence

\displaystyle f_1 = 1; f'_{n+1} = f_nf_{n+1} \text{ on } (0,1) \text{ and } f_{n+1}(0)=1.

Show that {\lim_{n\rightarrow \infty}f_n(x)} exists for every {x \in [0,1)} and determine the limit function.

Problem 10. Let {K} be an equilateral triangle in the plane. Prove that for every {p>0} there exists an {\varepsilon >0} with the following property: If {n} is a positive integer and {T_1,...,T_n} are non-overlapping triangles inside {K} such that each of them is homothetic to {K} with a negative ratio and

\displaystyle \sum_{\ell =1}^n \text{area}(T_\ell) > \text{area} (K)-\varepsilon,

then

\displaystyle \sum_{\ell =1}^n \text{perimeter} (T_\ell) > p.

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IMC 2016 – Day 2 – Problem 8

July 28, 2016 2 comments

Problem 8. Let {n} be a positive integer and denote by {\Bbb{Z}_n} the ring of integers modulo {n}. Suppose that there exists a function {f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n} satisfying the following three properties:

  • (i) {f(x) \neq x},
  • (ii) {x = f(f(x))},
  • (iii) {f(f(f(x+1)+1)+1) = x} for all {x \in \Bbb{Z}_n}.

Prove that {n \equiv 2} modulo {4}.

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IMC 2016 – Day 2 – Problem 7

July 28, 2016 Leave a comment

Problem 7. Today, Ivan the Confessor prefers continuous functions {f:[0,1]\rightarrow \Bbb{R}} satisfying {f(x)+f(y) \geq |x-y|} for all {x,y \in [0,1]}. Fin the minimum of {\int_0^1 f} over all preferred functions.

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IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

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IMC 2016 Problems – Day 2

July 28, 2016 Leave a comment

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

Problem 7. Today, Ivan the Confessor prefers continuous functions {f:[0,1]\rightarrow \Bbb{R}} satisfying {f(x)+f(y) \geq |x-y|} for all {x,y \in [0,1]}. Fin the minimum of {\int_0^1 f} over all preferred functions.

Problem 8. Let {n} be a positive integer and denote by {\Bbb{Z}_n} the ring of integers modulo {n}. Suppose that there exists a function {f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n} satisfying the following three properties:

  • (i) {f(x) \neq x},
  • (ii) {f(f(x))=x},
  • (iii) {f(f(f(x+1)+1)+1) = x} for all {x \in \Bbb{Z}_n}.

Prove that {n \equiv 2} modulo {4}.

Problem 9. Let {k} be a positive integer. For each nonnegative integer {n} let {f(n)} be the number of solutions {(x_1,...,x_k) \in \Bbb{Z}^k} of the inequality {|x_1|+...+|x_k| \leq n}. Prove that for every {n \geq 1} we have {f(n-1)f(n+1) \leq f(n)^2}.

Problem 10. Let {A} be a {n \times n} complex matrix whose eigenvalues have absolute value at most {1}. Prove that

\displaystyle \|A^n\| \leq \frac{n}{\ln 2} \|A\|^{n-1}.

(Here {\|B\| = \sup_{\|x\|\leq 1} \|Bx\|} for every {n \times n} matrix {B} and {\|x\| = \sqrt{\sum_{i=1}^n |x_i|^2 }} for every complex vector {x \in \Bbb{C}^n}.)

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IMC 2016 – Day 1 – Problem 2

July 27, 2016 Leave a comment

Problem 2. Let {k} and {n} be positive integers. A sequence {(A_1,...,A_k)} of {n\times n} matrices is preferred by Ivan the Confessor if {A_i^2 \neq 0} for {1\leq i \leq k}, but {A_iA_j = 0} for {1\leq i,j \leq k} with {i \neq j}. Show that if {k \leq n} in al preferred sequences and give an example of a preferred sequence with {k=n} for each {n}.

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IMC 2014 Day 2 Problem 5

August 1, 2014 2 comments

For every positive integer {n}, denote by {D_n} the number of permutations {(x_1,...,x_n)} of {(1,2,...,n)} such that {x_j \neq j} for every {1 \leq j \leq n}. For {1 \leq k \leq \frac{n}{2}}, denote by {\Delta(n,k)} the number of permutations {(x_1,...,x_n)} of {(1,2,...,n)} such that {x_i = k+i} for every {1 \leq i \leq k} and {x_j \neq j} for every {1 \leq j \leq n}. Prove that

\displaystyle \Delta(n,k) = \sum_{i = 0}^{k-1} {k-1 \choose i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}.

IMC 2014 Day 2 Problem 5

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