Tag Archives: binomial

2012-23 A solution

Prove that for each positive integer \(n\), there exist \(n\) real numbers \(x_1,x_2,\ldots,x_n\) such that \[\sum_{j=1}^n \frac{x_j}{1-4(i-j)^2}=1 \text{ for all }i=1,2,\ldots,n\] and \[\sum_{j=1}^n x_j=\binom{n+1}{2}.\]

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2012-6 Matrix modulo p

Let p be a prime number and let n be a positive integer. Let \(A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n} \) be a \(p^n \times p^n\) matrix. Prove that \( A^3 \equiv I \pmod p\), where I is the \(p^n \times p^n\) identity matrix.

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