# 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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# 2011-24 (n-k) choose k

Evaluate the sum $\sum_{k=0}^{[n/2]} (-4)^{n-k} \binom{n-k}{k} ,$ where [x] denotes the greatest integer less than or equal to x.

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# 2010-14 Combinatorial Identity

Let n be a positive integer. Prove that

$$\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n$$.

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Prove that $$\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2$$.