For a nonnegative integer n, let \(F_n(x)=\sum_{m=0}^n \frac{(-2)^m (2n-m)! \Gamma(x+1)}{m! (n-m)! \Gamma(x-m+1)}\). Find all x such that Fn(x)=0.
Category Archives: problem
2011-20 Double infinite series
For a real number x, let d(x)=minn:integer (x-n)2. Evaluate the following double infinite series:
. . . + 8 d(x/8)+4 d(x/4) + 2 d(x/2) + d(x) + d(2x) / 2 + d(4x)/4 + d(8x)/8 + . . .
2011-19 Irreducible polynomial
Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.
2011-18 Continuous Function and Differentiable Function
Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if
|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,
then g(x)=0 for all x∈I.
2011-17 Infinitely many solutions
Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.
2011-16 Odd Sets with Even Intersection
Let A1, A2, A3, …, An be finite sets such that |Ai| is odd for all 1≤i≤n and |Ai∩Aj| is even for all 1≤i<j≤n. Prove that it is possible to pick one element ai in each set Ai so that a1, a2, …,an are distinct.
2011-15 Two matrices
Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.
2011-14 Invertible matrices
For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)
(Due to a mistake, the problem is fixed at 3:30PM Friday.)
2011-13 Sums of Partial Sums
Let a1, a2, … be a sequence of non-negative real numbers less than or equal to 1. Let \(S_n=\sum_{i=1}^n a_i\) and \(T_n=\sum_{i=1}^n S_i\). Prove or disprove that \(\sum_{n=1}^\infty a_n/T_n\) converges. (Assume a1>0.)
2011-12 Determinant
Let M=(mi,j)1≤i,j≤n be an n×n matrix such that mi,j=i(i+1)(i+2)…(i+j-2). (Note that m1,1=1.) What is the determinant of M?