Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).
Category Archives: problem
2010-15 Characteristic Polynomial
Let A, B be 2n×2n skew-symmetric matrices and let f be the characteristic polynomial of AB. Prove that the multiplicity of each root of f is at least 2.
2010-14 Combinatorial Identity
Let n be a positive integer. Prove that
\(\displaystyle \sum_{k=0}^n (-1)^k \binom{2n+2k}{n+k} \binom{n+k}{2k}=(-4)^n\).
2010-13 Upper bound
Prove that there is a constant C such that
\(\displaystyle \sup_{A<B} \int_A^B \sin(x^2+ yx) \, dx \le C\)
for all y.
2010-12 Make a nonsingular matrix by perturbing the diagonal
Let A be a square matrix. Prove that there exists a diagonal matrix J such that A+J is invertible and each diagonal entry of J is ±1.
POW will resume on Sep. 3 Friday for Fall 2010
KAIST Problem of the Week will continue in the fall semester of 2010. The first problem of 2010 Fall will be posted online on Sep. 3, Friday 3PM.
2010-11 Integral Equation
Let z be a real number. Find all solutions of the following integral equation: \(f(x)=e^x+z \int_0^1 e^{x-y} f(y)\,dy\) for 0≤x≤1.
2010-10 Metric space of matrices
Let Mn×n be the space of real n×n matrices, regarded as a metric space with the distance function
\(\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|\)
for A=(aij) and B=(bij).
Prove that \(\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}\) is a closed set.
2010-9 No zeros far away
Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of \(f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}\) are in the disk |z|<M on the complex plane.
2010-8 Monochromatic Box
Let k be a postivive integer. Let f(k) be the minimum number n such that no matter how we color the integer points in {(x,y,z): 0<x,y,z≤n} with k colors, there always exist 8 monochromatic points forming the vertices of a box whose sides are parallel to xy- or yz- or xz- plane. Determine f(k).