Let 1≤a1<a2<…<ak<n be a sequence of integers such that gcd(ai,aj)=1 for all 1≤i<j≤k. What is the maximum value of k?
(Problem updated on Sep. 26, 8AM: gcd(ai,aj)=1.)
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Let 1≤a1<a2<…<ak<n be a sequence of integers such that gcd(ai,aj)=1 for all 1≤i<j≤k. What is the maximum value of k?
(Problem updated on Sep. 26, 8AM: gcd(ai,aj)=1.)
Let k>1 be a fixed integer. Let π be a fixed nonidentity permutation of {1,2,…,k}. Let I be an ideal of a ring R such that for any nonzero element a of R, aI≠0 and Ia≠0 hold.
Prove that if \(a_1 a_2\ldots a_k=a_{\pi(1)} a_{\pi(2)} \ldots a_{\pi(k)}\) for any elements \(a_1, a_2,\ldots,a_k \in I\), then R is commutative.
What is the value of the following infinite series?
\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)
Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.
Prove that a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.
(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)
KAIST Problem of the Week will continue in the fall semester of 2009. The first problem of 2009 Fall will be posted online on Sep. 4, Friday 3PM.
Let \(P_1,P_2,\ldots,P_n\) be n points in {(x,y): 0<x<1, 0<y<1} (n>1). Let \(r_i=\min_{j\neq i} d(P_i,P_j)\) where d(x,y) means the distance between two points x and y. Prove that \(r_1^2+r_2^2+\cdots+r_n^2\le 4\).
Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.
Does there exist a set of circles on the plane such that every line intersects at least one but at most 100 of them?
Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.
Suppose that * is an associative and commutative binary operation on the set of rational numbers such that
Prove that either