Suppose y(x)≥0 for all real x. Find all solutions of the differential equation \(\frac{dy}{dx}=\sqrt{y}\), y(0)=0.
Category Archives: problem
2009-17 Relatively prime sequence
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.)
2009-16 Commutative ring
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.
2009-15 Double sum
What is the value of the following infinite series?
\(\displaystyle\sum_{n=2}^\infty \sum_{m=1}^{n-1} \frac{(-1)^n}{mn}\)
2009-14 New notion on the convexity
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.)
POW will resume on Sep. 4, Friday for Fall 2009
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.
2009-13 Distances between points in [0,1]^2
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 \(r_1^2+r_2^2+\cdots+r_n^2\le 4\).
2009-12 Colorful sum
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.
2009-11 Circles and lines
Does there exist a set of circles on the plane such that every line intersects at least one but at most 100 of them?
2009-10 x and cos x
Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.