MathDB

1

Part of 2006 Taiwan National Olympiad

Problems(5)

Easy problem

Source: Taiwan NMO 2006 oral test

3/21/2006
Let AA be the sum of the first 2k+12k+1 positive odd integers, and let BB be the sum of the first 2k+12k+1 positive even integers. Show that A+BA+B is a multiple of 4k+34k+3.
number theory proposednumber theory
Symmetric

Source: Taiwan NMO 2006

3/21/2006
Positive reals a,b,ca,b,c satisfy abc=1abc=1. Prove that 1+3a+b+c≥6ab+bc+ca\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}.
inequalitiesinequalities proposed
Two circles

Source: Taiwan NMO 2006

3/21/2006
P,QP,Q are two fixed points on a circle centered at OO, and MM is an interior point of the circle that differs from OO. M,P,Q,OM,P,Q,O are concyclic. Prove that the bisector of ∠PMQ\angle PMQ is perpendicular to line OMOM.
geometrycircumcircleangle bisectorgeometry proposed
Equation: (x+y)/(x²-xy+y²) = 3/7

Source: Taiwan NMO 2006

3/21/2006
Find all integer solutions (x,y)(x,y) to the equation x+yx2−xy+y2=37\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}.
number theory proposednumber theory
Safes and keys

Source: Taiwan NMO 2006

3/21/2006
There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)
probabilityexpected valuecombinatorics proposedcombinatoricsRandom walkrandom walks