MathDB

8

Part of 2008 Mathcenter Contest

Problems(3)

sum [(1+ab)/(1+a)]^ {2008}>=4 if a,b,c,d>0 with abcd=1

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/10/2022
Let a,b,c,d∈R+a,b,c,d \in R^+ with abcd=1abcd=1. Prove that (1+ab1+a)2008+(1+bc1+b)2008+(1+cd1+c)2008+(1+da1+d)2008≥4\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4 (dektep)
inequalitiesalgebra
Goblin Tribe game

Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/11/2022
Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level kk (k≠).0k \not=). 0) comes up. Let's assume Goblin AA. Goblin AA explodes itself. Goblin A's Level becomes 00. After that, Level of Goblin kk next to Goblin AA clockwise gets Level 11. Prove that: 1.) If after that Goblin kk next to Goblin AA explodes itself and keep doing this, k′k' next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again. 2) 2.) If after that we can choose any Goblin whose level is not 00 to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning.
(gools)
combinatorics
\sum d(A_i,A_j)<10^{-2008}

Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/9/2022
Prove that there are different points A0  ,A1  ,⋯A2550A_0 \,\, ,A_1 \,\, , \cdots A_{2550} on the XYXY plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If d(Ai,Aj) d(A_i,A_j) represents the distance between Ai  ,AjA_i\,\, , A_j then ∑0≤i<j≤2550{d(Ai,Aj)}<10−2008 \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008} Note : {x} \{x \} represents the decimal part of x e.g. {3.16}=0.16 \{ 3.16\} = 0.16.
(passer-by)
geometrycombinatoricscombinatorial geometry