Let a,b,c∈R. Prove that ∑cyc(a3−b3)2+3∑cyc(a2−b2)2+6(a−b)(b−c)(c−a)(ab+bc+ca)≥0. (LightLucifer)
2011 Mathcenter Contest + Longlist
Round 1 (one and only)
sl = longlist
For natural n, define fn=[2n√69]+[2n√96] Prove that there are infinite even integers and infinite odd integers that appear in number f1,f2,…. (tatari/nightmare)
We will call the sequence of positive real numbers. a1,a2,…,an of length n when a1≥a1+a22≥⋯≥a1+a2+⋯+ann.Let x1,x2,…,xn and y1,y2,…,yn be sequences of length n. Prove that n∑i=1xiyi≥1n(n∑i=1xi)(n∑i=1yi). (tatari/nightmare)
At the 69 Thailand-Yaranaikian meeting attended by 96 Thai delegates and a number (unknown) from the Yaranakian country. Some time after the meeting took place, the meeting also discovered something amazing that happened in this meeting!! That is, regardless of whether we select at least 69 of Thai participants and select all the Yaranikian country participants who are known to Thais in the initial selection group, there is at least 1 person fo form a minority. They found in that minority, there was always 1 more Yaranikhians than Thais. Prove that there must be at least 28 of the Yaranaikian attendees who know the Thai delegates. (Note: In this meeting, none of the attendees were half-breeds. Thai-Yara Nikian) (tatari/nightmare)
Given x,y,z∈R+. Find all sets of x,y,z that correspond to x+y+z=x2+y2+z2+18xyz=1(Zhuge Liang)
Let x,y,z represent the side lengths of any triangle, and s=x+y+z2 and the area of this triangle be √s square units. Prove that s(1x(s−x)2+1y(s−y)2+1z(s−z)2)≥12(1s−x+1s−y+1s−z)(Zhuge Liang)
Find the function f:R−{0}→R such that f(x)+f(1−1x)=1x,∀x∈R−{0,1}(-InnoXenT-)
Let a,b,c∈R+. Prove that a11b5c5+b11c5a5+c11a5b5≥a+b+c(Real Matrik)
Let a,b,c∈R+ If 3=a+b+c≤3abc , prove that 1√2a+1+1√2b+1+1√2c+1≤(32)3/2(Real Matrik)
Longlist (the rest)
Let a,b,c∈R+ with abc=1. Prove that a3b3a+b+b3c3b+c+c3c3c+a≥12(1a+1b+1c) (Zhuge Liang)
Given k1,k2,...,kn∈R+, find all the naturals n such that k1+k2+...+kn=2n−31k1+1k2+...+1kn=3(Zhuge Liang)
Let p,q,r∈R with pqr=1. Prove that (11−p)2+(11−q)2+(11−r)2≥1 (Real Matrik)
Let a,b,c∈R+ with a+b+c=3. Prove that 2(ab+bc+ca)≤5+abc(Real Matrik)