MathDB

3

Part of 2014 Thailand TSTST

Problems(5)

Inequality with abc=1

Source: 2013 Thailand October Camp Inequalities Exam p3

3/7/2022
For all pairwise distinct positive real numbers a,b,ca, b, c such that abc=1abc = 1, prove that 1a+1b+1c+1(a+b+c+1)2+38(a2−b2)3+(b2−c2)3+(c2−a2)3(a−b)3+(b−c)3+(c−a)33≥1.\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}{(a+b+c+1)^2}+\frac{3}{8}\sqrt[3]{\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}}\geq 1.
inequalities
Sum of abc that a+b+c=2013

Source: 2013 Thailand October Camp Combinatorics Exam p3

3/7/2022
Let SS be the set of all 3-tuples (a,b,c)(a, b, c) of positive integers such that a+b+c=2013a + b + c = 2013. Find ∑(a,b,c)∈Sabc.\sum_{(a,b,c)\in S} abc.
combinatoricsalgebra
Sum of digit of 9^n is more than 9

Source: 2013 Thailand October Camp Number Theory Exam p3

3/7/2022
Let s(n)s(n) denote the sum of digits of a positive integer nn. Prove that s(9n)>9s(9^n) > 9 for all n≥3n\geq 3.
number theorysum of digits
Simplify terms

Source: 2013 Thailand October Camp Algebra and Functional Equations Exam p3

3/8/2022
Define ak=22k−2013+ka_k=2^{2^{k-2013}}+k for all integers kk. Simplify (a0+a1)(a1−a0)(a2−a1)⋯(a2013−a2012).(a_0+a_1)(a_1-a_0)(a_2-a_1)\cdots(a_{2013}-a_{2012}).
algebrasimplify
4 collinear orhocenters wanted, incenter of tangential ABCD related

Source: 2013 Thailand October Camp Geometry Exam p3

10/22/2020
Let OO be the incenter of a tangential quadrilateral ABCDABCD. Prove that the orthocenters of △AOB\vartriangle AOB, △BOC\vartriangle BOC, △COD\vartriangle COD, △DOA\vartriangle DOA lie on a line.
geometryincentercollinearorthocentertangential