MathDB

2

Part of 2014 Thailand TSTST

Problems(5)

finite number of solutions to a diophantine equation

Source: UW PT 101

10/13/2021
Prove that the equation x8=n!+1x^8 = n! + 1 has finitely many solutions in positive integers.
number theoryDiophantine equation
Sqrt(sum of a^2+b^2/a+b) \geq sum of sqrt(2ab/3a+b+2c)

Source: 2013 Thailand October Camp Inequalities Exam p2

3/7/2022
Let a,b,ca, b, c be positive real numbers. Prove that a2+b2a+b+b2+c2b+c+c2+a2c+a≥2ab3a+b+2c+2bc3b+c+2a+2ca3c+a+2b.\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.
inequalities
Geometric Inequality

Source: 2013 Thailand October Camp Algebra and Functional Equations Exam p2

3/8/2022
In a triangle ABCABC, let x=cos⁡A−B2,y=cos⁡B−C2,z=cos⁡C−A2x=\cos\frac{A-B}{2},y=\cos\frac{B-C}{2},z=\cos\frac{C-A}{2}. Prove that x4+y4+z2≤1+2x2y2z2.x^4+y^4+z^2\leq 1+2x^2y^2z^2.
geometric inequalitygeometryinequalities
There are 2 ‘i’ such that a_i<a_(i+1)

Source: 2013 Thailand October Camp Combinatorics Exam p2

3/7/2022
Find the number of permutations (a1,a2,...,a2013)(a_1, a_2, . . . , a_{2013}) of (1,2,…,2013)(1, 2, \dots , 2013) such that there are exactly two indices i∈{1,2,…,2012}i \in \{1, 2, \dots , 2012\} where ai<ai+1a_i < a_{i+1}.
combinatorics
pependicularity wanted, touchpoints of incircle related

Source: 2013 Thailand October Camp Geometry Exam p2

10/22/2020
In a triangle ABCABC, the incircle with incenter II is tangent to BCBC at A1,CAA_1, CA at B1B_1, and ABAB at C1C_1. Denote the intersection of AA1AA_1 and BB1BB_1 by GG, the intersection of ACAC and A1C1A_1C_1 by XX, and the intersection of BCBC and B1C1B_1C_1 by YY . Prove that IG⊥XYIG \perp XY .
geometryperpendicularincircle