MathDB

Problem 1

Part of 2017 Vietnamese Southern Summer School contest

Problems(3)

Asking for address

Source: Southern Summer School, gr. 10

7/9/2017
A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits, and if you permute its digits, you will have other 5 numbers. The interesting thing is that the sum of these 5 numbers is exactly 2017. That's all.". After a while, B can determine A's house number. And you, can you find his house number?
arithmeticalgebra
Weird inequality

Source: Southern Summer School, gr. 11

7/9/2017
Let x,y,zx,y,z be the non-negative real numbers satisfying xy+yz+zx≤1xy+yz+zx\leq 1. Prove that: 1−xy−yz−zx≤(6−26)(1−min⁡{x,y,z}).1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).
Inequalityalgebrainequalities
Convergent sequence of real numbers

Source: Southern Summer School, gr. 12

7/9/2017
Given a real number aa and a sequence (xn)n=1∞(x_n)_{n=1}^\infty defined by: {x1=1x2=0xn+2=xn2+xn+124+a\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right. for all positive integers nn. 1. For a=0a=0, prove that (xn)(x_n) converges. 2. Determine the largest possible value of aa such that (xn)(x_n) converges.
analysisSequencereal numberConvergence