MathDB

2

Part of 2003 National High School Mathematics League

Problems(2)

Choose the Figure

Source: 2003 National High School Mathematics League, Exam One, Problem 2

3/16/2020
If a,b∈R,ab≠0a,b\in\mathbb{R},ab\neq 0, the the possible figure of ax−y+b=0ax-y+b=0 and bx2+ay2=abbx^2+ay^2=ab is https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy80Lzc3NGNjZWNiN2ZjYzIxMTJlYWE5NDlmZmQ0ZjE1NzgwNmNhM2JiLnBuZw==&rn=MTI0MjQ1ODUyMTI0MjQyNTI0MjUyNS5wbmc=
Number Theory

Source: 2003 National High School Mathematics League, Exam Two, Problem 2

3/16/2020
Let the lengths of three sides of a triangle be l,m,n(l>m>n)l, m, n(l>m>n). If {3l104}={3m104}={3n104}\left\{\frac{3^l}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}, find the minimum value of the perimeter of the triangle. Note: {x}=x−[x]\{x\}=x-[x] and [x][x] denotes the integral part of number xx.
number theory