MathDB

3

Part of 2000 National High School Mathematics League

Problems(2)

Hyperbola

Source: 2000 National High School Mathematics League, Exam One, Problem 3

3/10/2020
A(−1,1)A(-1,1), B,CB,C are points on hyperbola x2−y2=1x^2-y^2=1. If △ABC\triangle ABC is a regular triangle, then the area of △ABC\triangle ABC is (A)33(B)332(C)33(D)63\text{(A)}\frac{\sqrt3}{3}\qquad\text{(B)}\frac{3\sqrt3}{2}\qquad\text{(C)}3\sqrt3\qquad\text{(D)}6\sqrt3\qquad
conicshyperbolageometry
Call One Another

Source: 2000 National High School Mathematics League, Exam Two, Problem 3

3/11/2020
There are nn people, any two of them have called each other at most once. In any group of n−2n-2 of them, anyone of the group has called with other people in this group for 3k3^k times, where kk is a non-negative integer (the value of kk is fixed). Find all possible integers nn.