MathDB

2

Part of 1997 National High School Mathematics League

Problems(2)

Regular Tetrahedron

Source: 1997 National High School Mathematics League, Exam One, Problem 2

3/4/2020
In regular tetrahedron ABCDABCD, E∈AB,F∈CDE\in AB,F\in CD, satisfying: ∣AE∣∣EB∣=∣CF∣∣FD∣=λ(λ∈R+)\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+). Note that f(λ)=αλ+βλf(\lambda)=\alpha_{\lambda}+\beta_{\lambda}, where αλ=<EF,AC>,αλ=<EF,BD>\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>. (A)\text{(A)} f(λ)f(\lambda) increases in (0,+∞)(0,+\infty) (B)\text{(B)} f(λ)f(\lambda) decreases in (0,+∞)(0,+\infty) (C)\text{(C)} f(λ)f(\lambda) increases in (0,1)(0,1), decreases in (1,+∞)(1,+\infty) (D)\text{(D)} f(λ)f(\lambda) is a fixed value in (0,+∞)(0,+\infty)
geometry3D geometrytetrahedron
Complex Number

Source: 1997 National High School Mathematics League, Exam Two, Problem 2

3/4/2020
For real numbers x0,x1,⋯ ,xnx_0,x_1,\cdots,x_n, there exists real numbers y0,y1,⋯ ,yny_0,y_1,\cdots,y_n, satisfying that z02=z12+z22+⋯+zn2z_0^2=z_1^2+z_2^2+\cdots+z_n^2, where zk=xk+iyk(k=0,1,⋯ ,n)z_k=x_k+\text{i}y_{k}(k=0,1,\cdots,n). Find all such (x0,x1,⋯ ,xn)(x_0,x_1,\cdots,x_n).
complex numbers