MathDB

3

Part of 1992 National High School Mathematics League

Problems(2)

Tetrahedron Problem

Source: 1992 National High School Mathematics League, Exam One, Problem 3

2/26/2020
Areas of four surfaces of a tetrahedron are S1,S2,S3,S4S_1,S_2,S_3,S_4. And the largest one of them is SS. λ=S1+S2+S3+S4S\lambda=\frac{S_1+S_2+S_3+S_4}{S}, then λ\lambda always satisfies (A)2<λ≤4(B)3<λ<4(C)2.5<λ≤4.5(D)3.5<λ<5.5\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5
geometry3D geometrytetrahedron
Area Smaller, Smaller...

Source: 1992 National High School Mathematics League, Exam Two, Problem 3

2/27/2020
In coordinate system, there are six points Pi(xi,yi)(i=1,2,⋯ ,6)P_i(x_i,y_i)(i=1,2,\cdots,6), satisfying: (1) xi,yi∈{−2,−1,0,1,2}x_i,y_i\in\{-2,-1,0,1,2\}. (2) For any three points, they are not collinear. Prove that there exists a triangle △PiPjPk(1≤i<j<k≤6)\triangle P_iP_jP_k(1\leq i<j<k\leq6), its area is not larger than 22.
geometry