MathDB

1

Part of 1993 National High School Mathematics League

Problems(2)

Problem of "0+0=0"

Source: 1993 National High School Mathematics League, Exam One, Problem 1

2/27/2020
If M={(x,y)∣∣tan⁡πx∣+sin⁡2πx=0},N={(x,y)∣x2+y2≤2}M=\{(x,y)||\tan\pi x|+\sin^2\pi x=0\},N=\{(x,y)|x^2+y^2\leq2\}, then ∣M∩N∣|M\cap N| is equal to (A)4(B)5(C)8(D)9\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}8\qquad\text{(D)}9
Simple Geometry

Source: 1993 National High School Mathematics League, Exam Two, Problem 1

2/28/2020
In convex quadrilateral ABCDABCD, only DD is an obtuse angle. Use some line segments to divide it into nn obtuse triangles. But on its sides (except A,B,C,DA,B,C,D ), there is no vertex of triangles we divided into. Prove that if and only if n≥4n\geq4, we can divide the convex quadrilateral into such nn triangles.
geometry