MathDB

1

Part of 2000 National High School Mathematics League

Problems(2)

Two Sets

Source: 2000 National High School Mathematics League, Exam One, Problem 1

3/10/2020
If A={x∣x−2≤0},B={x∣10x2−2=10x}A=\{x|\sqrt{x-2}\leq0\},B=\{x|10^{x^2-2}=10^{x}\}, then A∩(R\B)A\cap(\mathbb{R}\backslash B) is (A){2}(B){−1}(C){x∣x≤2}(D)∅\text{(A)}\{2\}\qquad\text{(B)}\{-1\}\qquad\text{(C)}\{x|x\leq2\}\qquad\text{(D)}\varnothing
Area in Geometry

Source: 2000 National High School Mathematics League, Exam Two, Problem 1

3/11/2020
In acute triangle ABCABC, D,ED,E are two points on side BCBC, satisfying that ∠BAE=∠CAF\angle BAE=\angle CAF. FM⊥AB,EN⊥ACFM\perp AB,EN\perp AC (M,NM,N are foot points). AEAE intersects the circumcircle of △ABC\triangle ABC at DD. Prove that the area of △ABC\triangle ABC and quadrilateral AMDNAMDN are equal.
geometrycircumcircle