MathDB

1996 National High School Mathematics League

Part of National High School Mathematics League

Subcontests

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Three Numbers

Let x∈(−12,0)x\in\left(-\frac{1}{2},0\right), α1=cos⁡(sin⁡xπ),α2=sin⁡(cos⁡xπ),α1=cos⁡(x+1)π\alpha_1=\cos(\sin x\pi),\alpha_2=\sin(\cos x\pi),\alpha_1=\cos (x+1)\pi, then (A)α3<α2<α1(B)α1<α3<α2(C)α3<α1<α2(D)α2<α3<α1\text{(A)}\alpha_3<\alpha_2<\alpha_1\qquad\text{(B)}\alpha_1<\alpha_3<\alpha_2\qquad\text{(C)}\alpha_3<\alpha_1<\alpha_2\qquad\text{(D)}\alpha_2<\alpha_3<\alpha_1

A Party of Some People

n(n≥6)n(n\geq6) people attend a party. It is known that (1) Anyone knows at least [n2]\left [\frac{n}{2} \right] people. (2) For any [n2]\left[\frac{n}{2}\right] people, either two people among them know each other, or among people else, there are two people know each other. Prove that there are three people in the nn people, they know one another.
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