1988 National High School Mathematics League

Part of National High School Mathematics League

Subcontests

(12)

1

Inequality

a,b

are real numbers, satisfying that

\frac{1}{a}+\frac{1}{b}=1

. Prove that for any

n\in\mathbb{Z}_+

(a+b)^{2n}-a^n-b^n\geq2^{2n}-2^{n+1}

1

Complex Plane

On complex plane, path equation of moving point

Z_1

|Z_1-Z_0|=|Z_1|

Z_0(Z_0\neq0)

is a fixed point. Another moving point

Z

ZZ_1=-1

. Find the path of

Z

and describe its location and shape.

1

Rectangle Rotation

Lengths of two sides of a rectangle are

\sqrt2,1

. The rectangle rotates a round around one of its diagonal. Find the volume of the revolved body.

1

Count Problem

Two teams take part in a competition. There are 7 members (numbered 1 to 7). Two member 1 start a competition first. The failer is sifted out, and the winner start a new competition with member 2 in the other team. ... When all members of a team are out, the competition ends. The number of possible situations is________.

1

Triangle Problem

\triangle ABC

\angle A=\alpha

CD,BE

are height on sides

AB,AC

\frac{|DE|}{|BC|}=

1

Binomial Expansion

In the binomial expansion of

(\sqrt{x}+2)^{2n+1}

, sum of coefficients that power of

x

is an integer is________.

1

Arithmetic Sequence

x\neq y

. Two sequences

x,a_1,a_2,a_3,y

b_1,x,b_2,b_3,y,b_4

are arithmetic sequence. Then

\frac{b_4-b_3}{a_2-a_1}=

1

All Lines

In rectangular coordinate system, define that if and only if both

x

y

-axis of a point are integers, we call it itegral point.

I

is set of all lines,

M

is set of lines that pass exactly one intengral point,

N

is set of lines that pass no itengral point,

P

is set of lines that pass infinitely many itengral points. Then, how many conclusions are right? (1)

M\cup N\cup P=I

N\neq\varnothing

M\neq\varnothing

P\neq\varnothing

\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4\qquad

1

3D Geometry Problem

Given three planes

\alpha,\beta,\gamma

. Intersection angle between any two planes are all

\theta

\alpha\cap\beta=a,\beta\cap\gamma=b,\gamma\cap\alpha=c

. Given two conditions: A:

\theta>\frac{\pi}{3}

a,b,c

share one point.

(\text{A})

A is sufficient but unnecessary condition of B.

(\text{B})

A is necessary but insufficient condition of B.

(\text{C})

A is sufficient and necessary condition of B.

(\text{D})

2

Three point sets

M,N,P

are three point sets on a plane.

M=\{(x,y)||x|+|y|<1\}

N=\{(x,y)|\sqrt{(x-\frac{1}{2})^2+(y+\frac{1}{2})^2}+\sqrt{(x+\frac{1}{2})^2+(y-\frac{1}{2})^2}<2 \sqrt2 \}

P=\{(x,y)||x+y|<1,|x|<1,|y|<1\}

\text{(A)}M\subset P\subset N\qquad\text{(B)}M\subset N\subset P\qquad\text{(C)}P\subset N\subset M\qquad\text{(D)}

\text{(A)(B)(C)}

Line Family

On the coordinate plane, is there a line family of infinitely many lines

l_1,l_2,\cdots,l_n,\cdots

, satisfying the following? (1) Point

(1,1)\in l_n

n\in \mathbb{Z}_{+}

n\in \mathbb{Z}_{+}

k_{n+1}=a_n-b_n

k_{n+1}

is the slope of

l_{n+1}

a_n,b_n

are intercepts of

l_n

x

y

k_nk_{n+1}\geq0

n\in \mathbb{Z}_{+}

2

Ellipse Problem

If the coordinate origin is inside the ellipse

k^2x^2+y^2-4kx+2ky+k^2-1=0

, then the range value of

k

\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1

Three Equal Division

\triangle ABC

P,Q,R

divides the perimeter of

\triangle ABC

into three equal parts.

P,Q\in AB

\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}

2

Three functions

We have three functions. The first one is

y=\phi(x)

. The second one is the inverse function of the first one. The figure of the third funcion is symmetrical to the second one about line

x+y=0

. Then, the third function is

\text{(A)}y=-\phi(x)\qquad\text{(B)}y=-\phi(-x)\qquad\text{(C)}y=-\phi^{-1}(x)\qquad\text{(D)}y=-\phi^{-1}(x)

Recurrence Sequence

Define sequence

(a_n):a_1=1,a_2=2,a_{n+2}=\begin{cases} 5a_{n+1}-3a_n,\text{if }a_n\cdot a_{n+1}\text{ is even}\\ a_{n+1}-a_n,\text{if }a_n\cdot a_{n+1}\text{ is odd} \end{cases}

Prove that for all

n\in\mathbb{Z}_+

a_n\neq0