MathDB

1

Define a New Calculation

x*y=ax+by+cxy

for all real numbers

x,y

, where

a,b,c

are uncertain. It is known that

1*2=3,2*3=4

. If there exists a real number

d

, for any real number

x

,

x*d=x

, then

d=

________.

9

1

Multiple of 11

In the given sequence

1,4,8,10,16,19,21,25,30,43

, sum of a few adjacent numbers in the sequence is a multiple of

11

. The number of such number sets is________.

8

1

The Number of the Solutions

The number of nonnegative solutions to the equation

2x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}=3

is________.

7

1

Triangle Problem

In

\triangle ABC

, if

A,B,C

are geometric series, and

b^2-a^2=ac

, then

B=

________.

6

1

Increasing Sequence or Dncreasing Sequence

Let

0<a<1

.

x_1=a,x_2=a^{x_1},\cdots,x_n=a^{x_{n-1}}

. Then sequence

(x_n)

\text{(A)}

Is an increasing sequence.

\text{(B)}

Is an decreasing sequence.

\text{(C)}

Increases when

n

is odd, decreases when

n

is even.

\text{(D)}

Decreases when

n

is odd, increases when

n

is even.

5

1

Complex Numbers

Let

Z,W,\lambda

be complex numbers,

|\lambda|\neq1

. Which statements are correct about the equation

\overline{Z}-\lambda Z=W

? I.

Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}

is a solution to the equation. II. The equation has only one solution. III. The equation has two solutions. IV. The equation has infinitely many solutions.

\text{(A)}

Only I and II.

\text{(B)}

Only I and III.

\text{(C)}

Only I and IV.

\text{(D)}

None of

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

.

4

2

Choose the Figure

Which figure can be the images of equations

mx+ny^2=0

and

mx^2+ny^2=1

(m,n\neq0)

? https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9kLzUzZTAwZjU1YzEyN2I3ZDJjNjcwNDQ2ZmQ5MDBmYWZlODAwNGU0LnBuZw==&rn=YWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYS5wbmc=

Geometry Inequality

Given 5 points on a plane. Let

\lambda

be the ratio of maximum value between the points to minimum value between the points. Prove that

\lambda\geq2\sin\frac{3}{10}\pi

.

3

2

Anti-trigonometric Function

If

\arccos\frac{4}{5}-\arccos\left(-\frac{4}{5}\right)=\arcsin x

, then

\text{(A)}x=\frac{24}{25}\qquad\text{(B)}x=-\frac{24}{25}\qquad\text{(C)}x=0\qquad\text{(D)}

No such

x

A Problem About Football Match

16 cities attend a football match. Each city has two teams: A and B. According to the rule, each team can compete with any other team of other cities at most once. After a few days, we find that except team A of city 1, the number of matches they've played are different. Then, how many matches have team A of city 1 played?

2

Both Analytic Geometry and 3D Geometry

PQ

is a chord of parabola

y^2=2px(p>0)

and

PQ

pass its focus

F

. Line

l

is its directrix. Projection of

PQ

on

l

is

MN

. The area of curved surface that

PQ

rotate around

l

is

S_1

, the area of spherical surface of the ball with diameter of

MN

is

S_2

, then

\text{(A)}S_1>S_2\qquad\text{(B)}S_1<S_2\qquad\text{(C)}S_1\geq S_2\qquad\text{(D)}

Not sure

3D Geometry Problem

In cube

ABCD-A_1B_1C_1D_1

,

E

is midpoint of

BC

,

F\in AA_1

, and

A_1F:FA=1:2

. Calculate the dihedral angle between plane

B_1EF

and plane

A_1B_1C_1D_1

.

1

2

Too Easy...

a,b

are real numbers (neither is

0

).Given two conditions: A:

a>0

. B:

a>b

and

a^{-1}>b^{-1}

. Then, which one of the followings are true?

(\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})

A is insufficient and unnecessary condition of B.

Analytic Geometry Problem

In rectangular coordinate system

xOy, A(x_1,y_1), B(x_2,y_2)

, where

x_1,y_1,x_2,y_2

are 1-digit-numbers. Intersection angle between

OA

and

x

-axis positive direction is larger than

\frac{\pi}{4}

, intersection angle between

OB

and

x

-axis positive direction is smaller than

\frac{\pi}{4}

. Projection of

A

on

y

-axis is

A'

, projection of

B

on

x

-axis is

B'

. Area of

\triangle OBB'

is

33.5

larger than

\triangle OAA'

. Find all 4-digit-number

\overline{x_1x_2y_1y_2}

.

1985 National High School Mathematics League

Subcontests

Define a New Calculation

Multiple of 11

The Number of the Solutions

Triangle Problem

Increasing Sequence or Dncreasing Sequence

Complex Numbers

Choose the Figure

Geometry Inequality

Anti-trigonometric Function

A Problem About Football Match

Both Analytic Geometry and 3D Geometry

3D Geometry Problem

Too Easy...

Analytic Geometry Problem