First, multiple-choice questions (7 points for each small question, ***42 points)

1, a, b and c are rational numbers, and the equation holds, so the value of 2a+999b+ 100 1c is ().

(a)1999 (b) 2000 (c) 2001(d) cannot be determined.

2. If 5a2+200 1a+9=0 and, the value of is ().

(A) (B) (C) (D)

3. It is known that in △ABC, ∠ACB=900, ∠ABC= 150 and BC= 1, then the length of AC is ().

(A) (B) (C) (D)

4. As shown in the figure, in △ABC, d is a point on the side of AC. In the following four cases, the case that △ABD∽△ACB is not necessarily true is ().

(A) (B)

(c) Abdul -ACB

5.① In the real number range, the roots of the unary quadratic equation are: ② In △ABC, if △ABC is an acute triangle; ③ In the sum of △ABC, A, B and C are three sides of △ABC and three sides of △ABC respectively. If so, the area s of △ ABC is greater than. In the above three propositions, the number of false propositions is ()

(A)0(B) 1(C)2(D)3

6. A shopping mall gives customers a discount, which stipulates that: ① shopping does not exceed 200 yuan at one time, and there is no discount; (2) If you are dissatisfied with 500 yuan after shopping in 200 yuan at one time, you will get a 10% discount on the marked price; (3) If the shopping exceeds 500 yuan at one time, 500 yuan will be given a discount according to Article (2), and the part exceeding 500 yuan will be 20% off. Someone went shopping twice and paid 168 yuan and 423 yuan respectively; If he only shops for the same goods once, the amount payable is ()

(A)522.8 yuan (B)5 10.4 yuan (C)560.4 yuan (D)472.8 yuan.

Fill in the blanks (7 points for each small question, 28 points for * * *)

1, the coordinate of a given point p in the rectangular coordinate system is (0, 1), o is the coordinate origin, ∠QPO= 1500, and the distance from p to q is 2, so the coordinate of q is.

2. It is known that two circles with radii of 1 and 2 are tangent to point P, so the distance from point P to the tangent of these two circles is.

3. If it is known to be a positive integer, then =

4. Positive integers. If you add 100 and 168 respectively, you can get two complete square numbers. This positive integer is.

Iii. Answering questions (***70 points)

1, there are three points A (0, 1), B (1, 3) and C (2, 6) in the rectangular coordinate system; It is known that the points on the straight line with the abscissa of 0, 1 2 are d, e and f respectively. The experimental value makes AD2+BE2+CF2 reach the maximum. (20 points)

(1) proves that if any integer is taken, quadratic functions always take integer values, then they are all integers;

(2) Write the inverse proposition of the above proposition, judge its truth and prove your conclusion. (25 points)

3. As shown in the figure, D and E are two points on the side of △ABC on BC, F is a point on the extension line of BC, ∠DAE=∠CAF. (1) Judge the positional relationship between the circumscribed circle of △ABD and the circumscribed circle of △AEC, and prove your conclusion; (2) If the radius of the circumscribed circle of △ABD is twice, BC=6, AB=4, find the length of BE.

Answer the question:

1, as shown in the figure, EFGH is an inscribed quadrilateral of a square ABCD, and the acute angle is sandwiched between two diagonal lines EG and FH.

The angle is θ, and ∠BEG and ∠CFH are both acute angles. It is known that EG=k, FH=, and the area of quadrilateral EFGH is S.

(1) Verification: sinθ =;

(2) Try to represent the area of a square.

2. Find all positive integers A, B, C, so that the equation about X,

All the roots of are positive integers.

3. At acute angle △ABC, AD⊥BC, D is vertical foot, DE⊥AC, E is vertical foot, DF⊥AB and F is vertical foot. O is the outer center of △ABC.

Verification: (1) △ AEF ∽△ ABC;

(2)AO⊥EF

4. As shown in the figure, in the quadrilateral ABCD, AC and BD intersect at point O, the straight line is parallel to BD, and the extension lines of AB, DC, BC, AD and AC intersect at points M, N, R, S and P respectively.

Verification: pmpn = prps