9. If, find the value of

TESTLAR

9. If , find the value of .

Answer
2. Let , where are constants. Given , find the value of .
Answer -17
3. If are non-zero real numbers, satisfying , prove that among there must be two opposite numbers.
4. If and , find the value of in terms of .
Answer
5. Given . Find the value of .
Answer 3
6. If , find the value of .
Answer

Lecture 3

1. Given that the equations in :

have a common solution. Find the common solution.
Answer -1/3
2. If positive numbers satisfy , solve the equation in

Answer 0,5
Given that the equation has positive integer solution, where is also a positive integer, find the minimum possible value of .

Answer 2

5. If and , find the value of

Answer
Given that the system of equations has no solution, where are integers between and 10 inclusive, find the number of the values of and
Answer 7
2. Solve the system of equations

Answer
3. Solve the system

Answer x=4, y=5, z=6 va x=-4, y=-5, z=-6
4. Find the values of such that the system of equations in and

has a positive integer solution .
Answer 12
5. Solve the system of equations

Answer x=0, y=1, z=3, u=5, v=7

Given , find the ratio .

Answer 1:2:3
5. Given , find the value of .
Answer
7. Given that , and . Prove that
8 Given , prove that

9. Given that , prove that at least one of is 2 .
10 Given that , prove that for any natural number

Testing Questions (B)
1. If (where are real numbers), then must be
(A) positive; (B) negative; (C) 0 ; (D) an integer.
2. Given and . Prove that

3. If , prove that .

4. If , find the value of .
Answer
5. (CHNMOL/2004) Given that the real numbers satisfy , find .
Answer 1 yoki -2
Testing Questions (B)

1. Quyidagi tenglamaning haqiqiy ildizlari yig’indisini toping .

Ans -2
2. (CHINA/2001) If is a factor of , find the value of .
ANS 31
3. Agar bo’lsa u holda quyidagi ifodaning qiymatini toping.
ANS 0
4. Factorize .
ANS

4. What is the minimum value of ?

ANS 4
7. If two real numbers and satisfy , find the value of .
ANS 0
8. Given that are integers. If , find the value of .
ANS 2
9. Given . Find the value of
ANS 4019
10. are two constants with . If the equation || has three distinct solutions for , find the value of .
ANS 3
Testing Questions (B)

4. Given that real numbers are all not zero, and . Find the value of , where

ANS 4013
5. The numbers are partitioned into two groups of 100 each, and the numbers in one group are arranged in ascending order: , and those in the other group are arranged in descending order. . Find the value of the expression

ANS 10000
8. (AHSME/1977) In If the points are on the sides and respectively, such that , then , in degrees, is
(A) , (B) , (C) , (D) (E) none of preceding.

9. If the lengths of three sides of a triangle are consecutive positive integers, and its perimeter is less than or equal to 100 , how many such acute triangles are there?

29
10. (A.HSME/1996) Triangles and are isosceles with , and intersects at If is perpendicular to , then is
(A) . (B) (C) . (D) (E) not uniquely determined.

Testing Questions (B)

1. (CHINA/1991) In is on the side , and the angle bisector of intersects at such that and . Then in degrees is
(A) , (B) (C) , (D) .
2. (CHINA/1998) In triangle . Extend to an arbitrary point . The angle bisectors of angle and intersect at , and the angle bisectors of and intersect at , and so on. The angle bisectors of and intersect at . Find the size of in degrees.
3 gradus
3. In are on , such that . Prove that .

4. In right-angled is on such that is on such that . Given that , find in degrees.

30 gradus
5. (MOSCOW/1952) In is on the side and is on the side , such that . Find in degrees.
30 gradus
7. In the right triangle are points on and respectively. Prove that

8. (CHNMOL/1990) is an isosceles triangle with . There are 100 points on the side . Write , find the value of .

Ans 400
9. In is the midpoint of are two points on and respectively, and . Prove that .

10. (CHINA/1996) Given that is an inner point of the equilateral triangle , such that . Find the length of the side of .

Ans
Testing Questions (B)

(SSSMO(J)/2003/Q8) is a chord in a circle with center and radius . The point divides the chord such that and . Find the length in .

Ans 25
2. (CHINA/1996) is a rectangle, is an inner point of the rectangle such that , find .
ANs
Determine whether such a right-angled triangle exists: each side is an integer and one leg is a multiple of the other leg of the right angle.

4. (AHSME/1996) In rectangle is trisected by and , where is on is on and . Which of the following is closest to the area of the rectangle

(A) 110 , (B) 120 , (C) 130 , (D) 140, (E) 150 .