This document contains 20 math problems and their solutions. The problems involve proofs of algebraic identities, calculating values based on expressions, determining maximum/minimum values, and finding values that satisfy given equations. The document provides step-by-step workings and explanations for determining the answers to the problems.

1. Page 1 of 9
Smart Solution
By NOVIAN NUR FATIHAH
1. Prove for any real numbers a, b apply
Proof :
2. Prove for any positive numbers a, b, c and d apply √ !
Proof :
√ √ √√ √ √
3. If a and b are arbitrary numbers, prove that !
Proof :
4. Calculate the value of :
√ √ √ √
Answer :

2. Page 2 of 9
√ √
√
√ √
So, √ √ √ √
= ( ) ( ) ( ) ( )
( )
5. Given are real numbers. If and
. Determine the maximum value!
Answer :
With abc formula will obtained :
√ √
So the maximum value is
√

3. Page 3 of 9
6. If , then specify the value of !
Answer :
This is a sequential multiplication.
So a = 6 and n = 36
7. If determine the value of a, b, x, and y!
Answer :
Because is the square of the digits unit numbers 0, 1, 4, 5, 6 or 9.
Mean b = 00, 11, 44, 55, 66 or 99
Quadratic number when divided by 4 the remaining 0 (for even) or 1 (for odd)
Numbers is divisible by 4 if the last 2 digits is divisible by 4, so b = 44
aabb = aa44 = 11 x a04 then a = 7
aabb = 7744 = 882
So a = 7, b = 4, x = 8, and y = 8
8. If where A and B are natural numbers, then determine the
values of A and B!
Answer :
[ ]
[ ] [ ]
[ ] [ ]

4. Page 4 of 9
[ ] [ ]
9. If , determine the value of A!
Answer :
( ) ( )
10. is a different natural numbers. If
then, determine the value of !
Answer :
11. Given x, y, z and t are real numbers that are not zero and satisfies the equation:
Determine the value of !
Answer:

5. Page 5 of 9
12. Determine the number of numbers consisting of a 4 digit number abcd that
satisfies the equation !
Answer:
So that the numbers : 1999, 2999, 3999, … , 9999
13. A function is expressed as
√
!
Determine the value of ( ) ( ) ( )
Answer :

6. Page 6 of 9
√ √
√ √
√ √
√ √
√ √
( ) ( )
( ) ( )
… …
… …
( ) ( )
------------------------------- +
= 1002
14. Given a and b are real numbers that are satisfy :
Determine the value !
Answer :
------------------------------------------ +
√ √
15. Defined
√ √ √
for all natural numbers. Determine
the value of !
Answer :

7. Page 7 of 9
√ √ (√ √ ) (√ √ √ )
√ √ √
√ √
Example,
√ √ √
√ √
√ √ √ √
(√ √ ) (√ √ ) (√ √ ) (√ √ ) (√ √ )
16. Given
Show the nearest integer of !
Answer :
( ) ( )
( ) ( ) ( )

8. Page 8 of 9
17. Find the value of ( ) ( ) ( ) !
Answer :
18. Prove that
Prove :
… … …
-------------------------------------------------------------- +
( )
19. If
then determine the formula for the value of A !
Answer :
∑ (∑ )

9. Page 9 of 9
Because,
( )
then :
∑ ∑ ∑ ∑
∑
∑
20. Determine the value of real numbers that satisfy the equation:
Answer:
If equation (1) times , equation (2) times and equation (3) times then
obtained:
by eliminating then obtained