Yusuf Sila: September 2011

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Quickly Square Two Digit Number : Up-Down Method

11:52 PM

Muhammad Yusuf , Posted in Math Trick , 0 Comments

Square a 2 Digit Number, for this example 37:

Look for the nearest 10 boundary
In this case up 3 from 37 to 40.
Since you went UP 3 to 40 go DOWN 3 from 37 to 34.
Now mentally multiply 34x40
The way I do it is 34x10=340;
Double it mentally to 680
Double it again mentally to 1360
This 1360 is the FIRST interim answer.
37 is "3" away from the 10 boundary 40.
Square this "3" distance from 10 boundary.
3x3=9 which is the SECOND interim answer.
Add the two interim answers to get the final answer.
Answer: 1360 + 9 = 1369

With practice this can easily be done in your head.

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Quickliy Multiply Any Number by 11

11:51 PM

Muhammad Yusuf , Posted in Math Trick , 0 Comments

You can directly write down the answer to any number multiplied by 11.

Take for example the number 51236 X 11.
First, write down the number with a zero in front of it.051236
The zero is necessary so that the rules are simpler.
Draw a line under the number.
Bear with me on this one. It is simple if you work through it slowly. To do this, all you have to do this is "Add the neighbor". Look at the 6 in the "units" position of the number. Since there is no number to the right of it, you can't add to its "neighbor" so just write down 6 below the 6 in the units col.
For the "tens" place, add the 3 to the its "neighbor" (the 6). Write the answer: 9 below the 3.
For the "hundreds" place, add the 2 to the its "neighbor" (the 3). Write the answer: 5 below the 2.
For the "thousands" place, add the 1 to the its "neighbor" (the 2). Write the answer: 3 below the 1.
For the "ten-thousands" place, add the 5 to the its "neighbor" (the 1). Write the answer: 6 below the 5.
For the "hundred-thousands" place, add the 0 to the its "neighbor" (the 5). Write the answer: 5 below the 0.
That's it ... 11 X 051236 = 563596

Practice it on paper first!

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Quickly To Multiply Any Two Digit Number by 11

11:48 PM

Muhammad Yusuf , Posted in Math Trick , 0 Comments

To multiply any two digit number by 11:

For this example we will use 54.
Separate the two digits in you mind (5__4).
Notice the hole between them!
Add the 5 and the 4 together (5+4=9)
Put the resulting 9 in the hole 594. That's it! 11 x 54=594

The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627
Practice it on paper first!

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Quickly Multiply Up to 20 x 20

11:46 PM

Muhammad Yusuf , Posted in Math Trick , 0 Comments

In just FIVE minutes you should learn to quickly multiply up to 20x20 in your head. With this trick, you will be able to multiply any two numbers from 11 to 19 in your head quickly, without the use of a calculator.
I will assume that you know your multiplication table reasonably well up to 10x10.
Try this:

Take 15 x 13 for an example.
Always place the larger number of the two on top in your mind.
Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
First add 15 + 3 = 18
Add a zero behind it (multiply by 10) to get 180.
Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
Add 180 + 15 = 195.

That is It! Wasn't that easy? Practice it on paper first!

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2011 Alabama Mathematics Contest : Algebra With Trigonometry

5:49 AM

Muhammad Yusuf , Posted in Algebra , Math Competitions , 0 Comments

1. Suppose x is a complex number satisfying the equation x + 1/x = 1. What is the value of x³ + 1/x³ ?

(A) −2 (B) −1 (C) 0 (D) 1 (E) 2

2. What is the number of sides of a regular polygon for which the number of diagonals is between 30 and 40 ?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

3. If a is a nonzero integer and b is a positive number such that ab² = log₁₀ b. What is the median of the set {0, 1, a, b,1/b}?

(A) 1/b (B) b (C) a (D) 1 (E) 0

3. Given the function f(x) = 2x⁴ − 3x³ + 4x² − 5x + 6, what is the sum of A,B,C,D, and E if f(x) = A(x − 1)⁴ + B(x − 1)³ + C(x − 1)² + D(x − 1) + E ?

(A) 16 (B) 17 (C) 18 (D) 19 (E) 20

4. If a polynomial function with real coefficients has 3 distinct x-intercepts, what is the maximal degree of the polynomial?

(A) 3 (B) 4 (C) 5 (D) 6 (E) None of these

5. Let S = i²ⁿ⁻²+i²ⁿ⁺², where n is an integer and i² = −1. The total number of possible distinct values of S is

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

6. Given the sequence {a_n}, where a₁ = 1 and a_n₊₁ = a_n + n for n ≥ 1. Find a₁₁.

(A) 50 (B) 55 (C) 60 (D) 65 (E) 66

7. If the age of a person 16 years ago was 5 times the current age of his son, and two years ago the sum of his age and his son’s age was 30, what is the age of his son now?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

8. For a certain integer n, 5n + 16 and 8n + 29 have a common factor larger than one. That common factor is

(A) 11 (B) 13 (C) 17 (D) 19 (E) 23

9. Function f satisfies f(x) + 2f(5 − x) = x for all real numbers x. The value of f(1) is

(A) 7/3 (B) 3/7 (C) 5/2 (D) 2/5 (E) None of these

10. How many 5 digit numbers that are divisible by 3 can be formed using 0,1,2,3,4,5 if no repetitions allowed ?

(A) 216 (B) 120 (C) 240 (D) 126 (E) 96

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Math Contest Problems

5:11 AM

Muhammad Yusuf , Posted in Math Competitions , Olympiad , 0 Comments

1. Three boys, A, B, and C, can work separately or together on a job. If working together, their efforts combine eficiently. It takes A working alone twice as long as it takes B working alone. It takes A four times as long as B and C working together to do the job. Also, it takes A two hours longer to do the job than it takes all three boys working together. How many hours does it take B to do the job?

2. We call a number ascending if each digit is greater than the digit that precedes it. For example, 457 is ascending, but 447 is not. How many ascending numbers are there between 400 and 5000?

3. Out of all polynomials with integer coe±cients which have both 1/2√2 and 1/2√3 as

roots, consider those of smallest possible degree. Among all of these, what is the smallest positive coe±cient which occurs in any of them?

4. A triangle has vertices at (0, 0), (4, 2), and (5, 1). What is the tangent of its angle at the vertex (4, 2)?

5. What is the smallest positive integer n such that the product 19999 × n ends in the four digits 2010?

6. A parallelogram has area 36 and diagonals of length 10 and 12. What is the length of its longest side?

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