# Monthly Archives: May 2012

## Problem Of the Day-2

Here is Yesterday Problem

Yesterday Winner was Akash Agarwal

You must be knowing that If you win the Problem of the day for 4 times in a week – then I will send you a Really interesting book on any topic you want 🙂

Today’s Problem :

Find all possible values of x satisfying :

[x]/[x-2] – [x-2]/[x] = (8{x} + 12)/([x-2][x])

(.) = Normal bracket

{.} = Fractional part function

[.] = GIF/Floor function

## Modular Arithmetic

**Modular arithmetic** is a notation and set of mathematics that were first introduced by Carl Friedrich Gauss. 🙂

The major insight is that equations can fruitfully be analyzed from the perspective of remainders. Standard equations use the ‘=’ sign. Modular arithmetic uses the ‘**≡**‘ sign. Two values that are ‘≡’ to each other are said to be congruent relative the modulus. In the case below, the modulus is 3. 🙂

Here’s an example of a modular equation:

7 ≡ 1 (mod 3).

By definition, this means that 3 divides 7 – 1.

**Definition 1**: a ≡ b (mod c) if and only if c divides a – b.

This definition tells us the following is true:

7 ≡ 1 ≡ 10 ≡ -2 (mod 3).

Now, one of the most interesting things about ‘≡’ is that it follows many of the same relations as ‘=’ . 🙂 🙂

**Notice 1**: For any value a,b,c,d,n where** a ≡ b (mod n) and c ≡ d (mod n): 🙂**

(a) **a + c ≡ b + d (mod n)**

** Proof :** We know that n divides (a + c) – (b + d) since this is equal to: (a -b) + (c – d).

(b) **a – c ≡ b – d (mod n)**

**Proof :** We know that n divides (a – c) – (b – d) since this is equal to: (a – b) – (c – d).

(c) **ac ≡ bd (mod n)**

**Proof :** We know that n divides ac – bd since this is equal to : c(a – b) + b(c – d).

(Q.E.D)

**Notice 2:** If **a ≡ b (mod n)** then:

(a)** a + c ≡ b + c (mod n)**

**Proof** : We know (a) since n divides a + c – (b + c) = a – b.

(b)** a – c ≡ b – c (mod n)**

** Proof** : We know (b) since n divides a – c – (b – c) = a – b.

(c) **ac ≡ bc (mod n)**

**Proof**: We know (c) since n divides ac – bc = c(a – b)

(Q.E.D)

**Corrolary 2.1**: **a ≡ d (mod n), b ≡ e (mod n), c ≡ f (mod n)**, then:

**a + b + c ≡ d + e + f mod n **

(1) We know that a + b ≡ d + e from above.

(2) We therefore know that (a + b) + c ≡ (d + e) + f.

(Q.E.D)

**Notice 3**: **a + b + c ≡ 0, a ≡ 0 (mod p), then b + c ≡ 0 (mod p).**

(1) a + b + c ≡ 0 (mod p) [Definition of ≡ ]

(2) b ≡ c (mod p) → a + b ≡ a + c (mod p) [See above]

(3) So, 0 ≡ a + b + c ≡ 0 + b + c ≡ b + c (mod p).

(Q.E.D)

I will cover modular arithmetic in depth 🙂 🙂

Today you must we wondering about its application ..

But after whole course You will surely love this topic

Thank you 🙂 🙂

If any doubts , then dont hesitate in asking Your doubts 🙂 🙂

I will surely love to solve your doubts

## Problem of the day !

We have started one more interesting feature of the blog

I will be posting problem of the day , and the best answer will be highly appreciated

If you win problem of the day for more than 4 times in a week then I will send you a mathematics very interesting E-Book

So here is Question 1 )

**QUESTION 1 )**

How many digits does the number 2^{1000} contain?

NOTE :

You have to give solution of your answer too

## Important theorems of Geometry(Triangles) – 1

Today we will be discussing about 2 important Triangle theorems which have very High applications in field of Geometry

1) Angle Bisector theorem

2) Stewart’s Theorem

**Angle bisector theorem :**

Introduction :

The **Angle Bisector Theorem** states that given triangle and angle bisector AD, where D is on side BC, then c/m= b/n . Likewise, the converse of this theorem holds as well.

Proof :

Because of the ratios and equal angles in the theorem, we think of similar triangles. There are not any similar triangles in the figure as it now stands, however. So, we think to draw in a carefully chosen line or two. Extending AD until it hits the line through C parallel to AB does just the trick:

Since AB and CE are parallel, we know that and . Triangle ACE is isosceles, with AC = CE.

By AA similarity, . By the properties of similar triangles, we arrive at our desired result:

c/m = b/n

**Stewarts Theorem** :

Introduction :

Given a triangle with sides of length opposite vertices , , , respectively. If cevian is drawn so that , and , we have that . (This is also often written , a form which invites mnemonic memorization, e.g. “A man and his dad put a bomb in the sink.”)

Proof :

Applying the Law of Cosines in triangle at angle and in triangle at angle , we get the equations

Because angles and are supplementary, . We can therefore solve both equations for the cosine term. Using the trigonometric identity gives us

Setting the two left-hand sides equal and clearing denominators, we arrive at the equation: . However, so and we can rewrite this as (A man and his dad put a bomb in the sink).

Thank you

Source :

Wikepedia

Art of problem solving

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