# Blog Archives

## CGMO-2012 (China Girls Math Olympiad 2012) Problem 8

Find the number of integers $k$ in the set $\{0, 1, 2,\cdots, 2012\}$   such that  $\binom{2012}{k}$ is a multiple of $2012$

## IMO 1983 – Problem 3

Let $N$ and $k$ be positive integers and  let $S$ be a set of  $n$ points in the plane such that

$(i)$ no three points of $S$ are collinear, and
$(ii)$  for any point $P$ of  $S$, there are at least $k$ points of $S$ equidistant from $P$

Prove that  $k$  $<$  $\frac{1}{2}$  $+$  $\sqrt{2n}$

Try the question …
Solution will be updated soon

## IMO 2012 problems

This year IMO problems !!!!

Problem 1 :

Given triangle  the point  is the centre of the excircle opposite the vertex  This excircle is tangent to the side  at , and to the lines  and  at  and , respectively. The lines  and  meet at , and the lines  and  meet at  Let  be the point of intersection of the lines  and , and let  be the point of intersection of the lines  and  Prove that  is the midpoint of .

Problem 2 :

Let ${n\ge 3}$ be an integer, and let ${a_2,a_3,\ldots ,a_n}$ be positive real numbers such that  ${a_{2}a_{3}\cdots a_{n}=1}$  Prove that

$\displaystyle \left(a_{2}+1\right)^{2}\left(a_{3}+1\right)^{3}\dots\left(a_{n}+1\right)^{n}>n^{n}.$

Problem 3 :

The liar’s guessing game is a game played between two players ${A}$ and ${B}$. The rules of the game depend on two positive integers ${k}$ and ${n}$ which are known to both players.

At the start of the game ${A}$ chooses integers ${x}$ and ${N}$ with ${1 \le x \le N.}$ Player ${A}$ keeps ${x}$secret, and truthfully tells ${N}$ to player ${B}$. Player ${B}$ now tries to obtain information about ${x}$ by asking player ${A}$ questions as follows: each question consists of ${B}$ specifying an arbitrary set ${S}$ of positive integers (possibly one specified in some previous question), and asking ${A}$whether ${x}$ belongs to ${S}$. Player ${B}$ may ask as many questions as he wishes. After each question, player ${A}$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any ${k+1}$ consecutive answers, at least one answer must be truthful.

After ${B}$ has asked as many questions as he wants, he must specify a set ${X}$ of at most ${n}$positive integers. If ${x}$ belongs to ${X}$, then ${B}$ wins; otherwise, he loses. Prove that:

1. If ${n \ge 2^k,}$ then ${B}$ can guarantee a win. 2. For all sufficiently large ${k}$, there exists an integer ${n \ge (1.99)^k}$ such that ${B}$ cannot guarantee a win.

Problem 4 :

Find all functions ${f:\mathbb Z\rightarrow \mathbb Z}$ such that, for all integers ${a,b,c}$ that satisfy ${a+b+c=0}$, the following equality holds:

$\displaystyle f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$

Problem 5 :

Let ${ABC}$ be a triangle with ${\angle BCA=90^{\circ}}$, and let ${D}$ be the foot of the altitude from ${C}$. Let ${X}$ be a point in the interior of the segment ${CD}$. Let ${K}$ be the point on the segment ${AX}$such that ${BK=BC}$. Similarly, let ${L}$ be the point on the segment ${BX}$ such that ${AL=AC}$. Let ${M}$ be the point of intersection of ${AL}$ and ${BK}$.

Show that ${MK=ML}$.

Problem 6 :

Find all positive integers ${n}$ for which there exist non-negative integers ${a_1, a_2, \ldots, a_n}$ such that

$\displaystyle \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.$

IMO 2012

Thank you and good luck for these delicious problems

## Bumper Problems

These are the set of 5 problems  each of  5 marks

The person who get more than 20 points will get a mathematics book on any topic he want…
So try these beautiful problems to test your mathematics abilities and getting new things…

Ways by which you can answer :
I am adding a form in the end of this post  , you can answer there ….

Problem 1)

The bisectors of the angles $A$ and $B$ of the $\bigtriangleup ABC$  meet the sides
$BC$ and $CA$ at the points $D$ and $E$ , respectively.
Assuming that $AE+BD =AB$, determine the angle $C$

Problem 2)

Given a  $\bigtriangleup ABC$ and  $D$ be point on side  $AC$ such that  $AB = DC$,
$\angle BAC= 60-2X$ ,   $\angle DBC= 5X$ and  $\angle BCA= 3X$
Find the value of  $X$

Problem 3)

If p and q are natural numbers so that

Prove that p is divisible by 1979 .

Problem 4)

Find highest degree n of 1991 for which 1991ⁿ  divides the number :

Problem 5)

Let ƒ(n) denote the sum of the digits of n. Let N = 4444⁴⁴⁴⁴
Find ƒ(ƒ(ƒ(n))))

You can use these symbols to write solutions more conveniently

 Mathematical Operators ∀ ∁ ∂ ∃ ∄ ∅ ∆ ∇ ∈ ∉ ∊ ∋ ∌ ∍ ∎ ∏ ∐ ∑ − ∓ ∔ ∕ ∖ ∗ ∘ ∙ √ ∛ ∜ ∝ ∞ ∟ ∠ ∡ ∢ ∣ ∤ ∥ ∦ ∧ ∨ ∩ ∪ ∫ ∬ ∭ ∮ ∯ ∰ ∱ ∲ ∳ ∴ ∵ ∶ ∷ ∸ ∹ ∺ ∻ ∼ ∽ ∾ ∿ ≀ ≁ ≂ ≃ ≄ ≅ ≆ ≇ ≈ ≉ ≊ ≋ ≌ ≍ ≎ ≏ ≐ ≑ ≒ ≓ ≔ ≕ ≖ ≗ ≘ ≙ ≚ ≛ ≜ ≝ ≞ ≟ ≠ ≡ ≢ ≣ ≤ ≥ ≦ ≧ ≨ ≩ ≪ ≫ ≬ ≭ ≮ ≯ ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ≺ ≻ ≼ ≽ ≾ ≿ ⊀ ⊁ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉ ⊊ ⊋ ⊌ ⊍ ⊎ ⊏ ⊐ ⊑ ⊒ ⊓ ⊔ ⊕ ⊖ ⊗ ⊘ ⊙ ⊚ ⊛ ⊜ ⊝ ⊞ ⊟ ⊠ ⊡ ⊢ ⊣ ⊤ ⊥ ⊦ ⊧ ⊨ ⊩ ⊪ ⊫ ⊬ ⊭ ⊮ ⊯ ⊰ ⊱ ⊲ ⊳ ⊴ ⊵ ⊶ ⊷ ⊸ ⊹ ⊺ ⊻ ⊼ ⊽ ⊾ ⊿ ⋀ ⋁ ⋂ ⋃ ⋄ ⋅ ⋆ ⋇ ⋈ ⋉ ⋊ ⋋ ⋌ ⋍ ⋎ ⋏ ⋐ ⋑ ⋒ ⋓ ⋔ ⋕ ⋖ ⋗ ⋘ ⋙ ⋚ ⋛ ⋜ ⋝ ⋞ ⋟ ⋠ ⋡ ⋢ ⋣ ⋤ ⋥ ⋦ ⋧ ⋨ ⋩ ⋪ ⋫ ⋬ ⋭ ⋮ ⋯ ⋰ ⋱ ⋲ ⋳ ⋴ ⋵ ⋶ ⋷ ⋸ ⋹ ⋺ ⋻ ⋼ ⋽ ⋾ ⋿ Exponents  :   ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁼ ⁽ ⁾ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋ ₌ ₍ ₎

## Problem of the day 3

Nice problem !

Functional Equations !

Q)
Determine all functions ƒ satisfying  the functional relations

ƒ(x) + ƒ(1/(1-x)) =( 2(1-2x)/(x(1-x))

where x is a real number x ≠ 0 , x ≠ 1