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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 ABC the point J is the centre of the excircle opposite the vertex A. This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L, respectively. The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of intersection of the lines AF and BC, and let T be the point of intersection of the lines AG and BC. Prove that M is the midpoint of ST.

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.

You may download the PDF version :
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 ….

Note : You have give solution of your answer too..

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 !

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

Post your answer in comments with whole solution 🙂
person who give best answer of problem of the day more than 4 times will be awarded a book