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such that 
is a perfect square for all integers 
and 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
to the right; to the left.
squares north of the initial square after some finite number of moves if it is initially facing:
red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and are the roots of the cubic equation 
. The value of 
is ...
so that 
and 
is ...
, 
, 
, 
has an average of 
and a median of 
. The largest number of such data is ...
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
. If and 
are squares of two consecutive natural numbers, then the smallest value of 
is ..., with 
, 
and 
. The points 
and 
lies on the line segment 
. with 
and 
. The measure of the angle 
is ...
meet 
for each natural number 
. The value of 
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ... with 
and 
. Point lies on the side so that 
. Suppose 
is a point on the side extension 
so that 
is perpendicular to . The point lies on the ray such that 
and 
. The large angle 
is ...
consists of integers with the following properties: For every three different members of there are two of them whose sum is a member of . The largest value of is ....
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of so that the game never ends for any initial square selection is ....
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
be non-zero real numbers. Define 
, then the number of all integers which lies in the range of is equal to.
be mid-points of 
respectively 
and since, 
and 
are isosceles 
is a kite,
and we also know, 
is isosceles
is altitude of 
,such, 
.........Let 
Therefore, 
, 
, 
, 
. Thus Midpoint of 
and midpoint of 
. Using distance formula we get 
.
such that 
are collinear. Let 
be the midpoint of 
, thus it is also the midpoint of 
. Now 
and 
. By pythagorean theorem, 
. And using MPT, the required answer is .
such that:
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
and 
.
.
be the altitude through 
.
and 
be the midpoint of 
be the intersection of the line and 
.
and 
,
and 
.
,
.
using 
and 
.
,
on 
,
.
by the Midpoint Theorem.
,
.
is a right triangle.