AoPS Academy
Would anyone tell me whether the number of ways for proving Nesbitt inequality more than one hundred ?
+1 w
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 ....
such that 
divides 
.
be a 
-dimensional inner product space of column vectors, where for 
and 
, the inner product of 
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](http://latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
, define a linear transformation 
on as follows:![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](http://latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](http://latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
. Then the dimension of the linear space formed by all linear transformations 
satisfying 
is 
over the field 
:
prove that
Hence, as a corollary, show
be a positive semi-definite matrix. Prove that the matrix 
, is also positive semi-definite for all 
.
with the aim to train for the first problem of . They play the following game. bacteria and choose a natural number . On each move, the player chooses any number from to . Then the player divides each bacterium into pants. Once chosen, the number cannot be chosen twice. If after any player's move the number of bacteria population is divisible by then that player loses. Determine who has the winning strategy depending on the given number if it's known that Amman starts first.
where 
is the inverse gamma function, and 
is the inverse of 
and 
then is this uniformly convergence on 
comment on uniformly convergence on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
where in general it is should be uniformly convergence.
and trying to solve
feet east and then makes a 
turn to the right and walks more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?
feet east and then makes a turn to the right and walks more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?
because he had moved at the start, then the ant started moving 
every turn.
feet east and then makes a turn to the right and walks more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?
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
with 
such that
.
to 
![\[I=\int\limits_{0}^{1}\frac{\ln x}{\sqrt{1-x^2}}\mathrm dx.\]](http://latex.artofproblemsolving.com/4/4/0/440e26c1aeee699f15a2564a5c9b63ccd94a4cb2.png)
obviously
