AoPS Academy
be reals such that 
. Prove that
Let be reals such that 
. Prove that
,
,
, 
, 
is not greater than 
.
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 ....
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
that satisfy 
, where is a real number. Prove that for all positive , 
be a set of integers satisfying the following property: For every positive integer 
and every set of coefficients 
, all integer roots of the polynomial 
are also elements of . It is given that contains all numbers of the form 
where 
are positive integers. Prove that contains all integers.![$f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$](http://latex.artofproblemsolving.com/8/b/d/8bdc04d75d0654da65463301f6fd0b0af4160b2b.png)
a polynomial whose roots 
are in geometric progression with ration 
. Find 
.
with maximums 
and 
. If 
, and 
can be expressed as 
where is a positive integer, find .
and 
the equation 
does not have real roots. Prove that 
.
be an acute-angled triangle with 
and 
be the altitude through 
with 
. Find the distance between the midpoints of 
and 
with the problem in the link. 
.
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
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
are the midpoints of 
respectively. Then 
and 
Also 
and 
Since 
we have 
Applying Pythagoras' theorem on 
we get 
to a point 
such that 
.
.
be the mid point of 
be the midpoint of 
.
too. Hence, 
.