[/quote]
,
,
.
,
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
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such that 
are the lengths of a right triangle.
with 
and 
prime number.
?
is the function integer part, hence 
.
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
.
chords, one of which has length 2025. Find the minimum possible value of 
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
,
denote the minimum value of the function 
.
and 
are equal positive numbers. Find the angle between the line 
and the 
-
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
be real numbers such that 
Find the maximum value of the expression 
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
ones are written in a circle. Petya and Vasya take turns making 
moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by 
.
consecutive numbers and increases each by 
.
sports, each awarding one gold medal to a winner. 
such that the experts can make their predictions so that at least 
of degree 
and 
intersect at exactly 
Y by
An odd integer is written in each cell of a 2009 2009 table. For 1 i 2009 let Ri be
the sum of the numbers in the ith row, and for 1 j 2009 let Cj be the sum of the
numbers in the jth column. Finally, let A be the product of the Ri, and B the product of
the Cj . Prove that A + B is different from zero.
I really need help on this. Can people give me hints? I don’t know where to start.
the sum of the numbers in the ith row, and for 1 j 2009 let Cj be the sum of the
numbers in the jth column. Finally, let A be the product of the Ri, and B the product of
the Cj . Prove that A + B is different from zero.
I really need help on this. Can people give me hints? I don’t know where to start.
