[/quote]
,
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
be real positive numbers such that: 
.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of 
for some positive integer 
.
be integer numbers 
satisfy that
.
is a perfect square number
can’t be any power 
of a positive integer number
such that:
such that for all 
,![\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]](http://latex.artofproblemsolving.com/a/a/1/aa1bce053788a3efa59819407b41cfe472f544d0.png)
such that 
such that
for all 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
and 
.
be a triangle. Circle 
passes through 
,
and 
again at points 
and 
respectively, and intersects segment 
at 
and 
such that 
and 
at 
at 
is parallel to 
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
Y by Adventure10, Mango247
Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex 
is on the perpendicular line through the center 
of the base of the prism (see figure). Let 
denote the side of the base, 
the height of the cell and 
the angle between the line 
and 
.
(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi.
(b) the total surface area of the cell is given by the formula
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is on the perpendicular line through the center 
of the base of the prism (see figure). Let 
denote the side of the base, 
the height of the cell and 
the angle between the line 
and 
.(a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi.
(b) the total surface area of the cell is given by the formula
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This post has been edited 1 time. Last edited by Peter, Oct 4, 2005, 4:47 PM
is equal to s. The sides UW, VX of these equilateral triangles are equal to 
.
be the intersection of the diagonals of the rhombus TUVW and Q the circumcenter identical with the orthocenetr of the equilateral triangle 
(the intersection of the plane of this triangle with the 3-fold rotational axis of the structure). From the right angle triangle 
with the side 
and the angles 
,
.
of the congruent trapezoids:
of these trapezoids is obviously equal to
.
(the side of the equilateral triangle 
,
.
]
