and 
.
are positive real numbers such that 
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
![\[\lfloor x \rfloor\]](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)
Prove that
of the pizza, where 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Is there an elegant way to prove this?
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
.
is stylish because the set 
can be partitioned as 
,
,
.
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
,
and 
although its supposed to be a quick comparison.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains 
points. Find the expected number of turns until Krithik wins.
be an isosceles triangle with 
Let 
be a point on the circumcircle of 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
Y by
Let 
be a family of equilateral triangles, all with parallel sides and of the same orientation. For each triangle If 
is its medial triangle. We define finally 
as the union of all 
and 
triangles as the union of all triangles. Prove what 
be a family of equilateral triangles, all with parallel sides and of the same orientation. For each triangle If 
is its medial triangle. We define finally 
as the union of all 
and 
triangles as the union of all triangles. Prove what 
.
is trivial.
.
.
.
angle with the side reflected over, the overlap of 
.![$[U] \leq 2[T]$](http://latex.artofproblemsolving.com/8/4/b/84be927cdfef30afa2bcdf10d84ce9482edfac20.png)
.![$3[U]\leq 6[T]$](http://latex.artofproblemsolving.com/6/f/3/6f30829fee966a90bfb347cfb247d401c7822480.png)
,![$3[U]=[S]+2[T]$](http://latex.artofproblemsolving.com/c/8/0/c80ea7d0d53d36f9e604afb8e5f25fbc1bafdbed.png)
because we overcounted the overlap between the three rhombi, ![$[S] \leq 4[T]$](http://latex.artofproblemsolving.com/1/1/f/11f2cad4132f872e35107eb5d882dd9647e4a600.png)
.
