[/quote]
,
and 
be 2 distinct positive integers.
be in an integer. Prove why ?
are non-negative reals such that 
.
.
a tangencial quadrilateral that is not a rhombus, let 
be lengths of tangents from 
to the incircle of the quadrilateral which center is 
.
be the midpoints of 
resp. Prove that![\[ \frac{MI}{IN}=\frac{a+c}{b+d} \]](http://latex.artofproblemsolving.com/e/7/0/e709acb8b17c0466a385ba8e7a89bb1bf16c72af.png)
is given. It turns out that for every positive integer number 
,
and 
are positive, they have the same number of divisors. Is it necessarily true that 
?
with 
be the intersection of 
and 
and define 
in a similar way. Furthermore, let 
.
,
are concyclic.
positive integers. Product of any one of them with sum of remaining numbers increased by 
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
,
is an homoemorphisme too.
,
.
be the second intersection point of the circumcircles of the triangles 
and 
.
,
and 
are the circumcenters of the triangles 
are concyclic.
where 
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
with maximums 
and 
.
,
can be expressed as 
where 
and 
.
Let 
.
Let 
.
Let 
.
Y by
p17. Let the roots of the polynomial 
be 
, and 
. What is the sum 
?
p18. Two students are playing a game. They take a deck of five cards numbered
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
p19. Compute the sum of all primes
such that 
is also prime.
p20. In how many ways can one color the
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
PS. You should use hide for answers. Collected here.
be 
, and 
. What is the sum 
?p18. Two students are playing a game. They take a deck of five cards numbered
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?p19. Compute the sum of all primes
such that 
is also prime.p20. In how many ways can one color the
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?PS. You should use hide for answers. Collected here.
This post has been edited 1 time. Last edited by parmenides51, Aug 11, 2023, 9:36 AM
to be 
Now with vietas we can find that 
Now 
So then 
= 
for which 
obviously doesn't work. 
works by inspection. For 
,
.
and 
which gives 
.
.
is 
.
with 
.![$\chi_{C_n}(k)=\sum_{m=0}^{n-1}(-1)^mk(k-1)^{n-m-1}=\sum_{m=0}^{n-1}(-1)^m[(k-1)^{n-m}+(k-1)^{n-m-1}]=(k-1)^n+(-1)^n(k-1)$](http://latex.artofproblemsolving.com/3/d/2/3d2dd6a85f16f7aa7e8cf99967a51eddcf7532f3.png)
