[/quote]
,
be real numbers such that 
.
of side length 
lies in the union of five equilateral triangles of side length 
,
is a polynomial of degree 
such that
.
.
is called good if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
is a quadratic residue modulo 
for every 
is considered a quadratic residue modulo 
such that 
.
be a triangle. Let 
be a circle passing through 
intersecting 
at 
and 
at 
.
be the intersection of 
and 
and 
and 
with the tangent to 
at 
.
be the second intersection of 
and 
and 
,
is the midpoint of 
.
is an arbitrary point on 
.
be the intersection of 
and 
.
intersects with the circumcircle of 
,
.
is the circumcenter of triangle 
.
.
where 
are integers such that 
for all 
.![\[
r_1 \ge r_a,\quad r_2 \ge r_b,\quad r_3 \ge r_c
\]](http://latex.artofproblemsolving.com/8/2/1/8214c052ab9ad2c979414f92b6707fa6de33664f.png)
because if any of the radii were smaller than these, the circles would not be able to reach and touch the incircle.![\[
\frac{r_1 + r_2 + r_3}{r} \ge \frac{r_a + r_b + r_c}{r}
\]](http://latex.artofproblemsolving.com/b/b/a/bbadb40e6aa197207abfa907981e0a4a88926969.png)
![\[
\frac{r_a + r_b + r_c}{r} = \frac{h_a - 2r}{h_a} + \frac{h_b - 2r}{h_b} + \frac{h_c - 2r}{h_c}
\]](http://latex.artofproblemsolving.com/5/3/2/5329fec1f02a728b5f34817e0ed43ded663516ae.png)
![\[
= \left(1 - \frac{2r}{h_a}\right) + \left(1 - \frac{2r}{h_b}\right) + \left(1 - \frac{2r}{h_c}\right)
= 3 - 2r \left( \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} \right)
\]](http://latex.artofproblemsolving.com/1/e/0/1e087ea87566fb4cf8cd2ae0cda3389247095eea.png)
![\[
\frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} = \frac{a + b + c}{2S}
\]](http://latex.artofproblemsolving.com/a/4/a/a4ab49131584cf2028270e7248fcbb5d2c6a7adb.png)
![\[
3 - 2r \cdot \frac{a + b + c}{2S} = 3 - \frac{4S}{a + b + c} \cdot \frac{a + b + c}{2S} = 3 - 2 = 1
\]](http://latex.artofproblemsolving.com/0/f/3/0f3ada5ca22deeecc1644d79138942caf04d5134.png)
![\[
\frac{r_1 + r_2 + r_3}{r} \ge 1
\]](http://latex.artofproblemsolving.com/7/1/e/71ea14c8fb8e3860a1db427a0afcdb3b2fd6254b.png)
a prime number with 
and 
when 
.
else 
.
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
and there is a morphism 
from the group 
to the monoid 
such that for all 
,
such that 
.
positive such that 
,
positive such that 
converges and 
diverges ?
is continuous, positive with 
,
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
.
but it is not clear that it is ok, even if 
,
diverges ?
converges ?
does not converge to 
,
Y by Mathematicsislovely, Mango247, Mango247
Let 
be a fixed positive integer. Prove that for every 
, there exist infinitely many 
and 
such that there are atleast 
primes among the following numbers given by
![\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]](//latex.artofproblemsolving.com/1/4/5/145bf6662b94ac521b905b028dff4ede7fd5fd18.png)
Proposed by Bimit Mandal
be a fixed positive integer. Prove that for every 
, there exist infinitely many 
and 
such that there are atleast 
primes among the following numbers given by![\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]](http://latex.artofproblemsolving.com/1/4/5/145bf6662b94ac521b905b028dff4ede7fd5fd18.png)
Proposed by Bimit Mandal
This post has been edited 1 time. Last edited by kapilpavase, Jan 25, 2021, 9:12 AM
be the first 
primes that are of the form 
.
.
and 
,
,
,
such that, there does not exist any 
with 
.
such that 
is not a 
th power modulo any 
.
where 
.
,
.
)
is asymptotically equivalent to,
Hence, for sufficiently large 
Thus, the total number of primes(counted with multiplicity) among the numbers of the form,
will be at least,
Therefore by pigeonhole principle there exists 
such that, the numbers of the form,
has at least,
primes.
be a fixed positive integer. Prove that for every 
,
and 
primes among the following ![\[m + 1, m + 2^t, m + 3^t, \cdots, m + N^t.\]](http://latex.artofproblemsolving.com/5/7/f/57f5384a94aa0df9cfdce5504e342dbabe78ffd6.png)
![\[\frac{n}{\varphi{n}} \ge 100Ct\]](http://latex.artofproblemsolving.com/1/1/b/11bd69bc1dc955000dbfd2c97adcbe864b241af6.png)
which is just possible by making 
.
isn't a 
power modulo 
(which is just CRT). Now we look at this 
table![\[
\begin{bmatrix}
1+n+a & 2^t+n+a & \dots & N^t+n+a \\
1+2n+a & 2^t+2n+a & \dots & N^t+2n+a\\
\vdots & \vdots & \ddots & \vdots \\
1+N^{100t}n+a & 2^t+N^{100t}n+a & \dots & N^t+N^{100t}n+a \\
\end{bmatrix}
\]](http://latex.artofproblemsolving.com/2/8/f/28fde65e00fc124236899bebd59e31082e90be7f.png)
Now by Prime Number Theorem on AP, we get that the number of primes in 
column is 
And hence the expected number of primes in each row is atleast ![\[\frac{C N^{100t}}{ \log N} \cdot \frac{N}{N^{100t}}=\frac{CN}{\log N}\]](http://latex.artofproblemsolving.com/b/1/3/b132cf47a94862e3a75801e0b10194017d742385.png)
As required.
