[/quote]
,
be a acute triangle with 
.
are the orthocenter and excenter. Let 
be a point on the same side of 
as 
,
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be the midpoint of 
.
are collinear.
be real numbers satisfying 
and 
Prove that 
Let 
and 
Prove that 
in the plane.Let 
be reflections of 
across the line 
be the midpoints of 
,
be the reflections of 
)
bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
?
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
,
.
so that 
The point 
on the segment 
satisfies 
the point 
on the segment 
satisfies 
and the point 
on the line 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
,
,
be positive integers from set 
such that every number from this set occurs exactly once. Is it possible that numbers 
all have different remainders upon division by 
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
,
is differentiable?
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 
.
be a number such that 
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.
