AoPS Academy
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)
. 
not divisible by 2. Prove 
(
is a subset, not necessarily a subgroup) that 
and 
that 
that
and 
be a function which is continuous on ![\( [0,1] \)](http://latex.artofproblemsolving.com/0/1/0/01026be294289be8f090c7b2586dde64a0d5b479.png)
and differentiable on 
, with 
. Assume that there is some 
such that 
. Prove that there exists some 
such that 
.
be a polynomial of degree with only real zeros and real coefficients.
we have 
. When does equality occur?
![$f:[0,1]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/9/d/0/9d00bb0f11e69022dded4a8baba1b496f66d272e.png)
,is a convex function and 
![\[\left(\int^{1}_{0}f(x)dx\right)^2\leq \dfrac{3}{4}\int^1_{0}f^2(x)dx\]](http://latex.artofproblemsolving.com/a/a/0/aa0ae3023a71f3452170023452601cc7df785eb9.png)
and 
invertible. Prove that if 
for any positive integer 
then 
given by 
and 
, for all 
. Show that 
Mathematical Gazette
is defined by 
and, for 
![\[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\]](http://latex.artofproblemsolving.com/b/c/6/bc6686c63a96c3be5575542d7ac120c2fa7e69b2.png)
Show that, for all , 
is an integer multiple of .
and 
be two points that are diametrically opposite to each other on a unit sphere. right square pyramids are fitted along the line segment 
, such that the apex and altitude of each pyramid 
, where 
, are 
and 
respectively, and the points 
are collinear. pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of . pyramids that can be fitted in the sphere, in terms of . tends to infinity.
be a real-valued function that is twice continuously differentiable throughout 
, and define![\[
\rho (x,y)=yh_x -xh_y .
\]](http://latex.artofproblemsolving.com/7/3/6/736993597c4b45cf77b9013185d2a4d931a7d231.png)
Prove or disprove: For any positive constants 
and 
with 
, there is a circle of radius whose center is a distance away from the origin such that the integral of 
over the interior of is zero.
,![\[
g_r(x) = \frac{1}{\tan^{r+1} x} \cdot \frac{d^r}{dx^r} (\tan x)
\]](http://latex.artofproblemsolving.com/6/e/7/6e7fd00178c7657a478f5d9fda80354f2a78816e.png)
![\[
\lim_{x \to \frac{\pi}{2}} g_r(x)
\]](http://latex.artofproblemsolving.com/3/a/5/3a577519fbeb3b419f6beef525f450fcd1fc9195.png)
