AoPS Academy
is a perfect-cube, a.k.a. 
:)
, however John cannot step foot on plots 
and 
. If John starts from point 
, how many ways can John get to point ? (Note : John can only move right or upwards)
be the zeros of the quadratic function 
. Compute for the numerical value of 
.
, let 
denote the set of real numbers 
satisfying ![\[ \left\lfloor x \right\rfloor \cdot \left( x - \left\lfloor x \right\rfloor \right) = kx. \]](http://latex.artofproblemsolving.com/f/3/d/f3d7d99dabc7a4ac62d0800dab951a6bb0282c90.png)
The set of positive real numbers 
such that has exactly 
elements is a half-open interval of length 
. Compute 
.
be an even integer and 
be the 
matrix of ones.
be the set of permutation matrices, 
.
is constitued of 2 elements to be determined.
be positive real numbers whose sum equals 
. For a natural number 
, let 
denote the number
for which 
holds. Prove that![\[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\]](http://latex.artofproblemsolving.com/e/4/6/e4626e17d9924353417a2b6a4ad6fb45d64a2675.png)
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
is analytic on a closed curve 
(i.e., 
is analytic in a region that contains ). Show that ![\[\int_\gamma \overline{f(z)} f'(z)\,\mathrm dz\]](http://latex.artofproblemsolving.com/c/3/7/c3784659f77e99585032e21fef95bd31a11c23e4.png)
is purely imaginary. (The continuity of 
is taken for granted.)
![$$\int_{0}^{1} \frac{d^{2025}}{dx^{2025}}\left[ e^{-x^2}\cdot \frac{d^{2025}}{dx^{2025}}\left[ e^{x^2} \right] \right]dx$$](http://latex.artofproblemsolving.com/1/e/0/1e0523315a7557fd7b68b452e6b671a797de6a59.png)
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
?
.
.
Solving this system, the minimum value for 
.
,
.
,
Solving this system, the minimum value for 
.
,
Solving this final system, the minimum value for 
.
.
,
and 
,
.
are coprime, we can easily get that 
,
.