be a triangle. The incircle of triangle 
at 
,
meets the incircle again at a point 
.
and 
meet the incircle of triangle 
and 
,
and 
are concurrent.
be an integer. We call a pair of integers 
g![\[0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b\]](http://latex.artofproblemsolving.com/e/8/d/e8df562e024b8476d69fdaea15338c27bb6c4eb8.png)
Prove that the number of 
.
written on the first page and 
empty pages. Suppose in each of the next 
be the largest possible number such that for any 
,
on the last page of her grimoire. Prove that there exists a positive integer 
,![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
.
intersect the incircle of triangle 
.
is perpendicular to line 
.
.
lies on segment 
is tangent to line 
.
lies on line 
such that 
.
.
be a function differentiable at 0 with 
.
be a smooth function on 
where 
for all 
and 
.
for all 
.
Where 
denotes the n-th Pell Number given by 
and 
invertible, for an infinite number of 
such that 
is a scalar
![\[
I = \int_0^1 \frac{(2^x + 2)\, dx}{1 + x^4}
\]](http://latex.artofproblemsolving.com/e/3/9/e391d4208a008146dc3922ff31ac1f544172b5b4.png)
using the trapezoidal rule with accuracy 
.
![\[
I = \int_0^1 \frac{(2x^2 + 1)\, dx}{\sqrt{(1 + x^2)(2 - x^2)}}
\]](http://latex.artofproblemsolving.com/3/7/4/3744a151b0383f98b63359cec6947395881e7496.png)
using the parabola method (Simpson's rule) with accuracy 
.
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 
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.
,
such that
Let
Find positive constants 
and 
such that
,
be normed spaces with 
is a family of continuous linear maps. Prove that the set![\[
X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\}
\]](http://latex.artofproblemsolving.com/e/c/d/ecd2cd83e90604ce128e8efb5e5b0748ad354ee3.png)
is either of first category in 
Y by anantmudgal09
Let 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](//latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
This post has been edited 1 time. Last edited by oVlad, Apr 12, 2025, 9:45 AM
and 
contains 
does it solve the problem?Since it contains the single element?
denotes the 
-
.
and 
are rather small, but all other 
'
not 
so it's useless information.
