AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be positive integers such that no one of them is divisible by any other member of the sequence. Then![\[
a_1 \geq 2^k,
\]](http://latex.artofproblemsolving.com/1/f/e/1fecb6d6811bfc4aff7ccefa20412f0a5b860066.png)
where 
is defined by the inequalities![\[
3^k < 2n < 3^{k+1}.
\]](http://latex.artofproblemsolving.com/2/8/f/28f12a76029727a828167f1cb8e118b84d710d62.png)
This estimate for 
is the best possible.
be a sequence of positive integers. Then![\[
\max \left( (a_i, a_j) \right) > \frac{38n}{147} - c,
\]](http://latex.artofproblemsolving.com/6/5/d/65df2fa2c65d90d0548cb3c11dd36ea42c4060e6.png)
where 
is a constant independent of 
, and 
denotes the greatest common divisor of 
and 
. This is the best possible estimate.
, there exists only degree polynomial 
satisfying 
and ![$ (x + 1)[f(x)]^2 - 1$](http://latex.artofproblemsolving.com/9/c/1/9c1e80d035c923dd13711bff624d8a1a3e6a3b15.png)
is an odd function.
be the inscribed circle of a triangle 
. Let 
and 
be the tangency points of and the sides 
and 
, respectively, and let 
and 
intersect at 
and 
, respectively, such that 
and are all distinct. The tangent line of at 
intersects 
at 
, the tangent line of at 
intersects 
at 
, and the tangent line of at M intersects 
at 
. Prove that 
and are collinear.
is a factor of 
, find the sum of 
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
be an acute triangle with 
. Denote by 
and 
points on the segment 
such that 
. 
is a point on segment 
. 
intersects 
and 
at 
and 
, respectively. The angle bisectors of 
and 
intersect at . If 
and 
, prove that 
is cyclic.
numbers can be expressed in 
digits in base 
, 
digits in base 
, and 
digits in base 
for some positive integers 
. How many such two-digit base numbers are there?
satisfies the following conditions:![\begin{align*}
&f(4xy)=2y[f(x+y)+f(x-y)]\text{ for all }x,y\in\mathbb R\\
&f(5)=3
\end{align*}](http://latex.artofproblemsolving.com/6/2/5/625b8046a00d07680dd2a8d2aa3231d5daa4fcf2.png)
.
we have 
. Let be a point in segment such that 
. Show that 
.
is a prime and 
,![\[
n! \sum_{pi+j=n} \frac{1}{p^i i! j!} \equiv 0 \pmod{p}.
\]](http://latex.artofproblemsolving.com/1/6/6/166f07aa9e69c0eb35833971acb9b04932fea087.png)
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
![\[\sum_{n=1}^{2025}n^5\]](http://latex.artofproblemsolving.com/0/8/3/08334297eaac7610645549ede6434e399000e480.png)
such that 
coprime, 
and 
.
and 
and join 
to 
is an equilateral triangle. Now, 
.
and as 
.
also 
.
.