[/quote]
,
and 
on a given circle meet at point 
.
be a point on segment 
and let 
meet the circle again at 
.
parallel to 
intersects 
at 
.
is parallel to 
.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
![\[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]](http://latex.artofproblemsolving.com/1/e/8/1e8b3e2d9a4d0917ec622449b058702700542f22.png)
for positive integers 
,
,
be prime, let 
be a positive integer, show that ![\[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]](http://latex.artofproblemsolving.com/9/d/4/9d409ce186320241eb358b5fbbb8a5a66120899c.png)
be a right triangle with a right angle in 
.
and 
lie in such a way that 
.
and 
be the projection of 
is half as long as 
be an isosceles triangle with 
and circumcircle 
.
.
be any point on 
with center 
intersects 
and 
,
,
,
.
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 the incentre of a non-equilateral triangle 
be the 
be the reflection of 
be the reflection of line 
in 
.
,
and line 
analogously. Let 
where 
is the circumcentre of triangle 
and 
.
.
such that 
is a perfect square and 
is a prime number.
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
such that for any real value of 
we have:
Y by
Points 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:12 AM
)
and say it's rational coordinates are (p,q) then move the point to ([p],[q]) where [•] denote the box function and we are done with all points with integral coordinates.
and 
is clearly 1, where the first point is a lattice point and the second one has rational coordinates.
