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.
of positive integers that satisfy the equation![\[
x + xy + xyz = 31.
\]](http://latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
be real numbers such that 
, and![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](http://latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.
people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.
touch 
, 
and 
at points 
, 
and 
respectively. The corresponding excircle touches the side at point 
. Let 
be the common point of 
and the incircle, closest to , and 
be the common point of 
and 
. Prove that 
.
be the 
-th term of the Fibonacci sequence (where 
and 
for 
). For each prime 
, let 
be the smallest integer such that is divisible by . Find the smallest possible value of 
.
, find the one with the smallest area.
to such that for all 
: 
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
+2 w
distinct points on a circle. Consider the set of distances between any two out of the points. What is the smallest size of this set?
such that for any positive integers 
, 
we have 
. Prove that for all positive integers 
, 
is divisible by 
.
be a set of 2025 positive real numbers. For a subset 
, define 
as the median of 
when all elements of are arranged in increasing order, with the convention that 
. Define![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set of 2025 positive real numbers, the following inequality holds:![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in .
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
: They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.
, Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](http://latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
and![\[f(xy)-f(x)f(y)=2022(f(x+y)-2xy-1),\forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/f/3/e/f3e2829f7ff34702893623cfd34e7def42c078d0.png)
lie in a square of area 
.
that intersect in a region of area not less than 
.