[/quote]
,
be an acute, scalene triangle, and let 
,
,
be the midpoints of 
,
,
,
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
,
,
such that
-
,
be a subset of 
vertices of the polygon. Show that if 
,
consisting of ![\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]](http://latex.artofproblemsolving.com/d/7/c/d7c9b2e32b9ed6d0c16be523df5acd83a7b782b2.png)
,
,
,
.
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 
.
denote the set of positive real numbers. Find all functions 
such that for each 
,
satisfying 
is called tasty if for every ordered pair of positive integers 
with 
at least one of 
and 
is a terminating decimal. Do there exist infinitely many tasty integers?
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
denote the set of all polynomials in three variables 
with integer coefficients. Let 
denote the subset of 
with 
.
for all non-negative integers 
satisfying 
.
of integer numbers, we define 
as the sum of the elements of 
and 
whose intersection is empty, whose union is the set 
and such that the product 
is a perfect square.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
Y by
Hello AoPS!
TeamsCode is proud to announce our Spring 2025 Virtual Programming Contest! It will take place on Saturday, April 5, from 11:00 AM to 5:30 PM Pacific Time through Youtube Livestream. We'll be featuring guest speaker Ion Barbus, an Analytics Manager in the Government and Public Services sector, from 4 to 5 PM.
Teams of up to 4 members will have 3 hours to solve programming problems. There will be two divisions: Novice and Advanced. Thousands of dollars worth of prizes will be distributed, through placement awards, raffle prizes, and more! Only pre-college participants are eligible for prizes.
To register for the contest, please visit: https://www.teamscode.org/contests/spring-2025. Join our Discord server here: https://discord.com/invite/Md36nAvMFM.
TeamsCode is proud to announce our Spring 2025 Virtual Programming Contest! It will take place on Saturday, April 5, from 11:00 AM to 5:30 PM Pacific Time through Youtube Livestream. We'll be featuring guest speaker Ion Barbus, an Analytics Manager in the Government and Public Services sector, from 4 to 5 PM.
Teams of up to 4 members will have 3 hours to solve programming problems. There will be two divisions: Novice and Advanced. Thousands of dollars worth of prizes will be distributed, through placement awards, raffle prizes, and more! Only pre-college participants are eligible for prizes.
To register for the contest, please visit: https://www.teamscode.org/contests/spring-2025. Join our Discord server here: https://discord.com/invite/Md36nAvMFM.
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