[/quote]
,
respectively 
are marked in such a way that 
and 
for 
.
,
people arranged in a circle, and 
coins are distributed among them, where each person has at least 
coins. Each person is then assigned a random index number in 
such that no two people have the same number. Then every minute, if 
is the number of minutes passed, the person with index number congruent to 
be a prime. We say that a sequence of integers 
is a 
,
such that whenever 
,
divides the sum![\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]](http://latex.artofproblemsolving.com/1/e/9/1e99c787677d83944ee00e370b7913b1c2728fcd.png)
and 
are 
is a 
board containing all the integers from 
to 
so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:
denote the largest integer such that for any coloring of a 
with two colors, there exists a monochromatic subgraph of 
.
we can find a natural 
such that for any integer 
,
?
with 
and 
,
be the midpoint of 
.
on the extension of 
past 
and point 
on segment 
such that 
.
be on the opposite side of 
from 
such that 
and 
.
intersect the circumcircle of 
again at 
,
intersect the circumcircle of 
again at 
.
,
are collinear.
be a point in the exterior of the circle 
of center 
and radius 
, and let 
, 
be the tangent segments from 
such that 
.Let the line from 
intersect the segment 
, and if the incircle of the triangle 
, then find the area of 
in terms of 
.
be the set of positive integers. Determine all functions 
such that 
is divisible by 
for all positive integers 
.
be a set of integers containing the numbers 
and 
.
belongs to 
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is 
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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