AoPS Academy
be the set of shapes with a regular polygon with sides of length 
. Surrounded with triangles with sides , 
, . Such that the part of the triangles with side length is on each of the sides, with the triangles outside of the regular polygon. Let 
be the probability that any two randomly chosen points in the shape , have the line segment between them being totally contained inside of , having no parts of the line located outside of the region . What is the average of the values in the set, 
. Write your answer as a fraction.
are integers, not all equal, and 
then what is the value of 
?
and 
are such that 
and 
. What is 
?
cents per mile with an initial fee of 
. Airline B charges 
cents per mile with an initial fee of 
. How many miles must a flight be if it costs the same for both Airline A and Airline B?
such that 
has an area of 
with 
. Point 
is on line 
with 
. What is the positive difference between the two possible areas of triangle 
?
is said to be 
if 
is a perfect square. Find the sum of all numbers.
is a nonconstant arithmetic sequence such that 
and the terms 
form a geometric sequence in that order. Find the smallest index 
for which 
.
. The dodgers have 6 pitchers of skill level 
. When a Padres pitcher with skill level 
is against a Dodgers pitcher with skill level 
, the probability that the Padres win that game is 
. The Padres have already determined that they will start their pitchers in the increasing order of skill level. In what order should the dodgers start their pitchers to maximize their expected wins? Give your answer as a sequence of six integers.
? (It’s 5 btw). In any such radical 
for some integers 
, the key is that 
such that 
is negligibly small. And adding this to 
squares away the square roots and cancels away the roots bits, leaving only integers behind. From this answer extraction is typically easy, perhaps a little module arithmetic is needed in combination with the binomial theorem.
? For example, what is the ones digit of 
? Is this feasible?
and the sum of their reciprocals is 
. What is the sum of these two numbers?. What is the ratio of the area of the circumscribed circle to the area of the inscribed circle?
, then 
-axis that is enclosed by the graph of the curve 
.
be a function that satisfies the following properties:
.
.
for all integers and 
.
. is located at the center of a square table with side 
. A coin with radius 
is randomly thrown onto the table. What is the probability that the coin comes to rest on the boundary of the circle?
hours, and the other in 
hours. How long in hours will they have to burn before one candle is 
times the length of the other?
and the sum of their reciprocals is 
. Find the largest integer.
, are placed so that a corner of one lies at the center of the other. Find the area of quadrilateral 
if 
.-team baseball league, each team plays each of the others 
times. The season ends, not in a tie, with each team the same number of games ahead of the following team. What is the greatest number of games that the last team could have won?
such that 
.
, find the six-digit number 
. is an even integer, express in terms of the number of solutions in positive integers of 
.
and 
for all 
. Find 
.
and 
, find 
in terms of and . Express your answer as a common fraction.
.
, then 
.
, the slope of the line 
.
, then we have an intersection 
as in the picture.
hence 
.
.
then 
is a pgm and 
is a pgm. Hence 
.
, then we have an intersection as in the picture. hence .. then is a pgm and is a pgm. Hence .
are isosceles trapezoids, otherwise OK!
are given to be equal, not the upstanding sides. See the picture below for clarification.
be real numbers . Prove that
,
that does not divide 
is 
is positive, so 
Thus, taking the equation modulo 
This congruence is equivalent to 
Testing some values of 
are possible solutions. However:
gives 
which has no integer solutions.
gives 
which has no integer solutions.
gives 
,
gives 
,
gives 
which is not a perfect square.
is 
,![\[n!+2023 \equiv 3 \pmod 4.\]](http://latex.artofproblemsolving.com/f/f/0/ff0f642f237717986890b2f7d34d695296f0ff1a.png)
However, 
This is contradiction, and there are no 
.
.
be real numbers such that 
.
