[/quote]
,
s.t. 
,
,
,
lie on a semicircle 
in this order with diameter 
,
.
and 
lie on segments 
and 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
,
of 
points in a plane 
-
points.
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer 
.
with real coefficients of degree 
such that the following two conditions hold:
(they are not necessarily distinct);
are 
.
is monic since the coefficient of 
is one.
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
around a circle so that the sum of every 
be a scalene triangle, and let 
.
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and 
meet at 
.
and 
are parallel.
be an interior point of a triangle 
meet the sides 
in 
respectively. Show that ![\[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]](http://latex.artofproblemsolving.com/7/5/9/759482bace109eb619aaea692625f7854faa9fec.png)
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
is situated inside the parallelogram 
such that 
.
.
Y by PikaPika999, Exponent11
Hello!
Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator)
1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.
2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.
3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.
4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.
5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.
6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.
7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.
8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.
9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.
10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator)
1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.
2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.
3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.
4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.
5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.
6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.
7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.
8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.
9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.
10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
This post has been edited 3 times. Last edited by mithu542, Apr 29, 2025, 9:30 PM
, and it follows that 
and the final probability is 
.
The distinction between overcount or not is whether arrangements with more sequences then necessary are counted more than once.
by considering one string 
and leaving the rest of the choices. We have 
Similarly, we can arrange multiple 
and 
Now, we have the probability of at least one sequence as 
ways to color the edges, so the denominator of the answer, in lowest terms, should divide 
does not divide 
for variables 
