[/quote]
,
.
in order to make it a perfect square?
is 
and 
.
is 
degrees, where 
and 
are the midpoints of 
and 
,
and 
.
and 
be positive real numbers such that 
.
and 
Prove that
is the midpoint of 
,
bisects 
and intersects 
at 
is the incenter of 
,
bisects 
and intersects the circumcircle of 
at 
,
is parallel to 
.
written on the board.At every step,Mikail chooses the two numbers 
and substitutes them with 
and 
where 
is the unchosen number on the board. Prove that at least 
negative number must be remained on the board at any step.
![$\sqrt[2]{2}$](http://latex.artofproblemsolving.com/1/8/6/1869ac018fb72a26a12ab14499cbf1b1c13b4a10.png)
,![$\sqrt[3]{3}$](http://latex.artofproblemsolving.com/1/7/a/17a6bb5dc11696b8650b6d3a11e770f861f4c326.png)
,![$\sqrt[5]{5}$](http://latex.artofproblemsolving.com/a/b/9/ab90f2d42dc24065e2a64988fe76f22eb9359799.png)
.
,
.
Y by
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
