Mathcounts marathon!!!!

[AwesomeToad]

NP

Suppose what is ?

[isabella2296]

We can factor out a from the top to get . The 41's cancel out, leaving us with , so .

New problem:

Andy has 3 identical peppermints and 2 identical candy canes. How many ways can he arrange the 5 candies on a table?

[$LaTeX$]

Andy can arrange them ways.

NP: The set has eight members. The
arithmetic mean of the set's members is 4.5 less than x. What is
the value of x ?

[steve123456]

(5 + 8 + 10 + 18 + 19 + 28 + 30 + x)/8=x-4.5. 5 + 8 + 10 + 18 + 19 + 28 + 30 + x =8x-36.x= 8x- 154. 7x=154, so x=22

NP: What value of x satisfies the equation 25^-2 = (5^48/x)/(5^[26/x] * 25^[17/x]) Sorry I can't use Latex.

[AwesomeToad]

[fishythefish]

And add a few braces...and fractions if you want:









NP:

In one of those "What color hats are the prisoners wearing?" logic problems, there are 10 (distinguishable) prisoners, and 1 red hat, 2 yellow hats, 3 green hats, and 4 blue hats. How many different orders of hats can there be when the prisoners are lined up in order? (Note: hats of the same color are not distinguishable.)

[Maybach]

Solution

There must be 10! ways to arrange the prisoners and 10!/2!3!4! combinations or 12,600.

NP

Find the value of 37+41.........+101.

[fishythefish]

Should we assume that you are just adding 4 each time? (There are multiple ways to write/solve this, but for simplicity's sake, I'll start the arithmetic series at .)





NP: What is the answer to my previous problem if there are 5 black prisoners and 5 white ones? (And those of the same color are not distinguishable.)

[brainomega]

solution

There are two variables- the color of the hat and the color of the skin. We just multiply the number of ways the hats can be arranged by the number of ways the prisoners can be arranged.

np

Bill drove to his work office at 50mph. Realizing that he forgot his portfolio, he rushed back at 65mph. After getting his portfolio, he drove to work at 70mph. What was his average speed for the drives? Round your answer to the nearest tenth.

[AndrewTom]

Solution

If the distance between home and office is , then the total distance is and the total time is . Dividing, we find that the average speed is to the nearest tenth.

new problem

On a long journey, a motorist drove at miles per hour for the first hours and then at miles per hour for the remainder of the journey. The average speed was miles per hour. How long did the journey take?

[goldenboy1.618]

Solution

The total time is 2 hours at 55 mph and hours at 70 mph.
The average speed is 60 mph, so the total distance is .
Likewise, the total distance is also .
Setting the two distances equal gives .
Simplifying yields , so the total time is hours.

NP

How many real numbers satisfy ?

[mathking123]

Sorry to interrupt, but when do Chapter competitions start?

[PowerOfPi]

Sometime in February (depending on Chapter), I believe.

[goldenboy1.618]

Guess I'll post a Solution.

The solutions are and . Thus there exist solutions.

NP

has AB = 15, AC = 20, and BC = .
Points X and Y are on AB and AC, respectively, such that BX = 8 and CY = 11. What fraction of the area of is inside quadrilateral BXYC? Express as a common fraction.

[mathking123]

I would just like to interject that as the author of this marathon, i'm glad that it has finally reached the 300 page mark.

Keep it up!

[pytheagle]

Since no one seems to be able to solve goldenboy1.618's problem (or is unwilling to expend the effort to do so and/or then to type up a solution into this thread), I will post a new problem

When you solve this problem

The sum of two positive integers is . The sum of their quotient and the reciprocal of their quotient is . What is the product of the two integers? (Source: MATHCOUNTS)

, please don't be so lazy as to post goldenboy1.618's problem, once again, as the new problem.

.

[PowerOfPi]

Solution

Translating into math, we get the following equations:

Multiplying the last equation by , we find that

But we also know that , so we can substitute:

Finally, we divide by to find

NP

Alice and Bob play a game involving a circle whose circumference is divided by equally-spaced points. The points are numbered clockwise, from to . Both start on point . Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves points clockwise and Bob moves points counterclockwise. The game ends when they stop on the same point. How many turns will this take? (Source: AMC)

[pytheagle]

Solution

We have the following:




Therefore, the answer is .

New Problem: Select any three-digit multiple of . Calculate the sum of the cubes of the digits of that number. This is now your new number. Now calculate the sum of the cubes of the digits of this new number. Continue this procedure of adding the cubes of the digits of the resulting number until you arrive at a number that is equal to the sum of the cubes of its digits. What is this number? (Source: MATHCOUNTS)

[the cliu]

answer

I really don't know if there is any other way besides guess and check.

NP

Heather and Isabelle play a two-person game which is won by the first person to accumulate 10 points. At each turn Heather gains a point with probability of . If she doesn't get a point then Isabelle gets a point. Isabelle is now ahead 9 to 6. What is the probability that Isabelle will win? Express your answer as a common fraction.

[goldenboy1.618]

Solution

The only way Heather can win at this point is if she scores 4 turns in a row. The probability that she scores 4 points now is .

Isabelle wins in any other circumstance, so her probability is .

NP

Joanna chooses a two digit positive integer. She multiplies the tens digit by 3, and she multiplies the units digit by 4. When Joanna adds these two quantities, the sum is her original number. What is Joanna's integer?