Mathcounts marathon!!!!

[pytheagle]

Solution

By definition of probability, we have Since we have two choices for every toss namely heads or tails, we have .

Let H be heads, T be tails, and _ represent a toss. Then, the desired outcomes are presented as follows:

1) H H H T _ _ _ _ _ _
2) T H H H T _ _ _ _ _
3) _ T H H H T _ _ _ _
4) _ _ T H H H T _ _ _
5) _ _ _ T H H H T _ _
6) _ _ _ _ T H H H T _
7) _ _ _ _ _ T H H H T

Notice that for all seven of the cases, we need to choose either H or T for the remaining six tosses leading to .

Hence, our answer is .

New Problem

A student's scores on five mathematics tests were and . The student needs an average score of eighty-one to receive a B grade on the report card. What is the lowest score the student can obtain on the one remaining test before report cards and still receive a B grade? (Source: MATHCOUNTS)

[greenhand]

[garfielddisco123]

Hi do not look, answer ans solution

(83+67+79+92+70+x)/6=81
83+67+79+92+70+x=486
391+x=486
x=95

[gaussintraining]

New Problem

The parallel sides of an isosceles trapezoid have lengths of units and units, and the perimeter is units. What is the area of the trapezoid?

[AndrewTom]

Solution

Length of equal sides = (74 - 19 - 29)/2 = 13. Perpendicular height = 12, by Pythagoras.

Area = 12(29+19)/2 = 288.

New problem

Prove that ax^2 + bx + c takes integral values for all values of x if and only if 2a, a+b and c are each integers.

[pytheagle]

The general term "MATHCOUNTS Level" does not extend to include problems in which proofs are required.

New Problem

How many strings of 3 letters do not include a B?

[cgyao15]

go calculate

NP
Find number of ways to spell: cgyao15cyao15 as 1 word.

[goldenboy1.618]

Solution

Assuming you meant cgyao15cgyao15, we have 14 letters to be arranged in permutations. Dividing by for repetition gives .

NP

If Kelly is 36 years old, then Mark is 29 years old. If Kelly is not 36, then Andrea is not 14. Andrea is indeed 14. Mark is either 29 or 45. How old is Mark? (From a school entrance exam)

[Maybach]

Solution

Um.....REALLY EASY

This means Kelly is 36 and Mark is 25. (Wow, some school entrance exam )

NP

In Texas, 10 students on the 2009 Mathcounts State Competition got 46/46. If only 4 people can make the national competition and one of them was already determined, how many ways can the remaining 3 people be selected from the remaining students?

[the cliu]

solution

Since one of them was already determined, there are 9 students left to distribute to 3 spots on the National Math Counts team. Since order doesn't really matter, our answer is

new problem

The probability that cliu will eat Chinese food for dinner is . The probability that cliu will eat Chinese food for lunch is . What is the probability that cliu will not eat Chinese food for lunch or dinner?

[Maybach]

Solution

Complementary counting:
1/9 * 11/13 = 11/117

NP

The probability that Maybach will have Chinese food for lunch is . The probability Maybach will have Chinese food for dinner is . What's the probability Maybach will have Chinese food for both lunch and dinner?(I always feel sick to my stomach after eating spicy food)

[greenhand]

solution

new problem

Find the focus and the directrix of the parabola

[AndrewTom]

This seems rather difficult for this level.

Solution

Comparing with , which has its focus at and directrix , by completing the square to get in the form , I get the focus at and the directrix as

New problem

Make with five s.

[Thunder365]

lol
Solution

100=111-11

NP

Carla is writing all the numbers from 0 to 99999 inclusive on a piece of paper. How many times does the digit "1" appear?

[cgyao15]

@goldenboy i didn't make a typo, cyao15 is a player too you know

[goldenboy1.618]

@cgyao15 Sorry for the confusion.

Solution

She writes 100,000 total numbers. This means she writes 1 in the units place 10,000 times, in the tens place 10,000 times, and so on for each place value, giving her a total number of ones.

NP

What is the degree measure of the smaller angle formed by a clock's hands at 8:20?

[AwesomeToad]

Solution

The minute hand is degrees past the and the hour hand is so the smaller angle formed is degrees

New Problem

Find all real solutions of

[AndrewTom]

Solution

Sketching the graph, or .

Alternatively, square both sides and factorise to get , giving the real roots .

NP

Solve .

[gauss1181]

The absolute values must be both 0. But that's not possible.

So there are in this equation.

NP:

Find the radius of the incircle of a triangle with side lengths 7, 24, and 25.

[Thunder365]

Solution:
A=RS
7-24-25 is a right triangle, so A=84. S=(7+24+25)/2=28. 84/28=3

NP:
Six students are going to sit in a row for Countdown Round in Mathcounts. Thunder wants to sit next to cliu. Athunder does not want to sit next to Alex. The other two people can sit anywhere. How many ways can these six students be seated?