ARML TST 2006(I didn't know where to put it sorry)

[stupidityismygam]

A Solution Race:

Pretty simple rules:
1) Solve 1 problem per post
2) Must Quote the original question
3) Must type up the complete solution not just the answer
4) Must be TEXed
5) First Person to post the solution gets a point
6) Once a solution is posted, don't post another solution (unless you believe the answer is wrong or you are already in the middle of typing it up)

TST ARML 2006

1. In the sequence of positive integers [unparseable or potentially dangerous latex formula] the value of [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] If [unparseable or potentially dangerous latex formula] write an equation which expresses [unparseable or potentially dangerous latex formula] explicitly in terms of [unparseable or potentially dangerous latex formula]

2. There are four different ordered pairs of positive integers [unparseable or potentially dangerous latex formula] with a less than b, which satisfy [unparseable or potentially dangerous latex formula] Find these four ordered pairs.

3. In isosceles traingle [unparseable or potentially dangerous latex formula] the bisectors of base angles [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] intersect at [unparseable or potentially dangerous latex formula] and the bisector of angle [unparseable or potentially dangerous latex formula]intersects [unparseable or potentially dangerous latex formula] at [unparseable or potentially dangerous latex formula] If [unparseable or potentially dangerous latex formula] find [unparseable or potentially dangerous latex formula]

4. A square root of [unparseable or potentially dangerous latex formula] Find the numerical value of a.

5. For all complex numbers x, the polynomial function P is defined by [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Write an equation which expresses [unparseable or potentially dangerous latex formula] explicitly in terms of [unparseable or potentially dangerous latex formula]

6. The area of triangle [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] Rectangles [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are drawn in the exterior of triangle [unparseable or potentially dangerous latex formula] and the area of each rectangle is [unparseable or potentially dangerous latex formula] [unparseable or potentially dangerous latex formula] is a point in the interior of triangle [unparseable or potentially dangerous latex formula] The sum of the areas of triangles [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] Find [unparseable or potentially dangerous latex formula]

7. The lengths of the sides of triangle [unparseable or potentially dangerous latex formula] are [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Find the numerical value of [unparseable or potentially dangerous latex formula]

8. Alphonse and Beauregard each roll one pair of fair cubical die, number from [unparseable or potentially dangerous latex formula] to [unparseable or potentially dangerous latex formula] Find the probability that Alphonse rolls a higher total than Beauregard.

9. In right triangle [unparseable or potentially dangerous latex formula] the length of hypotenuse is [unparseable or potentially dangerous latex formula] and the length of altitude [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] Square [unparseable or potentially dangerous latex formula] has [unparseable or potentially dangerous latex formula] on [unparseable or potentially dangerous latex formula] on [unparseable or potentially dangerous latex formula]and [unparseable or potentially dangerous latex formula]on [unparseable or potentially dangerous latex formula] Find the length of [unparseable or potentially dangerous latex formula]

10. In a certain series, the nth term is [unparseable or potentially dangerous latex formula] The sum of the first n terms of this series is [unparseable or potentially dangerous latex formula] Find the ordered pair of integers [unparseable or potentially dangerous latex formula]

11. There is a point [unparseable or potentially dangerous latex formula] on a circle whose diameter is [unparseable or potentially dangerous latex formula] The degree measure of arc [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula] If tangents through [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] intersect the tangent through [unparseable or potentially dangerous latex formula]at points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] respectively, then [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] is a trigonometric function. Find the common name for [unparseable or potentially dangerous latex formula]

12. In triangle [unparseable or potentially dangerous latex formula] is the center of the circumscribed circle and [unparseable or potentially dangerous latex formula] is the ceter of the inscribed circle. If [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] find [unparseable or potentially dangerous latex formula]

13. Find all values of [unparseable or potentially dangerous latex formula] which satisfy

[unparseable or potentially dangerous latex formula]

14. Two distinct numbers are randomly selected from the first [unparseable or potentially dangerous latex formula] positive integral perfect squares. Find the probability that these two numbers are both cubes of integers.

15. In parallelogram [unparseable or potentially dangerous latex formula] angle [unparseable or potentially dangerous latex formula] is acute and [unparseable or potentially dangerous latex formula] Point [unparseable or potentially dangerous latex formula] is on [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] A line through [unparseable or potentially dangerous latex formula] perpendicular to [unparseable or potentially dangerous latex formula] intersects [unparseable or potentially dangerous latex formula] at [unparseable or potentially dangerous latex formula] If [unparseable or potentially dangerous latex formula] find [unparseable or potentially dangerous latex formula]

16. In the rectangular coordinate plane, the line [unparseable or potentially dangerous latex formula] bisects the acute angle formed by [unparseable or potentially dangerous latex formula] and the positive x-axis. Find the numerical value of [unparseable or potentially dangerous latex formula]

17. In an arithmetic series, the sum of the first [unparseable or potentially dangerous latex formula] terms is [unparseable or potentially dangerous latex formula] the sum of the first [unparseable or potentially dangerous latex formula] terms is [unparseable or potentially dangerous latex formula] and the sum of the first [unparseable or potentially dangerous latex formula] terms is [unparseable or potentially dangerous latex formula] Write an equation expressing [unparseable or potentially dangerous latex formula] explicitly in terms of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]

18. The polynomials [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] have a common quadratic factor over the set of polynomials with integral coefficients. Find the ordered pair [unparseable or potentially dangerous latex formula]

19. Four students are taking a test of 100 questions of equal difficulty. Their respective probabilities of getting right answers are [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] What is the probability that at least one of the students will answer the first question correctly.

20. Given that [unparseable or potentially dangerous latex formula] and:

[unparseable or potentially dangerous latex formula]

find the value of the integer [unparseable or potentially dangerous latex formula]

21. A convex quadrilateral is inscribed in a circle so that one of its sides is a diameter of the circle. The lengths of the two sides of the quadrilateral which have exactly one end-point on this diameter are 7 and 20. If the length of the longest side of this quadrilateral is 25, find the length of the fourth side of the quadrilateral.

22. If [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] find a positive value for the product of [unparseable or potentially dangerous latex formula]

[bradp]

[zefuri]

[bradp]

[stupidityismygam]

[stupidityismygam]

[zefuri]

[stupidityismygam]

[jimdaaa]

19 is easy...i can't tex nor do i feel like following stephen's rules...

19. Four students are taking a test of 100 questions of equal difficulty. Their respective probabilities of getting right answers are 60\%, 50\%, 40\%, and 25\%. What is the probability that at least one of the students will answer the first question correctly.

1-.4*.5*.6*.75=.91

[zefuri]

[bradp]

only perfect cubes squared are both cubes and squares so they are

Hence, 4 choose 2 divide by 64 choose two which is equal to 1,64,729,4096

[unparseable or potentially dangerous latex formula]

i get the point ! :P

jk

[stupidityismygam]

[stupidityismygam]

[zefuri]

[zefuri]

[zefuri]

[stupidityismygam]

[zefuri]

[zefuri]

[stupidityismygam]

Pts:
I'll just edit it as it changes
Jim: 1
Jeffrey: 7 (dont get credit for #7 because it is like random and not right)
Brad: 2
Stephen: 8
NickR: 1

Problems unsolved:

9,11,15

[stupidityismygam]

[stupidityismygam]

[NickR]

Alright, this might not be the most elegant solution but here we go - #17

First, [unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Now, since [unparseable or potentially dangerous latex formula], the third equation now becomes...
[unparseable or potentially dangerous latex formula]

Now to rid us of that pesky [unparseable or potentially dangerous latex formula] term.

[unparseable or potentially dangerous latex formula] so then [unparseable or potentially dangerous latex formula]. This leads us to [unparseable or potentially dangerous latex formula].

Now our equation becomes [unparseable or potentially dangerous latex formula]

And I'll work on some other ones after foodtime...

EDIT: Scratch that solution for now as I made the false assumption that [unparseable or potentially dangerous latex formula] and that [unparseable or potentially dangerous latex formula] : P (stupid coincidence with the series I used to test) so I fail at life...