Click here to go to the NEW College Discussion Forum

I know that the AIME is tomorrow... so I looked on their website for some practice qs and was wondering if anyone could tell me a method to doing these besides trial and error (and w/o using a calculator) ...obviously you don't have to try all of them (or any) but any that you can help me with would be great... thanks

For how many values of k is 12^12 the least common multiple of the common integers 6^6, 8^8, and k?

Find the smallest prime number that is the fifth term of an increasing arithmatic sequence, all four preceding terms also being prime.

Find the sum of all positive integers n for which n^2-19n+99 is a perfect square.

The inscribed circle of triangle ABC is tangent to AB at P and its radius is 21. Given that AP=23 and PB=27, find the perimeter of the triangle.

The graph of y^2 + 2xy 40|x| = 400 partitions the plane into several regions. What is the area of the bounded region?

Let n be the number of ordered quadruples (x1, x2, x3, x4) of positive integers that satisfy S ⁹⁸ ₁ (w/ 4 on top of E thing) find n/100.

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9am and 10am and stay for exactly m minutes. The probability that either one arrives while the other is in the cafeteria is 40% and m=a-b (square root x), where a, b, and c are positive integers, and c is not divisible by the sqare of any prime. Find a + b + c.

By Eurostar (Eurostar) on Monday, March 24, 2003 - 10:28 pm: Edit

Just pick one to do for 3 hrs...
You will probably get it right, and thus a score higher than if you would attempt to do all 15.

By Brd (Brd) on Monday, March 24, 2003 - 10:52 pm: Edit

"Find the smallest prime number that is the fifth term of an increasing arithmatic sequence, all four preceding terms also being prime. "

There is no formula for the nth prime (should you chance upon one, you will be famous), so just start listing the first primes:

2,3,5,7,11,13,17,19,23,29,31,37,...

It doesn't take to much effort to see that 5, 11, 17, 23, 29, ... is an increasing arithmetic sequence, and I can't see another with a smaller fifth element. This question seems like a gimme; perhaps it is trying to snare people into automatically assuming that 7, 17, 27, 37, 47 gives the answer, without checking.

By Brd (Brd) on Monday, March 24, 2003 - 11:49 pm: Edit

"Let n be the number of ordered quadruples (x1, x2, x3, x4) of positive integers that satisfy x1 + x2 + x3 + x4 == 98. find n/100."

Ok, if it weren't for the odd part, this problem would be an easy combinatorics problem. So lets see if we can turn it into one. Let's represent our 4 odd numbers as:

x1 = 2*n1 + 1
x2 = 2*n2 + 1
x3 = 2*n3 + 1
x4 = 2*n4 + 1

where the ni's are any non-negative integers. Then we can restate the problem: how many (n1,n2,n3,n4) so that

(2*n1 + 1) + (2*n2 + 1) + (2*n3 + 1) + (2*n4 + 1) == 98?

After some trivial algebra. this s equivalent to:

n1 + n2 + n3 + n4 == (98 - 4)/2 == 47

But finding the number of ways any non-negative integers can add up to 47 is easy. You want to count the number of ways to distribute 47 "stars" (one for each of the positive integers up to 47) into 4 "bins" (some of which can be empty. Imagine these bins are separated by 3 "lines". Then there are 47 + 3 = 50 possible places to put all the "stars" and "lines". We just need to know how many ways there are to pick 3 places from 50, disregarding order.[1] That is "50 choose 3" or C(50, 3):

C(50, 3) == 50! / (47! * 3!) == (50 * 49 * 48) / (3 * 2 * 1) == (50 * 49 * 8)

but remember we actually want this number divided by 100, so it's really

(50 * 49 * 8)/100 == 49 * 4 == 196


[1] My explanation here is pretty lame -- this is now just a fact that I "remember". Check any book on combinatorics or probability for a better presentation this useful argument.

By Brd (Brd) on Tuesday, March 25, 2003 - 12:51 am: Edit

"The inscribed circle of triangle ABC is tangent to AB at P and its radius is 21. Given that AP=23 and PB=27, find the perimeter of the triangle. "

Draw the figure roughly, with AB as the base, with the other points of Q in between AC and R in between BC. Now draw three radii from the center of the circle, one to to each of the points of tangency OP, OQ, OR (and note these lines are perpendicular to the sides of the triangle). Now draw three lines from the center of the circle, one to each of the vertices of the triangle, OA, OB, OC. Notice you now have 6 small right triangles inside the original triangle. You can use pythagorus's theorem to find and the info you are given to deduce:

AP = 23
PB = 27
OA = Sqrt[970]
OB = 3*Sqrt[130]
AQ = 23
BR = 27

Note that AP = AQ = 23 and PB = BR = 27 (you could just note that the adjacent triangles are similar to conclude this immediately, but we will need OA and OB in a minute).

Then all we need is QC = RC = x

Label the angle ABC as theta. Then use some algebra and recall an identity:

Cos[theta] = Cos[(1/2)(2 theta)] = 2 (Cos[(1/2)theta])^2 - 1

But we know Cos[(1/2)theta], it is just PB/OB. So

Cos[theta] = 2(PB/OB)^2 - 1 = 16/15

Now finally plug this into the law of cosines:

(AC)^2 = (AB)^2 + (BC)^2 - 2(AB)(BC)Cos[theta]

which is

(23+x)^2 = 50^2+ (27+x)^2 - 2*50*(27+x)*(16/15)

and solve for x: x = 245/2

so the perimeter is 50 + 27 + x + 23 + x = 100 + 2x = 345.

By Brd (Brd) on Tuesday, March 25, 2003 - 01:22 am: Edit

Edit: BTW I found these problems online, and on the second one I worked, the online problem statement was:

"Let n be the number of ordered quadruples (x1, x2, x3, x4) of positive ODD integers that satisfy x1 + x2 + x3 + x4 == 98. find n/100."

You left out the "ODD" when you copied the problem, and I left out the "ODD" when I recopied your copy. But the problem I actually *worked* was the one I found online, with the "ODD" included. (The problem would be a dawdle without it).

Anywho, I am tired, I'll leave the others for someone else.

By Mekker3 (Mekker3) on Wednesday, March 26, 2003 - 07:54 pm: Edit

Thanks a bunch

By Brd (Brd) on Wednesday, March 26, 2003 - 08:22 pm: Edit

How'd you do?