SW-ARML Practice 9-16-12
Individual Problems
1. Compute the largest prime divisor of 15! – 13!.
[Hint: Use factoring.]
2. 3. A circle with center O and radius 1 contains chord AB of length 1, and point M is the midpoint of AB. If the perpendicular to AO through M intersects AO at P, compute the area of MAP.
[Hints: ABO is an equilateral triangle. Also, in a 30–60–90 triangle, the side facing the 30angle is half of the hypotenuse in length.]
3. Suppose that p and q are two-digit prime numbers such that p2– q2 = 2p + 6q + 8. Compute the largest possible value of p + q.
[Hints: Complete squares. X2 – Y2 = (X – Y)( X+ Y). The largest two-digit prime is 97.]
4. The four zeros of the polynomial x4+jx2+kx+225 are distinct real numbers in arithmetic progression. Compute the value of j.
[Hints: For any polynomial xn + axn-1 + + c (with leading coefficient 1), the sum of all of its zeros is equal to –a, and the product of all of its zeros is equal to (–1)nc. An arithmetic progression means there is a common difference between any two consecutive terms or, graphically, they are evenly spaced on the number line.]
5. If n is a positive integer, then n!! is defined to be n(n–2)(n – 4)2 if n is even and
n(n– 2)(n– 4) 1if n is odd. For example, 8!! = 864 2 = 384 and 9!! = 97531 = 945. Compute the number of positive integers n such that n!! divides 2012!!.
[Hints: The case when n is even should be obvious. Almost all odd solutions can be obtained by cleverly rearranging the terms of 2012!! and factoring out 2’s. 1007 = 19 53; 1009 is prime.]
6.One face of a 222 cube is painted (not the entire cube), and the cube is cut into eight 111cubes. The small cubes are reassembled randomly into a 222 cube. Compute the probability that no paintis showing.
[Hint: Focus on the 111 cubes that have painted faces.]
Team Problems
7.In ABC, mA = mB = 45 and AB = 16. Mutually tangent circular arcs are drawn centeredat all three vertices; the arcs centered at A and B intersect at the midpoint of AB. Compute the area of theregion inside the triangle and outside of the three arcs.
[Hints: Calculate the length of AC and BC. Proportionally compute the area of the three circular sectors.]
8. Compute the number of ordered pairs of integers (a, b) such that 1 < a 50, 1 b 50, and logba is rational.
[Hints: logba is rational if and only if a and b are integral powers of some integer d. Partition {2, 3, 4…., 50} accordingly.]
9. Suppose that 5-letter “words" are formed using only the letters A, R, M, and L. Each letter need not be used in a word, but each word must contain at least two distinct letters. Compute the number of such words that use the letter A more than any other letter.
[Hints: Depending on how many times the letter A is used, calculate the number of possibilities for each case. The number of distinct permutations of distinct objects, where the first object is used n1times, the second object n2 times, …., and the (last) kthobject nktimes, is equal to . For example, if X is used 3 times, Y is used 2 times, and Z is used 4 times, then there are possible ways to form a nine-letter “word” using X, Y, and Z according to the specified multiplicities.]
10.Positive integers a1; a2; a3, …. form an arithmetic sequence. If a1 = 10 and = 100, compute .
[Hints: means . For example, if a3 = 5, then = . Then, if a5 = 17, then = . The main goal is to figure out the common difference d. In an arithmetic sequence, an = a1 + (n – 1)d. Write out in terms of a2 and re-apply the formula to a2.]
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