Subcontests
(15)Minimum value of a sum
For t \equal{} 1, 2, 3, 4, define \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t, where ai∈{1,2,3,4}. If S_1 \equal{} 513 and S_4 \equal{} 4745, find the minimum possible value for S2. Distinct Pairs with Distinct Sums
From the set of integers {1,2,3,…,2009}, choose k pairs {ai,bi} with ai<bi so that no two pairs have a common element. Suppose that all the sums a_i\plus{}b_i are distinct and less than or equal to 2009. Find the maximum possible value of k. Lighthouse Distances
Four lighthouses are located at points A, B, C, and D. The lighthouse at A is 5 kilometers from the lighthouse at B, the lighthouse at B is 12 kilometers from the lighthouse at C, and the lighthouse at A is 13 kilometers from the lighthouse at C. To an observer at A, the angle determined by the lights at B and D and the angle determined by the lights at C and D are equal. To an observer at C, the angle determined by the lights at A and B and the angle determined by the lights at D and B are equal. The number of kilometers from A to D is given by qpr, where p, q, and r are relatively prime positive integers, and r is not divisible by the square of any prime. Find p\plus{}q\plus{}r, Game show with 3 prizes
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $1 to $9999 inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were 1,1,1,1,3,3,3. Find the total number of possible guesses for all three prizes consistent with the hint. Equilateral Triangle Tangency Points
Equilateral triangle T is inscribed in circle A, which has radius 10. Circle B with radius 3 is internally tangent to circle A at one vertex of T. Circles C and D, both with radius 2, are internally tangent to circle A at the other two vertices of T. Circles B, C, and D are all externally tangent to circle E, which has radius nm, where m and n are relatively prime positive integers. Find m \plus{} n.
[asy]unitsize(2.2mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);
pair Ep=(0,4-27/5);
pair[] dotted={A,B,C,D,Ep};draw(Circle(A,10));
draw(Circle(B,3));
draw(Circle(C,2));
draw(Circle(D,2));
draw(Circle(Ep,27/5));dot(dotted);
label("E",Ep,E);
label("A",A,W);
label("B",B,W);
label("C",C,W);
label("D",D,E);[/asy] Bill's Painted Stripes
Before starting to paint, Bill had 130 ounces of blue paint, 164 ounces of red paint, and 188 ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stirpe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.