PROBLEM 1:

Solution 1 (Fast): Expanding the equations and simplifying, we get y + 4x = x², and x + 4y = y². We can add the two equations and subtract the first from the second, and when we simplify, we get 5(x + y) = x² + y², and 3(y - x) = y² - x². Factoring y² - x² as a difference of squares and simplifying, we get y + x = 3. Plugging this into the first equation (5(x + y) = x² + y²), we get x² + y² = 5(3) = 15. Thus, our desired answer is (B).

~ Muhammad Sheriff

PROBLEM 2:

Solution 1 (Quadratics): Since the geometric sequence needs to have a constant ratio of less than 1 (or else the sum would be infinite), we can use the sum of an infinite geometric series formula which is a / (1 - r) where a is the first term and r is the constant ratio. We can set up an equation to find S which is (1 / r) / (1-r) because 1/r is the first term. We can simplify the equation into 1 / (r - r²). Now, we can set this equation to be equal to S. This means r - r² is equal to 1/S. Next, we can use the quadratic formula to find the constant ratio of r based on S. R is equal to (1 ∓ √(1 - 4(1 / S)))/2. Since the constant ratio has to be a real number, the least possible value of S must be 4 because (1 - 4( 1 / S)) must not be below zero or else we would get a non-real number. Thus, the answer is (E).
~ Roger Huynh

Solution 2 (Easy): We can use algebra to label the first term of the geometric series as x. We can then see that our common ratio is 1/x. Using the infinite geometric series summation formula, the sum of the series is x / ((x - 1)/ x). This simplifies to x² / (x - 1).  We can see that the smallest possible value of this expression is when x - 1 = 1 (dividing by something less than one is multiplying by something greater than one). Thus, the smallest possible value of S is 4, or (E).

~Muhammad Sheriff

PROBLEM 3:

Solution 1 (Vectors): We can use a "vector" to model XY . We first need to create our coordinate system. Let the bottom right corner of the cube of edge length 4 be the origin. Thus, XY has endpoints at  (4,0,0) and (0,4,10). We can now see that if x changes by -4, y changes by 4 and z changes by 10. We can model this as such: x changes by -1 ➨ y changes by 1 & z changes by 5/2 (this also works vice versa and etc). We need to find the coordinates of segment XY when z is 4 and 7. To make z = 4, z needs to change by 4, which means x changes by -8/5 and y changes by 8/5. Thus, the coordinates of the point contained in the bottom layer of the cube with side length 3 is (4 - 8/5, 0 + 8/5, 0 + 4) = (-12/5, 8/5, 4). Using this same strategy, we find the coordinates of the point in which line segment XY intersects is (-6/5, 14/5, 7). Using the 3D Pythagorean Theorem to find the distance between the two points, we get our desired answer, which is (A).

~Muhammad Sheriff

Solution 2 (Quick): We can calculate the distance from X to Y using 3 dimensional Pythagorean Theorem. The distance between X and Y is √(4² + 4² + 10²) which equals to 2√33. The square with a side length of three contains 3/10 of the distance. So, the distance contained by the side length 3 cube is (3√33)/5. The answer is (A).
~Roger Huynh

PROBLEM 4:

Solution 1 (Factoring): We can see that we have to find the number of unique values of xy such that  x²+y² = (xy)² has no solutions. We can move all terms to one side and factor the expression as (x² - 1)(y² - 1) = 1. We can see that as long as we set a value of x or y that is not 1 or -1, there is always a value of the other variable such that the expression is 1. Thus, the only ordered pairs that work are (x,y) = (1,1), (1,-1), (-1,1), and (-1,-1). The only unique values for the product of xy is 1 and -1. Thus, our desired answer is 2, or (C).

~Muhammad Sheriff

Solution 2 (Quick): We can test out a few values to figure out that |k| has to be smaller than 2 (If k = 2 then the circle and k = xy graph will intersect at (√2, √2). Thus, the only values of k that fits the condition are 1 and -1. (k can't be zero or else both graphs will intersect at (0,0)).  This makes our desired answer is 2, or (C).
~Roger Huynh

PROBLEM 5:

Solution 1 (Algebra): Let N = 5x for some integer x > 1. We can observe that the only values that work are "3x Green Balls, 1 Red Ball, 2x Green Balls", "3x + 1 Green Balls, 1 Red Ball, 2x - 1 Green Balls", "3x + 2 Green Balls, 1 Red Ball, 2x - 2 Green Balls"..., "5x Green Balls, 1 Red Ball, 0 Green Balls" and "2x Green Balls, 1 Red Ball, 2x Green Balls", "2x - 1 Green Balls, 1 Red Ball, 3x + 1 Green Balls", "2x - 2 Green Balls, 1 Red Ball, 3x + 2 Green Balls"..., "0 Green Balls, 1 Red Ball, 5x Green Balls". The first & second "case" has 2x + 1 permutations that work, and there are a total of 5x + 1 permutations. This makes the probabilty (4x + 2) / (5x + 1). We want to find the least possible value of x that satisfies (4x + 2) / (5x + 1) < 321/400. Multiplying both sides by (5x + 1)(400) and simplifying, we get 1600x + 800 < 1605x + 321, so 5x > 479. The least possible value of 5x that satisfies this is 480 since x has to be an integer, so x is 96. Since N = 5x, the least possible value of N that satisfies the inequality is 480. Thus, our desired answer is (A).

~Muhammad Sheriff

Solution 2 (Pattern recognition and Equations): We know that there are N+1 ways to place the red ball. The question asks for the least value of N such that P(N) is less than 321/400. If N = 5, P(N) = 1. If N = 10, P(N) = 10/11. If N=15, P(N) = 14/16. And if N = 20, P(N) = 18/21. If we continue this pattern, we can find out that P(N) = 1 - (N/5)-1)/(N+1) which is equal to (N(4/5)+2)/(N+1). Let's set this value to 321/400. Now, we have an equation of (N(4/5)+2)/(N+1) = 321/400. Multiply both sides by N+1 and you get N(4/5) + 2 = (321N + 321)/400. Now we multiply both sides by 400 and we get an equation of 320N + 800 = 321N + 321. Subtract both sides by 320N + 321 and we know that N = 479 if P(N) is equal to 321/400. For P(N) to be smaller, N must be larger so N must be equal to 480. Thus, the answer is (A).
~Roger Huynh

PROBLEM 6:

Solution 1 (Stars & Bars): Let us first derive the formula for the amount of terms that don't have to include all four variables,  then we can update it to include these constraints. We can see that the maximum sum of the powers of a given term can be N, and the minimum is 0 (if the term is 1), with all values in between being possible. Thus, we can see that the number of terms is the amount of ordered pairs (a, b, c, d)  that satisfy a +b +c +d <= N. Since the term has to have abcd in it, we can just modify the inequality to be  a +b +c +d <= N - 4  (Since there are N - 4 powers to "choose" from). To solve this, we can use stars and bars with an additional variable x that will be the "distance" required to get to 5. To find the number of quintuples, we can just use the non-negative stars and bars formula,  and this is equal to nCr(4 + N -  4,4) =  nCr(N, 4). This value is equal to 1001, and we can just test out values  to get N as 14, so our final answer is (B).

~Muhammad Sheriff

Solutions 2 (Multinomial Expansion):  Using the Multinomial Expansion, in the expansion of (x1 + x2 + x3 +  ... + xm)^n, each term corresponds to an exponent combination (e1, e2, e3,  ... em) = n and the number of such terms is n + m - 1 choose m - 1. Now, we need to use change of variables because the multinomial expansion only works for variables larger or equal to 0, not 1. We can define a' = a - 1, b' = b - 1, c' = c -1, d' = d - 1. So now, a', b', c', d' ≧ 0. Since a + b + c + d + e =  N, we get  (a' + 1) + (b' + 1) + (c' + 1) + (d'+1) + e = N ⇒ a' + b' + c' + d' + e = N - 4. Now that all variables are ≧ 0, we can apply the formula. Number of solutions = ((N-4) + 5 - 1 choose 5 - 1) = (N choose 4). We can now make an equation which is (N * (N-1) * (N-2) * (N-3))/24 = 1001. This means that (N * (N-1) * (N-2) * (N-3)) = 24024. We can factorize 24024 into 2^3 * 3 * 7 * 11 * 13. We can then figure out that the four integers are 11, 12, 13, and 14. Thus, N = 14. (A) is the answer.
~Roger Huynh