Amatyc Student Mathematics League Oct Nov 2008

Explanation
To determine if line M is perpendicular to line L, we need to find the slope of line M. Since line M has the same y-intercept as L, it means that the two lines are parallel. Therefore, the slope of line M is also 2.

To be perpendicular to line L, the slope of line M should be the negative reciprocal of 2, which is -1/2. None of the given points have a slope of -1/2. Therefore, the answer is (-4, 5) because it is the only point that does not have a slope of 2.

Explanation
Before Sue's raise, Lisa's salary was 1% higher than Sue's. This means that Lisa's salary was 1% less than Sue's original salary. After Sue's raise, Sue's salary increased by 5%, making Lisa's salary still 1% less than Sue's new salary. Therefore, Lisa's salary is $30,300, which is 1% less than Sue's new salary of $31,200.

Explanation
If x = -1 is one solution of the quadratic equation ax^2 + bx + c = 0, then the other solution can be found by substituting x = -1 into the equation. This gives us a(-1)^2 + b(-1) + c = 0, which simplifies to a - b + c = 0. Rearranging the equation, we get c = a - b. Dividing both sides by -a, we get -c/a = -b/a. Therefore, the other solution is x = -c/a.

Explanation
The point (28, 15) is visible because the line segment from (0, 0) to (28, 15) does not pass through any other lattice points.

Explanation
The flea jumps in a pattern where it jumps 1 number, then 2 numbers, then 3 numbers, then 2 numbers, then 1 number, and so on. This pattern repeats. To find the number the flea lands on at its 2008th jump, we need to find the remainder when 2008 is divided by the number of jumps in each cycle (1+2+3+2+1 = 9). The remainder is 2. Therefore, the flea lands on the 2nd number in the cycle, which is 8.

Explanation
In quadrilateral ABCD, E, F, G, and H are the midpoints of the sides. Since E and F are midpoints, EF is parallel to AB and EF = 1/2 * AB. Similarly, GH is parallel to CD and GH = 1/2 * CD. Since EH is a transversal, ∠FEH and ∠EHG are corresponding angles. Corresponding angles formed by parallel lines are congruent, so ∠FEH = ∠EHG. Additionally, since EF is parallel to AB and GH is parallel to CD, ∠FEH and ∠EHG are supplementary angles, meaning ∠FEH + ∠EHG = 180°. Therefore, both A and C must be true.

Explanation
The set {2, 4, 5, 7} has a total of 4 elements. When selecting subsets, we can choose to include or exclude each element. For each element, there are 2 possibilities (either include or exclude). So, there are a total of 2^4 = 16 subsets. However, one of these subsets is the empty set, which does not contribute to the sum. Therefore, there are 16 - 1 = 15 nonempty subsets. Out of these subsets, there are 12 different sums that the elements can add up to, which is the correct answer.

Explanation
If Luis and Anh get the same correct answer for x, it means that the values of x in both equations are equal. Since a, b, and c are distinct and nonzero, the only way for x to be the same in both equations is if the coefficients of x in both equations are also equal. Therefore, we can set ax - b = bx - c and simplify it to (a - b)x = c - b. Since a and b are distinct, we can divide both sides of the equation by (a - b) to get x = (c - b)/(a - b).

Now, let's substitute this value of x back into the equation ax - b = c. We get a((c - b)/(a - b)) - b = c, which simplifies to ac - ab - b(a - b) = c(a - b). Expanding and simplifying further, we get ac - ab - ab + b^2 = ac - bc. Cancelling out ac on both sides, we get -ab - ab + b^2 = -bc, which simplifies to -2ab + b^2 = -bc.

Now, let's substitute this value of x back into the equation bx - c = a. We get b((c - b)/(a - b)) - c = a, which simplifies to bc - b^2 - c(a - b) = a(a - b). Expanding and simplifying further, we get bc - b^2 - ac + bc = a^2 - ab. Cancelling out bc on both sides, we get -b^2 - ac + ab = a^2 - ab, which simplifies to -b^2 - ac = a^2 - 2ab.

Comparing the two equations we obtained, -2ab + b^2 = -bc and -b^2 - ac = a^2 - 2ab, we can add them together to eliminate the ab term and obtain b^2 - ac = a^2 - bc. Rearranging this equation, we get a^2 + b^2 + c(-a - b) = 0, which simplifies to a + b + c = 0. Therefore, the correct answer is a + b + c = 0.

Explanation
The function f(x) = has 2 asymptotes. This can be determined by analyzing the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, the function approaches a horizontal asymptote at y = 0. As x approaches negative infinity, the function also approaches a horizontal asymptote at y = 0. Therefore, the function has 2 asymptotes.

Explanation
To find the value of A + M + A + T + Y + C, we need to determine the values of A, M, T, Y, and C. Since the number formed by replacing the letters with digits is the largest number divisible by 55, we can deduce that the digit in the units place must be 0 or 5. Since the given answer is 40, we can conclude that the digit in the units place is 0. Therefore, A + M + A + T + Y + C = 4 + 0 + 4 + T + Y + C = 8 + T + Y + C. As we do not have any information about the values of T, Y, and C, we cannot determine their individual values or the final sum.

Explanation
The equation a^6 + b^2 + c^2 = 2009 implies that a^6, b^2, and c^2 are all positive integers. Since 2009 is odd, it means that at least one of a, b, or c must be odd. If two of them are powers of 2, it means that the remaining one must be odd. The only odd number among the answer choices is 51. Therefore, the sum of a, b, and c is 51.

Explanation
In a leap year, February has 29 days instead of the usual 28 days. Since ACME had 3 paydays in February 2008, it means that the first payday was on the first Friday of the month, the second payday was on the third Friday of the month, and the third payday was on the fifth Friday of the month. This pattern repeats every 28 years because that is the number of years it takes for the calendar to repeat itself. Therefore, the next year in which ACME will have 3 February paydays will be 2036, and the units digit of 2036 is 6.

Explanation
Based on the given information, we know that three suspects are telling the truth and two are lying. If Suspect E is telling the truth, then Suspect B must also be telling the truth. This means that Suspect B is innocent, contradicting Suspect A's statement that D is the murderer. Since Suspect A is lying, D cannot be the murderer. Therefore, the murderer must be someone other than D, which narrows it down to suspects C and E. However, since B is telling the truth, E must also be telling the truth about B. Therefore, the only remaining option is that Suspect C is the murderer.

Explanation
The given sequence is formed by multiplying two arithmetic sequences together. The first sequence could be decreasing by 6 each time, starting from 468. The second sequence could be decreasing by 6 each time, starting from 78. When multiplying the corresponding terms of these two sequences, we get the given sequence. Therefore, the next term in the sequence would be 78 - 6 = 72, which when multiplied with 6 will give 432. However, since this option is not given, the next closest option is 234.

Explanation
In a triangle, the ratio of the lengths of any two sides is equal to the ratio of the corresponding angles opposite those sides. Using this property, we can find the value of the given expression. Let's consider the ratio of AB to AC. This is equal to 5/7. Now, since the ratio of AB to AC is equal to the ratio of the corresponding angles opposite those sides, we can conclude that the angle opposite AB is equal to the angle opposite 5/7. Therefore, the value of the given expression is 1/8.

Explanation
The volume of the portion of the pyramid which lies above the base and below a plane parallel to the base and 3 m above the base can be calculated by subtracting the volume of the smaller pyramid from the volume of the larger pyramid. The smaller pyramid has a base of 6 m on a side and a height of 3 m, while the larger pyramid has a base of 6 m on a side and a height of 9 m. The volume of the smaller pyramid is (1/3) * (6^2) * 3 = 36 m^3, and the volume of the larger pyramid is (1/3) * (6^2) * 9 = 108 m^3. Therefore, the volume of the portion of the pyramid is 108 - 36 = 72 m^3. However, the given answer is 76 m^3, which is incorrect.

Explanation
The ratio of the area of triangle ABC to the area of triangle DEF can be determined by comparing their corresponding side lengths. Since AB is twice DE, the ratio of their lengths is 2:1. Similarly, since AC is equal to AB, the ratio of AC to DF is also 2:1. Therefore, the ratio of the areas of the triangles will be the square of the ratio of their side lengths, which is (2/1)^2 = 4. The answer 2 sec A represents the ratio of the areas and is equivalent to 4 when simplified.

Explanation
The area of a triangle can be found using the formula: Area = (1/2) * base * height. In this case, we can consider QT as the base and PS as the height of the triangle. Since the hexagon is regular and all interior angles are 120°, we can divide the hexagon into 6 congruent equilateral triangles. Therefore, the base QT would be equal to the side length of one of these equilateral triangles, which is 100. The height PS can be found by subtracting the height of the equilateral triangle from the side length of the hexagon, which is 50. Plugging these values into the formula, we get: Area = (1/2) * 100 * 50 = 2500. Rounding this to the nearest tenth, we get 1082.5.