| { |
| "summary": { |
| "published_entries": 226, |
| "published_unique": 220, |
| "final_rows": 223, |
| "answer_corrected": 11, |
| "no_change_corrections": 3, |
| "orphan_corrections": 2, |
| "retracted": 5, |
| "duplicate_removed": 6, |
| "added": 8 |
| }, |
| "conflicts": [ |
| { |
| "type": "retracted", |
| "problem": "A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$", |
| "published_answer": "4", |
| "final_answer": null |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$?", |
| "published_answer": "-13.5", |
| "final_answer": "-13.5" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?", |
| "published_answer": "8", |
| "final_answer": "\\frac{1}{2}\\csc{\\frac{1}{4}}" |
| }, |
| { |
| "type": "retracted", |
| "problem": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased?", |
| "published_answer": "7", |
| "final_answer": null |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "A shopper plans to purchase an item that has a listed price greater than $\\$100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\$30$ off the listed price, and Coupon C gives $25\\%$ off the amount by which the listed price exceeds\n$\\$100$. \nLet $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$?", |
| "published_answer": "50", |
| "final_answer": "50" |
| }, |
| { |
| "type": "added", |
| "problem": "A straight line joins the points $(-1,1)$ and $(3,9)$. Its $x$-intercept is:", |
| "published_answer": null, |
| "final_answer": "-\\frac{3}{2}" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is:", |
| "published_answer": "33\\frac {1}{3}\\%", |
| "final_answer": "8" |
| }, |
| { |
| "type": "added", |
| "problem": "At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to", |
| "published_answer": null, |
| "final_answer": "35\\%" |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?", |
| "published_answer": "24", |
| "final_answer": "24" |
| }, |
| { |
| "type": "added", |
| "problem": "For each positive integer $n$, let \n\n$a_n = \\frac{(n+9)!}{(n-1)!}$.\nLet $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is", |
| "published_answer": null, |
| "final_answer": "9" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "For real numbers $a$ and $b$, define $a \\diamond b = \\sqrt{a^2 + b^2}$. What is the value of $(5 \\diamond 12) \\diamond ((-12) \\diamond (-5))$?", |
| "published_answer": "13\\sqrt{2}", |
| "final_answer": "\\sqrt{2}" |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have?", |
| "published_answer": "23", |
| "final_answer": "23" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "If $991+993+995+997+999=5000-N$, then $N=$", |
| "published_answer": "25", |
| "final_answer": "$\\frac{b'}{a'}<\\frac{b}{a}$" |
| }, |
| { |
| "type": "added", |
| "problem": "If $\\log_6 x=2.5$, the value of $x$ is:", |
| "published_answer": null, |
| "final_answer": "36\\sqrt{6}" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "In the table shown, the formula relating \\(x\\) and \\(y\\) is:\n\\[\\begin{array}{|c|c|c|c|c|c|}\\hline x & 1 & 2 & 3 & 4 & 5\\\\ \\hline y & 3 & 7 & 13 & 21 & 31\\\\ \\hline\\end{array}\\]", |
| "published_answer": "y = x^2 + x + 1", |
| "final_answer": "2016" |
| }, |
| { |
| "type": "added", |
| "problem": "Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?", |
| "published_answer": null, |
| "final_answer": "\\frac{35}{27}" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\n\\[ (2 + i)^n = a_n + b_ni \\]for all integers $n\\geq 0$, where $i = \\sqrt{-1}$. What is\n\\[\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?\\]", |
| "published_answer": "\\frac{7}{16}", |
| "final_answer": "7" |
| }, |
| { |
| "type": "orphan_correction", |
| "problem": "Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$?", |
| "published_answer": null, |
| "final_answer": null |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$", |
| "published_answer": "\\frac{\\sqrt{2}}{2}", |
| "final_answer": "\\frac{\\sqrt{2}}{2}" |
| }, |
| { |
| "type": "retracted", |
| "problem": "Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the start do they pass each other?", |
| "published_answer": "47", |
| "final_answer": null |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth?", |
| "published_answer": "$1.15", |
| "final_answer": "1002" |
| }, |
| { |
| "type": "added", |
| "problem": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is:", |
| "published_answer": null, |
| "final_answer": "12" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of segment $AB$ is", |
| "published_answer": "70", |
| "final_answer": "6" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?", |
| "published_answer": "-8", |
| "final_answer": "\\pm 8" |
| }, |
| { |
| "type": "retracted", |
| "problem": "Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?", |
| "published_answer": "\\frac{25}{32}", |
| "final_answer": null |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used?", |
| "published_answer": "24000", |
| "final_answer": "\\frac{33}{7}" |
| }, |
| { |
| "type": "added", |
| "problem": "The number of points common to the graphs of\n$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is:", |
| "published_answer": null, |
| "final_answer": "4" |
| }, |
| { |
| "type": "retracted", |
| "problem": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or last.\n\nWhat is the average of the first and last numbers?", |
| "published_answer": "6.5", |
| "final_answer": null |
| }, |
| { |
| "type": "duplicate_removed", |
| "problem": "The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):", |
| "published_answer": "12", |
| "final_answer": "12" |
| }, |
| { |
| "type": "answer_corrected", |
| "problem": "The product $8 \\times .25 \\times 2 \\times .125 =$", |
| "published_answer": "$\\frac{1}{2}$", |
| "final_answer": "16" |
| }, |
| { |
| "type": "added", |
| "problem": "The sum of all integers between 50 and 350 which end in 1 is", |
| "published_answer": null, |
| "final_answer": "5880" |
| }, |
| { |
| "type": "orphan_correction", |
| "problem": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed?", |
| "published_answer": null, |
| "final_answer": null |
| } |
| ] |
| } |
|
|
