{"messages": [{"role": "user", "content": "For any finite set $S$ , let $|S|$ denote the number of elements in $S$ . Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,4,5\\}$ that satisfy \\[|A| \\cdot |B| = |A \\cap B| \\cdot |A \\cup B|\\]"}, {"role": "assistant", "content": "454"}], "solution": "454"}
{"messages": [{"role": "user", "content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its nonempty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for $\\{1, 2, 3, 6,9\\}$ is $9-6+3-2+1=5$ and for $\\{5\\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ ."}, {"role": "assistant", "content": "448"}], "solution": "448"}
{"messages": [{"role": "user", "content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $160$ points out of $300$ points attempted on the first day, and scored $140$ points out of $200$ points attempted on the second day. Beta who did not attempt $300$ points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\\frac{300}{500} = \\frac{3}{5}$ . The largest possible two-day success ratio that Beta could achieve is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?"}, {"role": "assistant", "content": "849"}], "solution": "849"}
{"messages": [{"role": "user", "content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$ ."}, {"role": "assistant", "content": "181"}], "solution": "181"}
{"messages": [{"role": "user", "content": "Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? "}, {"role": "assistant", "content": "968"}], "solution": "968"}
{"messages": [{"role": "user", "content": "Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal .  Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?"}, {"role": "assistant", "content": "525"}], "solution": "525"}
{"messages": [{"role": "user", "content": "Define an ordered triple $(A, B, C)$ of sets to be $\\textit{minimally intersecting}$ if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$ . For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\\{1,2,3,4,5,6,7\\}$ . Find the remainder when $N$ is divided by $1000$ . Note : $|S|$ represents the number of elements in the set $S$ ."}, {"role": "assistant", "content": "760"}], "solution": "760"}