aops_id int64 | problem string | best_solution string | problem_vector list | best_solution_vector list | last_modified string |
|---|---|---|---|---|---|
1,000,044 | Justin has a 55% chance of winning any given point in a ping-pong game. To the nearest 0.1%, what is the probability that he wins exactly 7 out of the first 10 points? | So if he has a 55% chance of winning, he conversely has a 45% chance of losing. The problem calls for him winning 7 times and losing 3, so his win percentage will be multiplied by itself 7 times and his losing percentage will be multiplied by itself 3 times so your expression should look like this $ 0.55^7*0.45^3$. | [
0.08697930723428726,
-1.2883906364440918,
-3.31341552734375,
-2.1786866188049316,
2.552647352218628,
-1.2810782194137573,
-3.2131125926971436,
7.319212913513184,
3.7486960887908936,
-2.5368223190307617,
7.924187183380127,
-2.4042491912841797,
6.686105728149414,
5.614565372467041,
-0.6990... | [
0.919487476348877,
-1.0458155870437622,
-1.7498561143875122,
0.5332849025726318,
-3.528571128845215,
-0.015989486128091812,
0.2593991458415985,
8.545211791992188,
7.129958629608154,
-4.414690017700195,
5.180262088775635,
-3.654730796813965,
2.190992832183838,
6.335718154907227,
-0.939518... | 2025-08-26 |
100,009 | "Evaluate\n\\[\n\\int_{\\frac{\\pi}{4}}^{\\frac{\\pi}{3}}\\frac{\\sqrt{\\sin x}+\\sqrt{\\cos x}+3(\\(...TRUNCATED) | "You may be right. I made this problem by the differentiation of $\\sin x \\sqrt{\\cos x}+\\cos x\\s(...TRUNCATED) | [3.5696184635162354,-3.917421579360962,5.230591773986816,1.709114670753479,0.4578912854194641,-7.831(...TRUNCATED) | [-1.926249623298645,-3.4507951736450195,1.3671175241470337,1.587105631828308,-0.9268237352371216,-7.(...TRUNCATED) | 2025-08-26 |
100,010 | "Billy Bob has a pet snail called Larry. The wall is 37 feet tall. Larry can climb 3 feet in one day(...TRUNCATED) | "[quote=\"mtms5467\"][hide]So basically Larry climbs 1ft/day. The day/date 37 days from June 2. (Oh (...TRUNCATED) | [-1.6345058679580688,0.8750608563423157,1.2080631256103516,5.76799201965332,-1.286815881729126,1.838(...TRUNCATED) | [-3.1719954013824463,3.6722984313964844,-0.39332446455955505,5.428849697113037,-3.256715774536133,1.(...TRUNCATED) | 2025-08-26 |
1,000,136 | "Deriving the Quadratic Formula\n\nProblem:\nDerive the quadratic formula.\n\nSolution:\nStart with (...TRUNCATED) | "Lol. I figured out how to do it this past year in 6th grade...\r\n\r\nMy math teacher never showed (...TRUNCATED) | [5.783517360687256,-0.5681543946266174,1.6161996126174927,4.2906174659729,2.6326303482055664,-6.3012(...TRUNCATED) | [7.1943535804748535,-5.314296245574951,3.37431263923645,3.7783281803131104,-0.010024062357842922,-9.(...TRUNCATED) | 2025-08-26 |
1,000,141 | "[b]Coin Problems[/b]\r\n\r\n[i]Tony has 11 more nickels than quarters. If the total value of his co(...TRUNCATED) | "there's a few ways to do problems like the second one that work for all positive integer number of (...TRUNCATED) | [2.063680648803711,-1.4086905717849731,2.245319366455078,-3.2822535037994385,-5.547669410705566,-3.8(...TRUNCATED) | [4.570125102996826,-3.631040573120117,0.9893410205841064,-5.473449230194092,-0.28434088826179504,-8.(...TRUNCATED) | 2025-08-26 |
100,015 | "A 6-letter car plaque is to be made using the letters \\(A,\\dots,Z\\) such that the letters are in(...TRUNCATED) | "[hide]I get $\\binom{26}{6}$. Choose any 6 letters and there exists a unique alphabetical arrangeme(...TRUNCATED) | [1.510438084602356,-4.618250846862793,4.103366851806641,0.2138204723596573,0.26913318037986755,-6.11(...TRUNCATED) | [0.9806643128395081,-3.395272970199585,4.214189529418945,0.3433818221092224,-1.8933871984481812,-4.5(...TRUNCATED) | 2025-08-26 |
100,019 | "Billy Bob has a huge garden. He picks a few flowers from it. There is one red flower, one blue flow(...TRUNCATED) | "[hide]Or you can count the number of total ways $4!=24$ and then subtract the number of ways the re(...TRUNCATED) | [2.8337879180908203,1.9365068674087524,5.488781929016113,1.8772523403167725,-0.04920530691742897,-2.(...TRUNCATED) | [3.3589131832122803,2.2523984909057617,4.945671558380127,1.2547461986541748,0.015567007474601269,-4.(...TRUNCATED) | 2025-08-26 |
100,023 | Simplify
\[
(1+x)(1+x^{2})(1+x^{4})(1+x^{8})\cdots
\]
for \(|x|<1\). | "[hide]When you multiply it out, you can see that the product is equal to\n$1+x+x^{2}+x^{3}\\dots$\n(...TRUNCATED) | [5.295506477355957,-1.9337574243545532,4.081794738769531,4.035449981689453,-3.5980660915374756,1.665(...TRUNCATED) | [4.299628257751465,-1.2035061120986938,4.512174606323242,1.610304832458496,-1.6633838415145874,2.743(...TRUNCATED) | 2025-08-26 |
1,000,249 | "Two players (You and Ben) are each arrested and placed in separate jail cells with no communication(...TRUNCATED) | "If all four possibilities are equally likely, then confessing is better:\r\n\r\nMe Ben Number of(...TRUNCATED) | [-0.20810171961784363,-4.6703901290893555,9.617036819458008,-0.7414187788963318,1.787200927734375,-6(...TRUNCATED) | [2.1278281211853027,-2.814142942428589,2.8729944229125977,-0.2436593770980835,2.5396640300750732,2.6(...TRUNCATED) | 2025-08-26 |
100,026 | "Let r and s be the roots of\n\\[\nx^{2}-(a+d)x+(ad-bc)=0.\n\\]\nProve that \\(r^{3}\\) and \\(s^{3}(...TRUNCATED) | "From ?vietta's? sums $r+s=a+d$ and $rs=ad-bc$. Thus $r^{3}+s^{3}=(r+s)^{3}-3rs(r+s)=(a+d)^{3}-3(ad-(...TRUNCATED) | [2.88655686378479,-5.038615703582764,2.6150460243225098,0.31835025548934937,-1.979373574256897,-5.23(...TRUNCATED) | [5.640284538269043,-3.797664165496826,4.356821060180664,0.26075446605682373,-3.817363739013672,-6.05(...TRUNCATED) | 2025-08-26 |
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