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$. | [
2.8030927181243896,
2.3136212825775146,
-1.2958351373672485,
-1.9936344623565674,
0.7724860906600952,
-0.9180781841278076,
-2.0114498138427734,
3.9851622581481934,
0.09174680709838867,
-0.4330829977989197,
0.372242271900177,
-0.48637667298316956,
-0.40065598487854004,
1.707797884941101,
... | [
1.9280145168304443,
2.939913272857666,
-0.24393343925476074,
-0.38710230588912964,
0.30727627873420715,
-0.24417300522327423,
-2.128354549407959,
4.120147228240967,
0.1987561583518982,
-0.019997013732790947,
1.0778592824935913,
-1.131377100944519,
-1.3441892862319946,
2.0876567363739014,
... | 2025-08-29 |
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) | [2.7128422260284424,3.717719554901123,3.5919721126556396,3.012800455093384,-0.7764116525650024,-1.28(...TRUNCATED) | [2.035567045211792,3.3168482780456543,1.5208553075790405,1.680232048034668,0.49179789423942566,-3.93(...TRUNCATED) | 2025-08-29 |
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) | [2.437530994415283,2.3186957836151123,-0.32914698123931885,3.550917148590088,0.14755621552467346,2.9(...TRUNCATED) | [0.6932055950164795,4.095274925231934,-0.2362677901983261,3.949507713317871,-0.9215804934501648,3.51(...TRUNCATED) | 2025-08-29 |
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) | [0.09950580447912216,2.898780584335327,1.2795394659042358,2.3857712745666504,0.7789515256881714,-3.6(...TRUNCATED) | [2.392263889312744,0.888561487197876,-0.22305071353912354,0.43031439185142517,1.0567017793655396,-0.(...TRUNCATED) | 2025-08-29 |
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) | [0.9209532141685486,2.537532329559326,0.8289640545845032,-0.41124844551086426,1.1612237691879272,-0.(...TRUNCATED) | [3.314828634262085,1.532874584197998,-0.2451361119747162,-1.12736177444458,-0.19789481163024902,-3.7(...TRUNCATED) | 2025-08-29 |
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) | [2.2758429050445557,2.8135111331939697,0.5543681383132935,2.2481179237365723,0.27129068970680237,-0.(...TRUNCATED) | [0.8686273694038391,2.195624828338623,0.7781714200973511,2.3987972736358643,-1.152292013168335,-2.87(...TRUNCATED) | 2025-08-29 |
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) | [4.367686748504639,1.2180286645889282,2.076432228088379,2.9230363368988037,-0.4840604364871979,0.734(...TRUNCATED) | [1.5911667346954346,3.1836299896240234,0.06413087248802185,1.7139215469360352,-0.021537721157073975,(...TRUNCATED) | 2025-08-29 |
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) | [3.719792604446411,2.548764944076538,1.6171727180480957,3.1904540061950684,-1.7015653848648071,-1.63(...TRUNCATED) | [2.7498693466186523,3.6081910133361816,-1.0554627180099487,2.8275842666625977,-1.6262929439544678,-2(...TRUNCATED) | 2025-08-29 |
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) | [2.7025845050811768,-1.7388453483581543,2.8729918003082275,0.05798490718007088,0.5387190580368042,-2(...TRUNCATED) | [1.945442795753479,0.19352975487709045,1.399412751197815,-0.08180882036685944,1.3471996784210205,-3.(...TRUNCATED) | 2025-08-29 |
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) | [2.7294540405273438,1.9211355447769165,2.197392702102661,4.631567001342773,-1.3012337684631348,-1.84(...TRUNCATED) | [1.5793049335479736,4.08672571182251,2.8744029998779297,3.157696008682251,0.3320061266422272,-3.5882(...TRUNCATED) | 2025-08-29 |
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