aops_id int64 2 2.31M | problem stringlengths 21 2.9k | best_solution stringlengths 8 29.4k | problem_vector listlengths 4.1k 4.1k | best_solution_vector listlengths 4.1k 4.1k | last_modified stringdate 2025-08-24 00:00:00 2025-08-24 00:00:00 |
|---|---|---|---|---|---|
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$. | [
3.570814847946167,
-1.7753154039382935,
-0.665418267250061,
-0.23375236988067627,
1.9694957733154297,
-1.1761784553527832,
-2.9528696537017822,
4.374476909637451,
-1.6055481433868408,
-0.9244315028190613,
4.8105034828186035,
2.7510228157043457,
5.853891849517822,
1.6139694452285767,
-2.4... | [
1.7708947658538818,
-5.836791038513184,
-0.3062914311885834,
0.06896131485700607,
-0.13791823387145996,
3.2823102474212646,
1.5323816537857056,
1.7637678384780884,
0.04532882198691368,
-3.2721052169799805,
3.677568197250366,
-3.1554181575775146,
3.772738218307495,
1.5303411483764648,
0.8... | 2025-08-24 |
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) | [-0.9829646348953247,-2.961660623550415,7.049641132354736,2.943843364715576,-1.6289494037628174,-5.6(...TRUNCATED) | [0.005187909118831158,-3.710606336593628,4.224234580993652,2.8036015033721924,-0.8013380169868469,-2(...TRUNCATED) | 2025-08-24 |
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) | [-2.098080635070801,-0.3907448351383209,-1.6729884147644043,5.783786773681641,-2.2908856868743896,4.(...TRUNCATED) | [-1.4371801614761353,3.6136295795440674,1.2283209562301636,2.120863676071167,-2.8960413932800293,2.2(...TRUNCATED) | 2025-08-24 |
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) | [0.9701810479164124,-0.5716570019721985,0.09008260071277618,3.866093397140503,1.5660951137542725,-2.(...TRUNCATED) | [4.2249674797058105,0.09290449321269989,0.4986517131328583,3.437798500061035,1.3503758907318115,-4.8(...TRUNCATED) | 2025-08-24 |
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) | [1.091335415840149,-1.76021146774292,4.271319389343262,-0.11592613905668259,-4.795106887817383,-4.63(...TRUNCATED) | [1.6146806478500366,-3.100133180618286,2.388460397720337,-6.496150970458984,-0.4887966811656952,-4.9(...TRUNCATED) | 2025-08-24 |
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) | [0.4449366629123688,-2.7237424850463867,3.228285551071167,1.6243022680282593,2.267239809036255,-1.27(...TRUNCATED) | [0.5394678711891174,-2.8471710681915283,1.969513177871704,-0.8629615902900696,1.2775065898895264,-2.(...TRUNCATED) | 2025-08-24 |
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) | [3.479153633117676,0.2953947186470032,2.407150983810425,2.228222608566284,-2.6047139167785645,-0.288(...TRUNCATED) | [-0.4807426631450653,0.059338074177503586,4.841386318206787,4.363824367523193,0.022590603679418564,0(...TRUNCATED) | 2025-08-24 |
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) | [1.9140053987503052,-2.1295340061187744,4.075040817260742,3.7389907836914062,-2.811946153640747,2.22(...TRUNCATED) | [5.38160514831543,1.3006267547607422,1.458322525024414,1.9071049690246582,-1.6588925123214722,2.7495(...TRUNCATED) | 2025-08-24 |
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) | [-2.6649794578552246,-1.9405810832977295,5.917922019958496,2.023343801498413,2.1261746883392334,0.21(...TRUNCATED) | [1.6224455833435059,1.1426305770874023,1.9818674325942993,-0.6865325570106506,0.4737430512905121,2.9(...TRUNCATED) | 2025-08-24 |
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) | [1.4639017581939697,-1.4736051559448242,3.2216315269470215,2.490168333053589,-3.6167309284210205,-2.(...TRUNCATED) | [3.8737406730651855,-2.2894341945648193,4.9870100021362305,0.6440514922142029,-2.431504011154175,-3.(...TRUNCATED) | 2025-08-24 |
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