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 2025-08-25 00:00:00 2025-08-25 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$. | [
-0.5415966510772705,
-0.4471573829650879,
-0.8446708917617798,
-0.973517656326294,
2.0166494846343994,
1.1591202020645142,
-2.652702569961548,
2.537790060043335,
-0.2503644526004791,
-1.0376265048980713,
3.784738063812256,
0.2834383547306061,
3.0496058464050293,
2.7120134830474854,
-1.22... | [
-0.14945341646671295,
-0.6359495520591736,
0.3701525330543518,
0.5619896054267883,
-1.4310364723205566,
2.5220203399658203,
-0.7751529216766357,
3.182236909866333,
0.9531182646751404,
-2.3101651668548584,
2.8149681091308594,
-0.36299630999565125,
0.465959370136261,
4.008862018585205,
-1.... | 2025-08-25 |
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) | [1.8249931335449219,-2.3721845149993896,3.3197240829467773,0.872027575969696,0.8807427883148193,-1.8(...TRUNCATED) | [-0.3368057608604431,-2.1904642581939697,2.485434055328369,0.9306263327598572,0.5297894477844238,-1.(...TRUNCATED) | 2025-08-25 |
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) | [-1.8792836666107178,0.8565375208854675,0.9974541664123535,2.6583175659179688,0.2608870565891266,3.4(...TRUNCATED) | [-2.3459439277648926,2.07476544380188,0.8504107594490051,2.5082578659057617,-0.7544269561767578,3.05(...TRUNCATED) | 2025-08-25 |
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) | [2.0913424491882324,0.04448004812002182,1.2797305583953857,1.8607616424560547,1.6554900407791138,-1.(...TRUNCATED) | [3.8289525508880615,-3.1585183143615723,1.9521594047546387,3.2345221042633057,0.8925213813781738,-3.(...TRUNCATED) | 2025-08-25 |
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) | [1.5186800956726074,0.015969419851899147,1.872541069984436,-1.388711929321289,-1.541198968887329,-0.(...TRUNCATED) | [3.168485403060913,-2.101365566253662,1.7768099308013916,-2.6053926944732666,1.1204946041107178,-1.5(...TRUNCATED) | 2025-08-25 |
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) | [-0.5985934734344482,-1.387424349784851,2.548039197921753,0.6973485946655273,1.9393223524093628,-1.2(...TRUNCATED) | [-0.9342586398124695,-1.0935840606689453,1.9004050493240356,0.4786365330219269,0.09614533185958862,0(...TRUNCATED) | 2025-08-25 |
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) | [0.33619776368141174,1.3778430223464966,3.100539445877075,1.9988372325897217,0.8490438461303711,1.44(...TRUNCATED) | [0.4154530465602875,1.6999996900558472,3.6031179428100586,0.7092217803001404,0.7998707890510559,-0.4(...TRUNCATED) | 2025-08-25 |
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) | [1.623451590538025,-1.564774990081787,2.4723780155181885,1.9586089849472046,-0.7616987228393555,3.04(...TRUNCATED) | [2.743098258972168,-0.30482661724090576,2.77850604057312,1.4165208339691162,-0.1648148149251938,2.96(...TRUNCATED) | 2025-08-25 |
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) | [-0.6221884489059448,-2.691359281539917,6.24393892288208,0.06021460145711899,1.8142298460006714,-1.6(...TRUNCATED) | [0.872802197933197,-1.3294340372085571,2.7414188385009766,-0.38841602206230164,1.4215285778045654,2.(...TRUNCATED) | 2025-08-25 |
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) | [0.7038238644599915,-2.1458656787872314,1.872026801109314,-0.15360446274280548,-0.7062354683876038,0(...TRUNCATED) | [1.9245648384094238,-1.421718955039978,2.9983694553375244,0.26316049695014954,-1.2755481004714966,0.(...TRUNCATED) | 2025-08-25 |
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