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$.
|
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] |
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
|
End of preview. Expand
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