aops_id
int64 2
2.31M
| problem
stringlengths 21
2.9k
| best_solution
stringlengths 8
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| problem_vector
listlengths 4.1k
4.1k
| best_solution_vector
listlengths 4.1k
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| last_modified
stringdate 2025-08-27 00:00:00
2025-08-27 00:00:00
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1,000,044
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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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-0.008535458706319332,
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0.01202875841408968,
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-0.0016930826241150498,
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-0.014433379285037518,
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-0.0001312073873123154,
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-0.011533038690686226,
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0.018635811284184456,
0.0013182745315134525,
-0.0022374934051185846
] |
2025-08-27
|
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)
| [0.009909790009260178,-0.012449627742171288,0.0161251500248909,0.006058339029550552,0.00023122517450(...TRUNCATED)
| [-0.006428204942494631,-0.010722576640546322,0.0040140170603990555,0.003906335216015577,-0.005830604(...TRUNCATED)
|
2025-08-27
|
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)
| [-0.007238154299557209,0.002288885647431016,0.0022486706729978323,0.016015999019145966,-0.0042119873(...TRUNCATED)
| [-0.01210505235940218,0.008677786216139793,0.002470973879098892,0.01616763323545456,-0.0099506685510(...TRUNCATED)
|
2025-08-27
|
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)
| [0.017451224848628044,-0.0017157908296212554,0.008866819553077221,0.015284442342817783,0.00677455728(...TRUNCATED)
| [0.018933268263936043,-0.015529998578131199,0.008531544357538223,0.011334807612001896,-0.00265378109(...TRUNCATED)
|
2025-08-27
|
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)
| [0.006747371982783079,-0.00358223426155746,0.008114206604659557,-0.00939754769206047,-0.017316635698(...TRUNCATED)
| [0.012300376780331135,-0.010700748302042484,0.0035853323061019182,-0.01750449277460575,-0.0025189407(...TRUNCATED)
|
2025-08-27
|
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)
| [0.004981253761798143,-0.01403488777577877,0.011357414536178112,-0.000719498610123992,0.001864324323(...TRUNCATED)
| [0.00031910522375255823,-0.007932323962450027,0.012443573214113712,-0.0021400863770395517,-0.0047115(...TRUNCATED)
|
2025-08-27
|
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)
| [0.007302251178771257,0.0060698469169437885,0.01649150811135769,0.005265043117105961,0.0012811287306(...TRUNCATED)
| [0.009624339640140533,0.0058858515694737434,0.016543781384825706,0.002637772588059306,0.000413437403(...TRUNCATED)
|
2025-08-27
|
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)
| [0.015368298627436161,-0.0071906172670423985,0.012421506457030773,0.010114409029483795,-0.0107725197(...TRUNCATED)
| [0.013290176168084145,-0.0024262184742838144,0.013543563894927502,0.0017325865337625146,-0.006009837(...TRUNCATED)
|
2025-08-27
|
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)
| [0.0009584046783857048,-0.013015174306929111,0.03151926398277283,-0.002742346143350005,0.00148224597(...TRUNCATED)
| [0.007301537320017815,-0.009212294593453407,0.013495930470526218,0.0015476205153390765,0.00538851972(...TRUNCATED)
|
2025-08-27
|
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)
| [0.00999057199805975,-0.014468141831457615,0.007835516706109047,0.000333522679284215,-0.007521555759(...TRUNCATED)
| [0.00911557674407959,-0.011665837839245796,0.012439913116395473,-0.00029639803688041866,-0.012386921(...TRUNCATED)
|
2025-08-27
|
End of preview. Expand
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