aops_id
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stringdate 2025-08-27 00:00:00
2025-08-28 00:00:00
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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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] |
2025-08-27
|
100,009
|
Evaluate
\[
\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sqrt{\sin x}+\sqrt{\cos x}+3(\sqrt{\sin x}-\sqrt{\cos x})\cos 2x}{\sqrt{\sin 2x}}\;dx.
\] You may be right. I made this problem by the differentiation of $\sin x \sqrt{\cos x}+\cos x\sqrt{\sin x}$ as shyong showed.
|
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] |
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, but at night he slips down 2 feet. Larry starts the climb on Sunday, June 2, 2006. On what day (day of the week and date) will Larry finish the climb? [quote="mtms5467"][hide]So basically Larry climbs 1ft/day. The day/date 37 days from June 2. (Oh wait...June 2, 2006 is a Friday...)
[hide="Assuming June 2, 2006 is a Sunday"]Saturday July 8[/hide]
[hide="Assuming he means July 2 (actually a Sun.!)"] Monday Aug 7[/hide][/hide][/quote]
It's a classic trick question. After 34 days, he's at 34 ft. And on the 35th day, he reaches the top, because he won't slide the 2 feet back down.
|
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] |
2025-08-27
|
1,000,136
| "Deriving the Quadratic Formula\n\nProblem:\nDerive the quadratic formula.\n\nSolution:\nStart with (...TRUNCATED)
| [-0.0002122835285263136,-0.04541593790054321,0.0046520717442035675,0.04031360149383545,-0.0005049610(...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)
| [-0.00006175064481794834,0.007315790746361017,0.026415608823299408,0.034050799906253815,0.0002337306(...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)
| [-0.000032167423341888934,-0.05821501463651657,0.010568084195256233,0.049778323620557785,-0.00017727(...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)
| [0.00030263312510214746,-0.06200506165623665,0.004736138507723808,0.005954334978014231,0.00155126536(...TRUNCATED)
|
2025-08-27
|
100,023
| "Simplify\n\\[\n(1+x)(1+x^{2})(1+x^{4})(1+x^{8})\\cdots\n\\]\nfor \\(|x|<1\\). [hide]When you multip(...TRUNCATED)
| [-0.0000482597533846274,-0.008691142313182354,0.030783245339989662,0.05414757505059242,-0.0001775505(...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)
| [-0.000031315295927925035,-0.0027431652415543795,0.03295576944947243,-0.02672015316784382,3.97694066(...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)
| [0.0000723194534657523,-0.012164993211627007,0.017007797956466675,0.005845030304044485,0.00057241902(...TRUNCATED)
|
2025-08-27
|
End of preview. Expand
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