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5 votes

Solving $\int_0^1(\ln x)^{2024}dx$

If what you said is right and: $$ I(a)=\int_0^1 \ln^a(x) dx\implies I(a)=-aI(a-1) $$ Just becomes a recurrence relationship: $$ I(a)={(-1)^n}n!\binom{a}{n} I(a-n)\implies I(a)=(-1)^a a! I(0) $$ And ...
Masd's user avatar
  • 1,021
4 votes
Accepted

Creative Algebra Net Problem Solving Question

Since you have $4$ linear equations in $3$ unknowns, it's an overdetermined system and, thus, potentially inconsistent, but this answer shows this isn't the case. First, to avoid dealing with ...
John Omielan's user avatar
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3 votes
Accepted

Tetrahedron analogue of a triangle Cevians property

First, a proof for equation $(1)$ will be given. Then, an analogous method will be employed for the case of a tetrahedron. Part 1: Proof for $(1)$ For a triangle $\triangle XYZ$, let $[XYZ]$ denote ...
Euclid's user avatar
  • 1,458
2 votes

Probability Question on Stanford Math Tournament

Alternative approach: There is a smiley face on day 10 if and only if there are an even number of flips in the 9 days between day 1 and day 10. For $~k \in \{0,1,2,3,4\},~$ the probability of having ...
user2661923's user avatar
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2 votes
Accepted

Proving opposite statements in inequalities

Your calculations were correct but you were incorrect that it yields the original inequality iff $ab+bc+ca\le a+b+c$. Let the difference between RHS and LHS be $\Delta\ge0$, then your last line ...
Beans's user avatar
  • 46
2 votes
Accepted

Solving $\int_0^1(\ln x)^{2024}dx$

The function isn't even defined at $0$ so how can it be on the lower bound? Is it implied that it means the limit as $x$ approaches $0$? Yup, exactly. More broadly, if $f$ has a "problem point&...
PrincessEev's user avatar
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1 vote

Solving $\int_0^1(\ln x)^{2024}dx$

You have the Integral with the right value. Your confusion is with the lower bound. Here are 2 ways to confirm that the lower bound is indeed $0$. Change of variables : The Integral of $\ln x$ is $L=x ...
Prem's user avatar
  • 12.3k
1 vote

IMO 2024 PROBLEM 3: Verification required for unconventional solution

At the end, you seem to be claiming that both $a_1, a_3, \ldots$ and $a_2, a_4, \ldots$ are eventually periodic, which is false. For example, consider the sequence 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ... ...
Ted's user avatar
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1 vote

Probability Question on Stanford Math Tournament

You can visualize this with a tree. Draw one node, and color it black (smiley face). Then, draw three nodes out of it, if the original node was black, color two black, otherwise, just color 1. Each ...
Robertmg's user avatar
  • 1,883
1 vote

Probability Question on Stanford Math Tournament

Here is how I would approach it: We have a Markov process. $P_n\pmatrix {\text {smiley}\\\text {no smiley}} =\pmatrix{\frac 23& \frac13\\ \frac 13&\frac 23}P_{n-1}\pmatrix {\text {smiley}\\\...
user317176's user avatar
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1 vote

Probability Question on Stanford Math Tournament

A quick observation of running numbers for $n = 3$ it reveals that The number of times there are flips (by which I mean a smiley face to no smiley face) within these events. Besides the first day, ...
Satish Ramanathan's user avatar
1 vote

Seeking "900 Geometry Problems" Book – Any Leads on Its Whereabouts?

I think you heard it a little wrong. The title is "110 Geometry Problems for the International Mathematical Olympiad" authored by Titu Andreescu & Cosmin Pohoata Check out : https://www....
Prem's user avatar
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1 vote

IMO 2024 p-3,Sequence of Counts - Are Odd or Even Terms Eventually Periodic?

Solution Interesting combinatorics in disguise of sequence. The key is to prove that sufficiently large numbers appear finitely many times in such sequence, and such number of appearance should be ...
Saucitom's user avatar
  • 307

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