Solving an AIME functional equation (how do we guess a solution) @blackpenredpen

Uploaded April 2026 | Updated September 2026, 1 week ago

We will solve a hard functional equation problem from the 2012 American Invitational Mathematics Examination (AIME) . Given f(0)=1, f(x)*f(2x^2)=f(2x^3+x), and f(2)+f(3)=125, we need to find f(5). This is a question asked by one of my viewers!

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Solving an AIME functional equation (how do we guess a solution)

$100 integrals part 2 (EXTREME PASSION!) 100 integrals part 2! Including integration by parts, u-substitution, trig substitution, partial fraction decomposition, reverse power rule for integrals, integral reduction formulas, Weierstrass substitution, improper integrals, special functions, Feynmans technique of integration, integration with power series, the Gaussian Integral, double integrals, Jacobian, polar coordinate, change the order of integration, and more! Get the file & answer key on my Patreon 👉 https://www.patreon.com/posts/102752235 100 integrals (original) 👉 https://youtu.be/dgm4-3-Iv3s Thanks to Nathaniel Barnett for the time stamps! 0:00 100 integrals part 2! Q1, 0:00:46 integral of 2022 Q2, 0:03:03 integral by trig identities and u-sub Q3, 0:08:23 integral by trig sub Q4, 0:12:34 integral by u-sub and integration by parts Q5, 0:17:02 Q6, 0:19:42 Q7, 0:30:07 Q8, 0:33:45 Q9, 0:37:58 Q10, 0:45:16 Q11, 0:48:30 Q12, 0:53:16 Q13, 0:59:06 Q14, 0:58:34 Q15, 1:00:41 Q16, 1:04:56 Q17, 1:07:27 Q18, 1:12:27 Q19, 1:14:42 Q20, 1:18:50 Q21, 1:22:30 integral of x^n*e^x Q22, 1:25:14 integral of (ln(x))^n Q23, 1:27:40 integral of sin^n(x) Q24, 1:37:44 integral of tan^n(x) Q25, 1:42:58 integral of sec^n(x) Q26, 1:52:14 Q27, 1:54:45 Q28, 1:57:45 Q29, 2:01:49 Q30, 2:04:49 Q31, 2:24:38 integral of sec(x) by Weierstrass Substitution Q32, 2:27:17 Q33, 2:31:25 Q34, 2:35:00 Q35, 2:37:20 Q36, 2:42:02 Q37, 2:44:24 Q38, 2:50:18 Q39, 2:51:53 Q40, 2:56:12 Q41, 3:00:21 Q42, 3:04:07 Q43, 3:06:45 Q44, 3:11:58 Q45, 3:19:02 Q46, 3:24:19 Q47, 3:28:36 Q48, 3:31:54 Q49, 3:33:44 Q50, 3:43:40 Q51, 3:49:16 Q52, 3:55:44 Q53, 3:59:50 Q54, 4:03:18 integral of x/(1+e^x) Q55, 4:08:21 integral of e^x/(1+x) Q56, 4:09:21 integral of e^e^x Q57, 4:10:27 integral of ln(ln(x)) Q58, 4:12:40 Q59, 4:14:28 Q60, 4:19:48 integral of the Lambert W function Q61, 4:25:39 improper integral of 1/x^p from 1 to inf Q62, 4:28:52 Q63, 4:29:32 Q64, 4:32:55 Q65, 4:35:29 the Gaussian Integral Q66, 4:50:04 Q67, 4:55:42 Q68, 4:58:36 integral of ln(x) from 0 to 1 Q69, 5:02:56 Q70, 5:10:32 Q71, 5:13:24 integral of ln(x)/(x-1) from 0 to 1 Q72, 5:21:08 integral of (x-1)/ln(x) from 0 to 1 Q73, 5:28:08 integral by symmetry Q74, 5:34:14 integral by even and odd function properties Q75, 5:43:16 Q76, 5:51:17 Q77, 5:52:58 Q78, 6:00:28 the Dirichlet integral, the integral of sin(x)/x from 0 to inf Q79, 6:08:10 Q80, 6:13:28 Q81, 6:37:48 double integral & change the order of integration Q82, 6:45:45 Q83, 6:49:24 Q84, 6:54:00 Q85, 6:57:25 Q86, 7:03:50 Q87, 7:07:33 Q88, 7:11:11 Q89, 7:17:27 Q90, 7:21:37 Q91, 7:28:52 Q92, 7:37:47 Q93, 7:55:15 Q94, 8:07:00 Q95, 8:10:24 Q96, 8:21:05 Q97, 8:22:55 Q98, 8:26: 30 Q99, 8:29: 59 Q100, 8:33:33 Q101, 8:38:32 the BIG integral from the Berkeley Math Tournament$

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Solving an AIME functional equation (how do we "guess" a solution)