The German Math Olympiad Problem That 99% of Students Get Wrong!

Added: 2026-05-10

[00:00:01]Welcome to Smart Math. Today, I am solving this math Olympic problem in a very unique way.

[00:00:06]So, our equation is 3 ^ m - 4 ^ m, which is equals to square root of 12 ^ m - 16 ^ m.

[00:00:15]So, as our equation is 3 to the power of m - 4 to the power of m, that is equals to square root of 12 ^ m - 16 ^ m, whole square root.

[00:00:28]So, now I am canceling out square root. So, I'm taking both side square.

[00:00:34]Both side square, the square and square root canceled out. So, our equation will be 3 3 ^ m >> [clears throat] >> 4 ^ m whole square, that is equals to 12 ^ m - 16 ^ m.

[00:00:52]So, where as we know that 12 is equals to 3 into 4 ^ m, and 16 is equals to 4 into 4 ^ m, that is 4 squared.

[00:01:07]So, replace this term into 3 ^ m 3 to the power of m - 4 to the power of m whole square, that is equals to 12 is 3 into 4 whole power m, and 16 is 4 squared whole power m.

[00:01:27]So, this means that 3 power m, 4 power m whole square, that is equals to uh 3 into 4 is 3 into 4 is 3 ^ m into 4 ^ m 4 power m whole square.

[00:01:51]So, this is the equation of this is a minus b a minus b whole square.

[00:01:59]So, a minus b whole square that is equal to a minus b power one into a minus b power one.

[00:02:08]So, I'm wrote that 3 m 3 m minus 4 m into 3 m minus 4 m which is equal to 3 to the power of 3 m and power of m 4 to the power of m minus 4 to the power of m whole square.

[00:02:35]So, our equation will be So, now I am let 3 m that is equal a and 4 m that is equal b to simplify more.

[00:02:51]So, I'm replacing 3 m is a minus 4 m is b.

[00:02:58]So, a minus b again.

[00:03:02]So, this term is 3 is a and b into b minus the four b's four square.

[00:03:13]So, this is plus goes opposite sign negative and positive.

[00:03:18]So, this is a called a minus b into a minus b minus ab minus ab into and plus b square that is equal to zero.

[00:03:40]So, a minus b into a minus b Now, I am taking minus b common minus b common means they give a minus b remainder.

[00:03:53]So, that is equal to zero.

[00:03:56]Now, I am taking A minus B A minus B both side.

[00:04:00]A minus B. So, this time I'm I'm common A minus B.

[00:04:05]A minus B common.

[00:04:07]So, they gave me A minus B into minus minus B bracket is equals to zero.

[00:04:18]So, our equation is A minus B into A minus B minus B and is equals to zero.

[00:04:27]So, now A minus B into A minus 2 B. That is equals to zero.

[00:04:38]This means that A minus B is equals to zero or A minus 2 B. That is equals to zero.

[00:04:46]This term minus here goes up same positive.

[00:04:50]A is equals to B or A is equals to 2 B. Minus here goes positive. So, this is called case one and this is called case number two.

[00:05:07]So, now case number one solved.

[00:05:09]So, now case number one is A is equals to B.

[00:05:13]So, I am recall A is 3 to power of M and B is 4 to the power of M.

[00:05:23]This means that A is 3 M which is equals to B is 4 M.

[00:05:30]So, now I am taking both side divided by by 4 M divided by by 4 M.

[00:05:35]So, this term cancel out into one and this term is 3 M divided by by 4 M. That is equals to 4.

[00:05:44]So, power is will be same. So, I am taking whole power 3 to the power of a 3 divided by by 4 whole power m that is equals to 1.

[00:05:55]So, now I'm using that something have zero power that is equals to 1.

[00:06:02]This means that 1 is equals to if I'm taking 3 divided by 4 whole power zero which is equals to 1.

[00:06:10]So, now I'm I'm replacing 3 divided by by 4 whole power m that is equals to 3 divided by by 4 whole power zero.

[00:06:22]So, this time this means that if I uh I'm if I I'm know that a to the power of b and a to the power of c that means that base are same means power is also same.

[00:06:35]This include that b is also equal to c.

[00:06:39]So, this is also same.

[00:06:41]So, power will be also same. This means m is equals to zero.

[00:06:47]So, value of m is zero. Now, case number two.

[00:06:52]So, case number two is a is equals to 2b. So, we got a is 3m into two is and b is 4 power of m. So, 3 to the power of m that is equals to 4 uh two uh four if I'm taking it both side divided by by 4m.

[00:07:15]Both side divided by by 4m. So, 4m numerator denominator cancel. And 3 divided by 4 I I'm taking whole power m.

[00:07:25]That is equals to 2.

[00:07:28]So, to solve uh more I'm taking log both side.

[00:07:33]So, log both side mean uh we using formula whether log of a to the power of b that is equals to B log of A.

[00:07:44]So, this means that M log of 3 / 4 M log of 3 / 4 that is equals to log of 2.

[00:07:56]So, now I am both side divide by log of 3 / 4 log of 3 / 4 this time this time cancel.

[00:08:09]This means that M is equals to log of 2 / by log of 3 / 4.

[00:08:21]So, as we know that log of A / B B, that is log of A minus log of B.

[00:08:38]So, this conclude that M, that is equals to log of 2 / by log of 3 minus log of 4.

[00:08:53]That's the value of second value of M.

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