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Not an answer, but doesn't really fit in a comment.

Glorfindel ran a script to find and try to upload linked images to Stack Exchange's Imgur repository (those images won't be removed by Imgur). His answer includes a link to a (very long) output of results. Searching for "mathoverflow" jumps to the results for MathOverflow (and also meta.mathoverflow). Many images were successfully uploaded. There are only a handful of uncertainties or failures. Here they are (I "snipped" out the successful replacements):

## mathoverflow.net

### Successful replacements

[snip]

### Uncertainties

| Frequency | URL | Replacement | Post IDs |
|----------:|-----|-------------|----------|
| 1 | https://i.imgur.com/KFkbXCz.gif | Not uploaded but archived as https://web.archive.org/web/20230503101817/https://i.imgur.com/KFkbXCz.gif | 291829 |
| 1 | https://i.imgur.com/p7zUONx.gif | Not uploaded but archived as https://web.archive.org/web/20230503101846/https://i.imgur.com/p7zUONx.gif | 375339 |

### Failures

| Frequency | URL | Failure | Post IDs |
|----------:|-----|---------|----------|
| 1 | https://i.imgur.com/76yeO.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type application/octet-stream returned for https://web.archive.org/web/20150427162752///i.imgur.com/76yeO.png | 39637 |
| 1 | https://i.imgur.com/7ctju.gif | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/Gnpdn.png | Not found in Wayback Machine. | 25897 |
| 1 | https://i.imgur.com/RleJK.png | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/bj5zs.png | Not found in Wayback Machine. | 30136 |
| 1 | https://i.imgur.com/cZKaB.gif | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 24272 |
| 1 | https://i.imgur.com/pChFw.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 56138 |
| 1 | https://i.imgur.com/xAZZb.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 38236 |
| 1 | https://imgur.com/F1x6j.png | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Not found in Wayback Machine. | 35780 |
| 1 | https://imgur.com/drlwRgv | DownloadFailedException: Image not found on imgur image page. | 168300 |
| 1 | https://imgur.com/xjxWV.gif | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Not found in Wayback Machine. | 24272 |

## meta.mathoverflow.net

### Successful replacements

[snip]

### Failures

| Frequency | URL | Failure | Post IDs |
|----------:|-----|---------|----------|
| 1 | https://i.imgur.com/76yeO.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type application/octet-stream returned for https://web.archive.org/web/20150427162752///i.imgur.com/76yeO.png | 5551 |
| 1 | https://i.imgur.com/cZKaB.gif | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | https://i.imgur.com/pChFw.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | https://i.imgur.com/xAZZb.png | com.stackexchange.toolbox.editor.ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Broken image, an error image is served instead. | 5551 |
| 1 | http://imgur.com/ | ProcessImageException: com.stackexchange.toolbox.editor.DownloadFailedException: Invalid Content-Type text/html returned for https://web.archive.org/web/20230509011314/https://imgur.com/ | 773 |

Here is my attempt to make easier links to the MO and meta.MO posts (some of them have several broken/missing images, I just put one link for each post):

1. DONE Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

   I have removed the images and moved the links (and full citation) to the actual article to the top of the question, and pointed out the section and Lemma that the images portrayed.

2. Numerical instability using only Heun's method on a simple PDE

   Uploaded the one available GIF to stack.imgur.com. Left the other two broken links as they are, since no copies are found on the Wayback Machine.

3. When to throw away data

4. What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?

   This gif is on imgur.com. It's too big to upload to stack.imgur.com (about 4.6MB). It seems to be in the Wayback Machine

5. Why does the area function of a parallelogram have a nonintuitive geometric solution?

6. DONE Topological scaling (?)

   The answer included a reference to the paper that the (now lost) image was taken from. I copied the image again and uploaded it.

   (The image has two parts marked (a) and (b), and caption text. I'm not sure if the original included all of that, or maybe just the (b) image. I decided to include all of it.)

7. Which subsets of the plane are similar to all their affine images?

   This gif is on imgur.com. It's too big to upload to stack.imgur.com. It seems to be in the Wayback Machine

8. Symmetries of the Julia sets for $z^2+c$

9. How to formulate such problem mathematically? (line continuation search)

10. The hexagonal property of Pascal's triangle

    Replaced image with tex (as suggested in chat)

11. OK What is the status of the imgur images on MathOverflow? (This question asks about difference between i.imgur.com and i.stack.imgur.com, the links mentioned there are just examples.)

12. OK How to post a file (This answer contains only a link to imgur.com, not to a specific picture.)