I've looked around the net and the problem is that something doesn't support my maximum resolution. Either it's the graphics card or the HDMI cable. I bought this PC only 2 years ago. I'm using an AMD A-10 with AMD HD 8670d graphics.

Your iGPU is okay (see full specs here - https://www.amd.com/en-us/products/graphics/desktop/oem/8670#), but you need HDMI cable at least version 1.3 - https://en.wikipedia.org/wiki/HDMI#Version_comparison

Yes, older HDMI cables max out at 1920x1200, as does single-link DVI and DisplayPort 1.1 and older. You either need a more recent HDMI cable as triock suggests or DisplayPort 1.2 cable to go to higher resolutions. I've got a 2560x1600 display which I've successfully connected to the computer using a DisplayPort 1.2 cable as well as dual-link DVI.

If you have to buy a new cable I'd recommend going the DisplayPort 1.2 route to keep the HDMI port freed up in case you ever want to have it connected to a television simultaneously as TVs generally have HDMI on them but not DisplayPort.

Hope this helps.

My integrated graphics doesn't have Display Port on it.

Um ... what's the difference between these HDMI cables, I suppose I should look at the text printed down the side of them?

Wikipedia: HDMI Versions
Audioholics: Understanding the Different HDMI Versions (1.0 to 2.0)

Basically, you get a higher bandwidth/bitrate with more recent versions, enabling higher resolutions at the same framerate, or higher framerates at the same resolution, more audio channels, as well as extra features.

Hey lads, I ended up buying a $50 USD graphics card, so I can get the high resolution now. The 2560x1440 resolution works fine on Linux even when plugged into the AMD A-10 integrated graphics, but it doesn't work in Windows! So stupid. I blame AMD. I had all the drivers and it still wouldn't work. Now I have to use a stupid graphics card that is actually worse than my A-10. Oh well.