Is there an equivalent to background-size: cover and contain for image elements?

I have a site with many pages and different background pictures, and I display them from CSS like:

body.page-8 {
    background: url("../img/pic.jpg") no-repeat scroll center top #000;
    background-size: cover;
}

However, I want to show different (fullscreen) pictures on one page using <img> elements, and I want them to have the same properties as the above background-image: cover; property (the images cant be displayed from CSS, they must be displayed from the HTML document).

Normally I use:

div.mydiv img {
    width: 100%;
}

Or:

div.mydiv img {
    width: auto;
}

to make the picture full and responsive. However the picture shrinks too much (width: 100%) when the screen gets too narrow, and shows the body’s background-color in the bottom screen. The other method, width: auto;, only makes the image full size and does not respond to the screen size.

Is there a way to display the image the same way that background-size: cover does?