Add Product to Cart with Price Change

If you want to add a product to the cart after change its price, then you have to write an Observer that listens the checkout_cart_product_add_after or checkout_cart_update_items_after event. The code is same except checkout_cart_product_add_after is called for only one item and checkout_cart_update_items_after is called for all items in the cart.

/**
 * @param Varien_Event_Observer $observer
 */
public function myDiscount(Varien_Event_Observer $observer)
{
    $item = $observer->getQuoteItem();
    if ($item->getParentItem()) {
        $item = $item->getParentItem();
    }

    //discount 20% off
    $discount = 0.20; 

    // Check if the discount isn't applied over and over while refreshing
    $specialPrice = $item->getOriginalPrice() - ($item->getOriginalPrice() * $discount);

    if ($specialPrice > 0) {
        $item->setCustomPrice($specialPrice);
        $item->setOriginalCustomPrice($specialPrice);
        $item->getProduct()->setIsSuperMode(true);
    }
}