To repair the problem we ordered a new tall cone chimney flashing for a sloped roof with collar and installed it appropriately. First we installed ice and water shield around the entire chimney area, then installed the chimney flashing and shingles over the flashing. The water will now run down the shingles onto the flashing and further from the flashing onto the shingles below. The appropriate overlap of items is installed in a common sense way consistent with the flow of water given gravity. The collar is installed and prevents an wter hitting the chimney to run down the collar and onto the shingles. The collar area is then caulked only where there is potential for the collar to leak. The tall cone is caulked only where the two parts are welded. This is an appropriate installation of a chimney flashing. If you are having a chimney like this installed be sure to consult a roofing professional as it is possible that the person installing the chimney is unaware of proper installation.