When you planted the azaleas, did you loosen the rootball well so roots could reach out and begin to establish into the surrounding soil? If they are planted from containers with the rootball intact in its pot shape, it can be very hard to wet the potting medium again once it dries out...the soil immediately around the plant can be moist but not the roots themselves. You don't say what form of fertilizer you are using 'once a month'. Azaleas and rhododendrons have low nutritional needs compared to many shrubs, it's easy to burn their fine surface roots, or damage them scratching in a granular type fertilizer.

First, if it was not planted properly the root ball may be totally dry and root bound. Container plants at garden centers have been force fed and get root balls with roots circling around. If this root ball ever gets dry, it will not absorb water unless it is dug up and soaked in a bucket of water. If the circling roots are not cut and opened up, they will eventually girdle the plant. If the root ball was not dry and it was planted too deeply or in an area with poor drainage, it will develop root rot. This is usually fatal. Lace bug is easy to spot on the underside of the leaves. It affects plants in too much sun. Rarefind Nursery has an excellent fact sheet on planting: Here is a link that might be useful: Tree & Shrub Planting Guide

Thanks all for your suggestions. These three azaleas are on a lake shore, but they're probably about a foot above the water line (there's a hard edge to the shore at this point). There's a large willow tree just behind them. But I stuck my finger in the soil at the base of all three, and the soil seemed to be evenly moist, not sodden, around all three. It has been a dry summer here, however, so it's possible they've gotten less moisture than they've been accustomed to. On the other hand, there is an irrigation system for the adjacent lawn which may hit the shrubs (some or all of them? I've never seen it run). The irrigation was, until recently, running daily. So possibly the affected one is getting a different water level on the foliage than the others. They also are in the sun only from about 10am-12:15pm this time of year - it's hard shade from a building in the morning and dappled shade from the tree for the rest of afternoon. Is that level of sun enough to scald? Thanks again for all your thoughts on this! It's reassuring that nobody seems to think this looks like an insect or fungal problem.

Brown tips are most often a sign of over watering. Does container have a drainage hole ? Are you letting top of soil dry before watering ? Best not to fertlize until leaf tip problems are resolved. Judging from your pic....the soil looks wet now.