Let me ask you a question: If you were facing a serious problem in your own life -- a dangerous illness, for example, or an unexpected crisis in your family -- wouldn't you want people to pray for you? Of course you would. The Bible says, "Pray for each other so that you may be healed" (James 5:16).
Does this mean God always answers our prayers exactly the way we wish He would, or exactly when we wish He would? No, not necessarily. But the Bible still tells us to pray, because God is able to use our prayers in ways we may never understand this side of heaven. If Jesus, the divine Son of God, needed to pray about every situation He faced, don't we need to pray even more? The Bible tells us to pray about everything: "Pray in the Spirit on all occasions with all kinds of prayers and requests" (Ephesians 6:18).
You say that sometimes a problem gets solved "without divine intervention" -- but how do you know God wasn't at work? What seemed to be a normal, everyday solution actually may have been His way of solving it. God is at work even in the ordinary things of life. Never forget: God loves us, and He has promised to hear the prayers of His people.
Recently a man wrote me telling how Christ had freed him from a terrible drug habit. "People were praying for me," he said, "and God answered their prayers." Ask God to bless others through your prayers.