I'm glad you mentioned the story of The Monk and the Murderer. It's a great example here of how God can answer prayer in unexpected ways. It's also a reminder that we should always pray for others, even if we don't know them personally.

GOD RESPECTS OUR FREE WILL AND DOESN'T INTERFERE WITH OUR DECISIONS - Free will is certainly one aspect of Divine theology, but so is Divine Providence, where God deems it necessary to intervene in human (individual) affairs.

A veritable explanation of both concepts can be illustrated by the normal classroom scene. The professor gives out assignments as requirements for passing the class. The student has the free will to do the assignments or not. He can choose to be diligent or irresponsive. The professor has no say in this.

But the Professor can decide on which grade to give by intervening in the educational process! The decisions of the student have consequences that are enforced by the schoolmaster.

By analogy, each man has the freedom to live according to righteous principles. And God has the Divine right to hold the person to account for his decisions.

Now, to give further analogy, if the student, along the way, needs help with the assignments, he can go to the professor for further instruction! He has the freedom to ask, or to sweat it out on his own.

Such is the power of prayer! Man's free will is not violated by Divine assistance brought on by prayer requests. Professors like to teach, and God likes to help his children. Providence flow from a caring God.

Further, the professor knows ahead of time that a diligent student will pass, and a wayward one will fail. Such preknowledge does not violate free will, nor legitimatize a sense of fatalism or predestination.

Study on...pray on when necessary...ask for intervention to tough problems. He would want you to. (Biblically: Ask, seek, knock)

Is praying futile given that God respects our Free Will and is omniscient?

The short answer is no.

Sometimes one has to be tenacious in prayer to get a response. Like a good Father, God desires to listen to his children’s supplications, especially when it comes to the health and welfare of individuals.

Even Jesus taught us how to pray. He taught the Apostles to pray the Our Father!

God never interferes with our free will. I am convinced that that is why most of us do not win the lottery. God knows what is best for us and winning a million dollars would be detrimental to our salvation.

I am sure God hears our prayers. However sometimes they are answered in a way we may never know. Sometimes God is more apt to answer our prayers for others than for ourselves.

The story of The Monk and the Murderer is a great example of God answering prayer on behalf of another person. Just a couple days before his execution, a condemned inmate James Malicoat wrote to his Religious friend: “You will see, prayer is never in vain.“

If Alice asks Bob for a cup of sugar, and Bob hands her a cup of sugar, how does that violate Alice's free will? The free will is in asking for help.

Scripture speaks, at various times, of people casting lots. The idea is to surrender a choice to a process which God is assumed to control, for example, rolling dice to select between several candidates for a leadership position. Now, if God causes the dice to come up a certain number... has that violated free will? First, it was your will (or the will of the group engaging in the process) to turn over the decision in the first place, as well as whether to honor the result. However, God has chosen an answer, thus falsifying your belief that God is not able to do so. Consider also Matthew 10:19 / Mark 13:11, wherein God promises to give words to those in need. The idea that providing something is necessarily somehow a violation of the recipient's free will is not only nonsense, but is clearly contrary to Scripture.

(Note: I don't recommend this as a test-taking strategy for the reasons cited in DJ's answer!)

Consider, again, your claim:

God cannot force me to go to the store, buy bread, and bring it home.

...but God clearly has the power to do so; the question is whether or not such exercise violates your free will. Already, if you've asked for bread, this seems questionable. But what if you're shopping for something else and bread accidentally falls into your cart? What if a "sudden muscle spasm" caused it to be knocked into your cart? What if you were worrying about your upcoming performance review and absentmindedly drove to the grocery store? What if a stranger walks up to you and says "say, I just bought this loaf of bread, but I'm not going to be able to eat it; would you like it?" At what point, especially if you asked for bread, is your free will actually violated?

You also said (in a comment):

If I am reminded by something to study, its because I saw it and processed it in my own brain, God can't remind me without breaking my free will.

This is clearly nonsense. Let's go back to bread. You need bread, but you're walking home, past the bakery, without it. Suddenly, a pretty leaf blows in front of you. You turn your head to watch, because it's pretty. It drifts in front of the bakery's sign. "That's right," you say to yourself, "I need to buy bread". Clearly, God could have caused the leaf to attract your attention. Did doing so violate your free will?

No, it did not... because if it did, if your "free will" is utterly dictated by such vagaries as most people would label "chance", then you don't have 'free will'. This is a critical philosophical point; if free will (volition) exists, it is either God's alone, or it isn't unique. There is no way for you to have volition which precludes external influences, because in order to have that, you would have to deny volition to everyone else. Now, a strict materialist would argue that free will is an illusion, which I don't believe (and I don't think you do either), but is at least a logically consistent view. The idea that your will cannot be influenced by outside factors, however, is only logically valid if your mind is the only thing that genuinely exists.

Let's say you pray for good grades. While walking up the stairs, you drop your book and it "coincidentally" lands open to a section you realize will be on the test, but you don't know well. Yes, you "saw it and processed it in [your] own brain", but the book opening to that particular page was an Act of God. Yet, as shown above, that act did not violate your free will.

The matter of "free will" for humans is invariably looked at from the (limited) point of view of the humans who already believe they have free will. The matter of God's free will can only be viewed from the point of view of God, who has it. Correctly, you also add that God is omniscient - he has infinite knowledge, knowing the end from the beginning. That is why one of his titles, Alpha and Omega, means "The Beginning and the End" (Revelation 21: & 22:13). That alone should give us pause for thought before making any claims about humans having "free will" in the sense generally understood. Praying for things, or events to occur, is never futile when we pray for God’s omniscient Free Will to be the answer, over and above our own desires to exercise our supposed ‘free will’.

But this question is about prayers of petition to the Alpha and the Omega. Is it futile to ask God for things, or for events to happen, in light of this matter of free will? Should we not just stick to prayers of worship?

This answer gives two biblical examples (no illustrations needed). They both have to do with people who were already part of God's 'family' on earth. They already had faith in God and knew to obey God as their Sovereign. That is actually the prerequisite for desiring to know God's will so as to do it.

The first is in 2 Kings 2:23-25, but a biblically literate reader would already know the covenant curse God had promised to bring on any of his people who remained hostile to him. The pertinent bit reads, "I will send wild animals against you, and they will rob you of your children..." (Leviticus 26:21-22). This account shows the prophet Elisha being mobbed by teenagers from the nearby town of Bethel that was the cult centre of apostasy of the northern kings. Elisha did no more that petition God to bring down a curse on those young hooligans. He did not ask for bears to come out of the woods to maul 42 of them. But that is what God chose to do, in answer to Elisha's prayer of faith for deliverance from a nasty situation.

The second is about events in 1 Samuel 9:1-20, well put in this quote:

“The enormous task of the believer is to rationalise the seeming caprice of random circumstances, and to trace purpose in what all the world would otherwise call the fortuitous flow of atoms. To rise above the science of existence to the spirituality of it is no easy task. And sometimes we get it wrong.

But the God who ordained the work of redemption on its gargantuan scale is also the providential Lord of every detail in His children’s lives, even for example, down into such domesticities as Saul, son of Kish, needing to go and search at one crucial time and in one crucial direction, for his father’s lost donkeys!” 1 Samuel 9:1-20 Pastoral letter by Rev. Guy R. Finnie, January 1997

Now, the intriguing thing about this quote is that God's people had asked the prophet Samuel to give them a human king, as all the nations had. Samuel was grieved, knowing that such a request of God was a rejection of God, who alone was to be their King. The will of the people was not glorifying to God. Yet when Samuel prayed to God about the peoples' petition, God told him, "Listen to them and give them a king."

The anointing of Israel's first king, Saul, required his father's donkeys to wander off and get lost. Of course, God being sovereign, he could have chosen to do that another way. But the point is that God determined what would happen, when, and to whom, even though Saul had no notion of ever being king. God caused his will to be done, working through the (wrong) will of the nation. Does that not give food for thought?

It certainly puts "human free will" in a different light! Yet, going back to the matter of prayer (which is what this question is all about) Samuel prayed to God, seeking God's will to be done in that grievous situation. One godly man prayed, and the course of the nation was changed, all triggered by some donkeys wandering off and getting lost.

Perhaps the main point of those examples is not the truism, "Be careful what you pray for", but "Pray for things and situations where God is expected to sovereignly cause his will to be done, howsoever he chooses." Praying for things, or events to occur, is never futile when we pray for God’s omniscient Free Will to be the answer, over and above our own desires to exercise our supposed ‘free will’.

There are a few points in favor of "petitioning the Lord" in our prayers:

1. For every commandment given, there are blessings conditional upon obedience to that law (see Deuteronomy 11:26-27, for example). Even if God were to give you a completely arbitrary command, deciding to keep that law would result in you being blessed. As you refer to, Jesus Christ commands us to pray in a way that clearly includes praying for our specific needs:

9 After this manner therefore pray ye: Our Father which art in heaven, Hallowed be thy name.

10 Thy kingdom come. Thy will be done in earth, as it is in heaven.

11 Give us this day our daily bread.

12 And forgive us our debts, as we forgive our debtors.

13 And lead us not into temptation, but deliver us from evil: For thine is the kingdom, and the power, and the glory, for ever. Amen.

If we fulfill this commandment, we can be assured of being blessed because of it.

2. Prayer is an opportunity to align our will with God's. Obviously, we cannot "change the mind" of one whose "ways are higher than [our] ways, and [His] thoughts than [our] thoughts," but through the process of counseling with the Lord in our desires, He can change our hearts to trust in His potentially different alternatives.

3. Omniscience does not necessarily mean that the future is predetermined. Just because God knows the future does not necessarily mean that he knows exactly what grade you will get on your test, for instance--it may mean that he knows if you stay up playing video games tonight you'll get a D, and if you go to sleep on time but you studied for your history class instead, you'll get a C, and if you study for math and then wake up early to cram history you'll get B+'s in both, etc. What exactly the mind of an all-knowing God is like is tricky to imagine, and I generally prefer to stay out of that (for now, at least), but it's worth at least remembering that our ability, as humans, to understand the ideas of time and space on the plane God operates on are limited, at best, and that our understanding of free will, omniscience, foreknowledge, and destination, might end up looking like a child trying to understand quantum mechanics in the next life.

(My answer has gotten so long and unwieldy that I have no idea if it's even coherent anymore. I hope it will be useful to you in parts, even if it doesn't hang together as a whole.)

You really have two questions here.

...if I ask for a good grade on my exam, God cannot force me to study, or choose my answers. How can God help me without breaking my free will?

1. How can God make good things happen for me without forcing me to act in certain ways?

Furthermore, He already knows the future and exactly what grade I will get (which is a part of His perfect plan), so why should I pray for a certain grade?

2. Why should we expect that our prayers will change anything? Isn't God just going to give us whatever He wants to give us anyway?

I will attempt to answer the second question first, then come back around to the first one.

Why should we expect that our prayers will change anything?

First, let's discuss how free will works in "ordinary" actions, and then we'll move on to prayer.

Free will in "ordinary" actions

Will and outcomes

Suppose I pick up a salt shaker and move it to the other side of the table. Nobody made me do it. It was my own free will. I can honestly say, "I chose to move the salt shaker."

On the other hand, suppose that I went to pick up the salt shaker and found that I couldn't, because some prankster had glued it to the table. Would that mean that the prankster had taken away my free will? Strictly speaking, the answer is no. I still made my free choice to try to move the salt shaker; it was only my goal that was frustrated. Free will does not include a guarantee of desired outcomes.

But even in this case, my free will had some effect on the world around me, even if it wasn't the effect I wanted. The salt shaker has new fingerprints on it; the glue holding it to the table is somewhat looser than it was before; my arm is in a different position, and the muscles in that arm are now more tired. My own thoughts, too, are different because of the choice I made. So my choice was far from meaningless.

Human will within God's will

God did not have to create a universe where my choices had any effect whatsoever, let alone the effects that I intended. But He did. God created and sustains a universe in which salt shakers can (in most cases) be moved by human muscles. In theory, He could at any moment alter the universe so that salt shakers can only be moved by His express command, and the thought of trying to move a salt shaker never even crosses our minds. But He chooses not to. He permits our free-will choices to have an actual effect on the world around us. To use a phrase I once heard (which was paraphrasing St. Thomas Aquinas), "God grants us the dignity of being causes"—that is, the "cause" in "cause-and-effect".

So even when I am doing something "by myself", I am only able to do it because God is cooperating with me. And yet this does not take away my free will. It is a partnership.

Analogy to prayer

Prayer is not that different from trying to move that salt shaker. When we pray, we are taking action in the hope of causing some particular effect. Compared to picking up a salt shaker, the connection between cause and effect is far more mysterious and hidden from us. But the basic logic is the same: we do something, and then, through God's cooperation, the thing we wanted might happen.

God didn't have to set things up this way. He could have created a universe where prayer has no effect whatsoever. But instead he gave us the gift of prayer, and encouraged us to use it. He grants us the dignity of being causes.

"But there's a difference," you might be saying. "Answering a prayer doesn't follow a general rule like the laws of physics. God is personally making a decision as to what will happen. It's his will, not mine."

Suppose you take a test, and afterwards, you petition your professor to give you partial credit on an answer you got half-right. The professor does. Does that mean your will didn't matter? It's true that the professor ultimately made the choice, but it wouldn't have happened if you hadn't chosen to petition him.

"That's still different," you might be saying. "My professor is a human being. Humans can change their mind. God doesn't change. So whatever his will already is, that's what will happen."

Part of God's will is that our choices should have effects. That includes choosing to pray.

Suppose your professor had a private rule that anyone who comes to him asking for partial credit, and who can explain their reasoning, gets that credit. So your petition didn't really change his mind; his mind was already set. This is analogous to how God can answer our prayers without Himself changing.

On God's foreknowledge

Furthermore, He already knows the future and exactly what grade I will get (which is a part of His perfect plan), so why should I pray for a certain grade?

You could just as easily ask, "He already knows exactly what grade I will get, so why should I study?" In both cases, as I hope I've already made clear, you are choosing to take an action with the aim of improving your grade.

You are basically arguing in favor of predestination: that the future is set in stone, and our own actions have no impact on it whatsoever. But this is wrongheaded. The future is a result of our actions. God is a witness to that future, because he is outside of time, but that does not mean he has taken away our free will and forced the future to come out that way.

To quote C. S. Lewis:

[God] does not "foresee" you doing things tomorrow; He simply sees you doing them: because, though tomorrow is not yet there for you, it is for Him. You never supposed that your actions at this moment were any less free because God knows what you are doing. Well, He knows your tomorrow's actions in just the same way—because He is already in tomorrow and can simply watch you. In a sense, He does not know your action till you have done it: but then the moment at which you have done it is already "Now" for Him.

^{—Mere Christianity by C. S. Lewis, Book IV, Chapter 3: "Time and Beyond Time"}

I also dispute the idea that it is "God's perfect plan" that you should score, say, 83% on your exam, as opposed to 82% or 84%. God allowing something to happen is very different from God causing it to happen, or even being glad it happened. When we start to believe that every little thing and every little detail is "part of God's plan", it leads us down a weird rabbit hole to a twisted, trying-to-be-God's-robot form of Christianity in which we earnestly attempt to discern if the Almighty wants us to put ketchup or mustard on this particular hot dog. God wants us to be good and to draw near to him in love, not to fill out his checklist of Every Event That I Have Planned For This Specific Day In History (which he refuses to let us see).

How can God make good things happen for me without forcing me to act in certain ways?

...if I ask for a good grade on my exam, God cannot force me to study, or choose my answers. How can God help me without breaking my free will?

Your performance on the exam is certainly dependent on your own choices and efforts, and God is (most likely) not going to whisper the answers in your ear if you don't study. But your performance can never be entirely dependent on those efforts. You don't have perfect control over reality, or even your own body. There is more than enough room for God to work. In fact, you need God's help to succeed, in the same way that a fish needs water to swim.

However, that doesn't mean that you will notice Him helping.

In C. S. Lewis's The Screwtape Letters, the senior demon Screwtape explains things this way to his "nephew" Wormwood, who has been assigned to tempt a particular human:

Don't forget to use the "heads I win, tails you lose" argument. If the thing he prays for doesn't happen, then that is one more proof that petitionary prayers don't work; if it does happen, he will, of course, be able to see some of the physical causes which led up to it, and "therefore it would have happened anyway", and thus granted prayer becomes just as good a proof as a denied one that prayers are ineffective.

You, being a spirit, will find it difficult to understand how he gets into this confusion. But you must remember that he takes Time for an ultimate reality. [...] If you tried to explain to him that men's prayers today are one of the innumerable co-ordinates with which the Enemy [i.e. God] harmonizes the weather of tomorrow, he would reply that then the Enemy always knew men were going to make those prayers and, if so, they did not pray freely but were predestined to do so. And he would add that the weather on a given day can be traced back through its causes to the original creation of matter itself—so that the whole thing, both on the human and on the material side, is given "from the word go". What he ought to say, of course, is obvious to us; that the problem of adapting the particular weather to the particular prayers is merely the appearance, at two points in his temporal mode of perception, of the problem of adapting the whole spiritual universe to the whole corporeal universe; that creation in its entirety operates at every point of space and time, or rather that their kind of consciousness forces them to encounter the whole, self-consistent creative act as a series of successive events.

^{—The Screwtape Letters by C. S. Lewis, Letter XXVII}

In other words, God crafts the universe, including all of time, partly in response to human prayers. From our limited perspective, the effect may come "before" the cause. From His perspective, it is perhaps like one of those modern sculptures where taut wires hold the other components in a particular arrangement, with our prayers being the wires.

Again, God's ability to witness our actions throughout time does not deprive us of free will, as Screwtape also discusses:

Why that creative act leaves room for their free will [i.e. why God chooses to do things the way he does] is the problem of problems, the secret behind the Enemy's nonsense about "Love". How it does so is no problem at all; for the Enemy does not foresee the humans making their free contributions in a future, but sees them doing so in His unbounded Now. And obviously to watch a man doing something is not to make him do it.

^{—The Screwtape Letters by C. S. Lewis, Letter XXVII}

"But how do I know that God didn't just manipulate me into praying?" you might ask. "That would violate my free will."

What do you mean by "manipulate"? If you're saying that God presented you with a set of circumstances where you were 100% guaranteed to choose to pray...well, the whole idea of free will is that there is no such thing as a 100% guaranteed choice. If a man puts a gun to your head and says "Give me all your money", you still have the option of refusing, even if the consequences would be unpleasant.

Technically, the existence or non-existence of free will can never be proven in the way that we are accustomed to proving things today. We can't run any experiments; we can't turn back the clock and see if we can make a previous choice differently. Even if we had a time machine, we would know that we'd made that choice before, so we wouldn't be reproducing the original scenario.

But we all act like we have free will. We all live our lives as though our decisions are our own. Even the determinists. In other words, we take it on faith.

Aside from all that...

A word of warning. The point of a prayer of petition is not to get God to grant our wishes like a genie. We should petition God to give us what we need to live the way he wants us to live. And you could live a life pleasing to God without ever taking an exam, let alone passing one.

II. PRAYER OF PETITION

[...]

2631    The first movement of the prayer of petition is asking forgiveness, like the tax collector in the parable: "God, be merciful to me a sinner!"¹⁰⁵ It is a prerequisite for righteous and pure prayer. A trusting humility brings us back into the light of communion between the Father and his Son Jesus Christ and with one another, so that "we receive from him whatever we ask."¹⁰⁶ Asking forgiveness is the prerequisite for both the Eucharistic liturgy and personal prayer.

2632    Christian petition is centered on the desire and search for the Kingdom to come, in keeping with the teaching of Christ.¹⁰⁷ There is a hierarchy in these petitions: we pray first for the Kingdom, then for what is necessary to welcome it and cooperate with its coming. This collaboration with the mission of Christ and the Holy Spirit, which is now that of the Church, is the object of the prayer of the apostolic community.¹⁰⁸ It is the prayer of Paul, the apostle par excellence, which reveals to us how the divine solicitude for all the churches ought to inspire Christian prayer.¹⁰⁹ By prayer every baptized person works for the coming of the Kingdom.

2633    When we share in God's saving love, we understand that every need can become the object of petition. Christ, who assumed all things in order to redeem all things, is glorified by what we ask the Father in his name.¹¹⁰ It is with this confidence that St. James and St. Paul exhort us to pray at all times.¹¹¹

¹⁰⁵ Lk 18:13.
¹⁰⁶ 1 Jn 3:22; cf. 1:7-2:2.
¹⁰⁷ Cf. Mt 6:10, 33; Lk 11:2, 13.
¹⁰⁸ Cf. Acts 6:6; 13:3.
¹⁰⁹ Rom 10:1; Eph 1:16-23; Phil 1:9-11; Col 1:3-6; 4:3-4, 12.
¹¹⁰ Cf. Jn 14:13.
¹¹¹ Cf. Jas 1:5-8; Eph 5:20; Phil 4:6-7; Col 3:16-17; 1 Thess 5:17-18.

—Catechism of the Catholic Church, Second Edition

Though, to give a word of warning to my word of warning, that doesn't mean you can't pray for success in your exams. Screwtape again:

On the seemingly pious ground that "praise and communion with God is the true prayer", humans can often be lured into direct disobedience to the Enemy who (in His usual flat, commonplace, uninteresting way) has definitely told them to pray for their daily bread and the recovery of their sick. You will, of course, conceal from him the fact that the prayer for daily bread, interpreted in a "spiritual sense", is really just as crudely petitionary as it is in any other sense.

^{—The Screwtape Letters by C. S. Lewis, Letter XXVII}

Here's an example of how God can bring something about that is part of his plan, without violating anyone's free will.

Let's take a guy named Bob. God, being omniscient, knows everything about Bob, including what Bob would would freely choose to do under any set of circumstances. This type of knowledge is referred to as Middle Knowledge.

Specifically, in this case, God knows that on a particular Sunday, if it rains, then Bob will accept his friend's invitation to go to church that day, where God knows he will be so moved by the sermon that he converts and is saved. On the other hand, if it stays sunny, Bob will go to see the footy instead, missing the chance, and will end up dying an atheist, unsaved. Now God can decide possibly the biggest part of anyone's life - whether they are saved or not - simply by making it rain or stay sunny on a particular day.

So while God may not directly alter our free will, he can certainly alter the world around us according to His will and plan, shaping the outcome as he sees fit without forcing anyone to do anything. This view of divine providence is called Molinism.

As to the other part of your question, why pray if everything is already planned: well, God took our prayers into account when He formed His plan. It may be that before creation, He knew that Bob's friend would pray for him and thus decided to save Bob; if Bob's friend had not prayed for Him, God would have foreknown that too and possibly left Bob in his atheism.

Edit: This is no way hard determinism, which posits that there is a causal link between the circumstances one is in/the various factors, and the actions one takes, ie your decision is forced by one thing or another. I am saying no such thing, quite the opposite. If knowledge of future actions is equivalent to hard determinism, this leads one very quickly to Open Theism. Knowing that a person will do something, or would do something in other circumstances, in no way causes that to necessarily be. There is no causal link.

There are lots of complexities around prayer, but this is not one of them.

It's true that many of the things involved in you getting bread to eat involve your own decisions, but many do not. For example the shops around you may have run out of bread or there may be a worldwide bread shortage. Those things are not in your control, but they are in God's. You are essentially praying about those things outside your control. God can arrange for there to be bread in the stores when you go to buy.

Now if you were to pray that you get your daily bread, but you never go to a bread shop or try to buy bread; or if you were to pray for good grades but never study or attend lectures or even turn up to the exam, then that is showing contempt for God. There is no paradox involved when you don't get them.

As for God's foreknowledge, yes he knows the grades you will get. But he also knows if you will pray or not. And he is quite capable of taking that into account.