Did David write Psalm 20 as a sort of a prayer for King Saul? Do you have any insight into this Psalm? -- Vedran Pintek (Zagreb)

I do not think there is enough information in this short psalm, or in its heading, to know for sure. Let's look at the text and see what leaps out:

1. To the choirmaster. A Psalm of David. The LORD answer you in the day of trouble! The name of the God of Jacob protect you! 2. May he send you help from the sanctuary, and give you support from Zion! 3. May he remember all your offerings, and regard with favor your burnt sacrifices! Selah 4. May he grant you your heart's desire, and fulfill all your plans! 5. May we shout for joy over your victory, and in the name of our God set up our banners! May the LORD fulfill all your petitions! 6. Now I know that the LORD will help his anointed; he will answer him from his holy heaven with mighty victories by his right hand. 7. Some boast of chariots, and some of horses; but we boast of the name of the LORD our God. 8. They will collapse and fall, but we shall rise and stand upright. 9. Give victory to the king, O LORD; answer us when we call.

* Verse 1 includes the heading. This is part of the Psalm proper in the Hebrew verse-numbering system. Here we see that David was the author/composer. "To the choirmaster" is probably some sort of musical direction, just like selah in verse 3. 
* The victory he is praying for does not yet appear to have taken place, although he is confident and faithful (vv.6-8).
* Victory is linked with faith and sactifice (vv. 2-3). Prayer is of the essence (v.9).
* In Old Testament theology, the Lord's anointed (v. 6) could be a prophet, priest, or a king. We know it is a king from verse 9. Which king? Since the Psalm is written by King David, and it was extemely common for people to refer to themselves in the third person, he is almost certainly praying for victory in his own life.
* He is not referring to a spiritual battle. It appears this victory he seeks is a military one. 
* Verse 7 is the most famous verse in Psalm 20, and one of the most well known in the entire Psalter! What do we put our trust in? Military might, or the Lord? We must choose one or the other.

That is what I see just looking casually at the Psalm. To answer your immediate question, no, it is not a prayer for Saul to be victorious. During much of Saul's kingship and acquaintance with David, the king was pursuing David, trying to kill him. Had David prayed for victory, it could well have led to his own death!

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