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Charles was 2nd in the other two exams.
The total number of points is 20+10+9=39, so X+Y+Z=13. If someone got three first places, he would have got at least 20 points, so 3X≥20 or X>6. In a similar way, if someone had got three third places, he couldn't have more than 9, so 3Z≤9 or Z≤3. Thus, the only possible values for (X,Y,Z) are (7,5,1), (7,4,2), (8,4,1), (8,3,2), and (9,2,1).
* (7,5,1) or (8,3,2) don't allow Bert getting 10 points.
* (7,4,2) or (9,2,1) don't allow Charles getting 9 points.
So, the only possibility is (8,4,1) and then Albert managed two 1sts and a 2nd (in Algebra), Bert had a 1st and two 3rds, and Charles got two 2nds (thus, Geometry) and a 3rd.
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