Excerpt from Yale University Entrance Examinations in Mathematics: 1884 to 1898

1. Prove that if in any triangle one side be greater than another, the angle opposite that side will be greater than the angle opposite the other.

2. Determine how many sides the polygon has, the sum of whose interior angles is equal to the sum of its exterior angles. Explain your method.

3. (a) How does the perpendicular from the center of a circle upon a chord divide the chord and the arc it subtends? (b) How is an angle inscribed in a circle measured? (c) How is each angle between two intersecting chords of a circle measured? (d) What is the locus of the center of a circle whose circumference passes through two given points? Give proof of your answer.

4. Divide a given finite straight line into any given number of equal parts and prove your construction.

5. Given two arcs of circles of 30° whose radii are 1 ft. and 2 ft. respectively; compare their lengths and the areas of the corresponding sectors; and calculate the length of the first and the area of its corresponding sector.

6. If two sides of a triangle be given, what angle will these sides form when its area is a maximum? Give proof.

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