1. What is the slope of a line perpendicular to y = (2/3)x + 5?

2. The Gauss-Bonnet theorem: for a closed orientable surface, ∫∫K dA = 2πχ where χ is Euler characteristic. For a torus:

3. A sphere fits perfectly inside a cone (base circle=sphere great circle, apex touches sphere). If r=5, what is the height of cone?

4. Descartes' Circle Theorem: four mutually tangent circles with curvatures k₁,k₂,k₃,k₄ satisfy (k₁+k₂+k₃+k₄)²=2(k₁²+k₂²+k₃²+k₄²). For three unit circles tangent to each other (k=1 each), the inner circle has curvature:

5. The inscribed angle theorem: an inscribed angle is __ of the central angle.

6. The area of a cyclic quadrilateral with consecutive sides 3,4,5,6 is (Brahmagupta's formula):

7. What is the circumference of a circle with area 49π?

8. An angle bisector divides a 70° angle into two equal parts. Each part is:

9. A cube has vertices at (0,0,0) and (1,1,1). The space diagonal has length:

10. A cylinder of radius 5 and height 12 has a cone of the same base removed from inside. Remaining volume:

11. Two parallel lines are cut by a transversal. Corresponding angles are:

12. Surface area of a sphere with radius 4:

13. What is the length of the diagonal of a 6×8 rectangle?

14. The area of a cyclic quadrilateral with sides 3,4,5,6 and semiperimeter s=9 (using Brahmagupta's formula √((s-a)(s-b)(s-c)(s-d))):

15. Two angles of a quadrilateral are each 80°. The other two are equal. What are they?

16. A rectangle has area 72. If length is doubled and width halved, new area:

17. An arc of 90° in a circle of radius 10 has chord length:

18. What is the perpendicular distance from point (0,0) to the line 4x+3y=10?

19. A regular polygon has interior angle 156°. How many sides?

20. In a triangle with circumradius R and inradius r, if the area is K and semiperimeter s:

21. The coordinates of the reflection of (5,-3) over the line y=x:

22. The diagonal of a square is 4√2. Side of square:

23. The surface area of a cone with r=6, slant height l=10 is:

24. Angle in a triangle opposite side of length 7, when the other sides are 5 and 8. Using law of cosines, cos of this angle is:

25. Volume of a pyramid with square base 6 and height 9:

26. What type of triangle has all angles less than 90°?

27. What is the area of the triangle with sides 9, 40, 41?

28. What is the distance from (−1, 3) to (4, −9)?

29. A regular icosahedron has 20 faces. Using Euler's formula with 30 edges, vertices=

30. The circumradius of equilateral triangle with side 6: