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Questions 1-5: Algebra

1. Solve the equation x3−3x2+2x−1=0.
2. Find the sum of all the real roots of the equation x4+4x3+6x2+4x+1=0.
3. Let a and b be real numbers such that a+b=1 and a2+b2=2. Find the maximum value of ab.
4. Let f(x)=2x3+3x2+4x+1. Find the number of points of intersection of the graphs of f and f′(x).
5. Let P(x) be a polynomial of degree 3 with real coefficients. Suppose that P(1)=P(−1)=0, and that P(x)>0 for all x∈(−1,1). Find the minimum value of P(0).

Questions 6-10: Geometry

6. In triangle ABC, let D be the midpoint of BC, and let E be the foot of the altitude from A to BC. Prove that AE+BD>AC.
7. Let ABCD be a trapezoid with AD parallel to BC. Let M be the midpoint of AD, and let N be the midpoint of BC. Prove that MN is parallel to AB and CD.
8. Let O be the center of the circle with equation x2+y2=1. Let P be a point on the circle, and let D be the foot of the altitude from P to OA. Prove that PD is a tangent to the circle.
9. Let ABC be a triangle with circumcircle ω. Let P be a point on ω, and let D be the foot of the altitude from P to BC. Prove that ∠APD=∠ABC.
10. Let ABC be a triangle with circumcircle ω. Let M be the midpoint of BC, and let P be a point on ω such that ∠PMC=90∘. Prove that ∠AMB=∠ACM.

Questions 11-15: Probability

11. Two dice are rolled. Find the probability that the sum of the two rolls is a multiple of 3.
12. A fair coin is flipped 10 times. Find the probability that the number of heads is exactly 5.
13. A deck of 52 cards is shuffled thoroughly. Find the probability that the top card is an ace, given that the second card is a king.
14. Three people are each given three cards from a standard deck of 52 cards. Find the probability that each person has exactly one card of each suit.
15. A bag contains 10 red marbles and 20 blue marbles. Two marbles are drawn without replacement. Find the probability that the first marble is red and the second marble is blue, or vice versa.

Questions 16-20: Trigonometry

16. Find the value of sin15∘.
17. Find the value of cos75∘.
18. Find the value of tan45∘.
19. Find the value of sin135∘.
20. Find the value of cos225∘.