Problem 4: If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, then the larger of the two numbers is
Problem 5: In trapezoid ABCD, sides AB and CD are parallel, and diagonal BD and side AD have equal length. If β DCB=110β and β CBD=30β, then β ADB=
Problem 10: The lines L and K are symmetric to each other with respect to the line y=x. If the equation of line L is y=ax+b with aξ =0 and bξ =0, then the equation of K is y=
Problem 12: If p,q and M are positive numbers and q<100, then the number obtained by increasing M by p% and decreasing the result by q% exceeds M if and only if
Problem 13: Suppose that at the end of any year, a unit of money has lost 10% of the value it had at the beginning of that year. Find the smallest integer n such that after n years the unit of money will have lost at least 90% of its value. (To the nearest thousandth log10β3 is .477.)
Problem 14: In a geometric sequence of real numbers, the sum of the first two terms is 7 and the sum of the first six terms is 91. The sum of the first four terms is
Problem 19: In β³ABC,M is the midpoint of side BC,AN bisects β BAC,BNβ₯AN and ΞΈ is the measure of β BAC. If sides AB and AC have lengths 14 and 19, respectively, then length MN equals
Problem 20: A ray of light originates from point A and travels in a plane, being reflected n times between lines AD and CD, before striking a point B (which may be on AD or CD) perpendicularly and retracing its path to A. If β CDA=8β, what is the largest value n can have?
Problem 22: How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form (i,j,k), where i,j and k are positive integers not exceeding 4?
Problem 23: Equilateral β³ABC is inscribed in a circle. A second circle is tangent internally to the circumcircle at T and tangent to sides AB and AC at points P and Q. If side BC has length 12, then segment PQ has length
Problem 25: In triangle ABC in the adjoining figure, AD and AE trisect \Varangle B A C. The lengths of BD,DE and EC are 2,3 and 6, respectively. The length of the shortest side of β³ABC is
Answer Choices:
A. 210β
B. 11
C. 66β
D. 6
E. not uniquely determined by the given information
Problem 26: Alice, Bob and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six.
Problem 27: In the adjoining figure triangle ABC is inscribed in a circle. Point D lies on AC with DC=30β, and point G lies on BA with BG>GA. Side AB and side AC each has length equal to the length of chord DG and β CAB=30β. Chord DG intersects sides AC and AB at E and F, respectively. The ratio of the area of β³AFE to the area of β³ABC is
Problem 30: If a,b,c,d are the solutions of the equation x4βbxβ3=0, then an equation whose solutions are d2a+b+cβ,c2a+b+dβ,b2a+c+dβ,a2b+c+dβ is
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