For problems and detailed solutions to each of the 1979 AHSME problems, please refer below:
Problem 1: If rectangle ABCD has area 72 square meters and F and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is
Problem 3: In the adjoining figure, ABCD is a square, ABE is an equilateral triangle and point E is outside square ABCD. What is the measure of β AED in degrees?
Problem 5: Find the sum of the digits of the largest even three digit number (in base ten representation) which is not changed when its unit's and hundred's digits are interchanged.
Problem 10: If P1βP2βP3βP4βP5βP6β is a tegular hexagon whose apothem (distance from the center to the midpoint of a side) is 2, and Qiβ is the midpoint of side PiβPi+1β for i=1,2,3,4, then the area of quadrilateral Q1βQ2βQ3βQ4β is
Problem 12: In the adjoining figure, CD is the diameter of a semi-circle with center O. Point A lies on the extension of DC past C; point E lies on the semi-circle, and B is the point of intersection (distinct from E) of line segment AI: with the semi-circle. If length AB equals length OD, and the measure of β EOD is 45β, then the measure of β BAO is
Problem 14: In a certain sequence of numbers, the first number in the sequence is I, and, for all nβ₯2, the product of the first n numbers in the sequence is n2. The sum of the third and the fifth numbers in the sequence is
Problem 15: Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being p:1 in one jar and q:1 in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is
Problem 16: A circle with area A1β is contained in the interior of a larger circle with area A1β+A2β. If the radius of the larger circle is 3, and if A1β,A2β,A1β+A2β is an arithmetic progression, then the radius of the smaller circle is
Problem 17: Points A,B,C and D are distinct and lie, in the given order, on a straight line. Line segments AB,AC and AD have lengths x,y and z, respectively. If line segments AB and CD may be rotated about points B and C, respectively, so that points A and D coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?
Problem 21: The length of the hypotenuse of a right triangle is h, and the radius of the inscribed circle is r. The ratio of the area of the circle to the area of the triangle is
Problem 23: The edges of a regular tetrahedron with vertices A,B,C and D each have length one. Find the least possible distance between a pair of points P and Q. where P is on edge AB and Q is on edge CD.
Problem 24: Sides AB,BC and CD of (simple) quadrilateral ABCD have lengths 4,5 and 20, respectively. If vertex angles B and C are obtuse and sinC=βcosB=53β, then side AD has length
Problem 25: If q1β(x) and r1β are the quotient and remainder, respectively, when the polynomial x8 is divided by x+21β, and if q2β(x) and r2β are the quotient and remainder, respectively, when q1β(x) is divided by x+21β, then r2β equals
Problem 27: An ordered pair (b,c) of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation x2+bx+c=0 will not have distinct positive real roots?
Problem 28: Circles with centers A,B and C each have radius r, where 1<r<2. The distance between each pair of centers is 2. If Bβ² is the point of intersection of circle A and circle C which is outside circle B, and if Cβ² is the point of intersection of circle A and circle B which is outside circle C, then length Bβ²Cβ² equals
Problem 30: In β³ABC,E is the midpoint of side BC and D is on side AC. If the length of AC is 1 and β BAC=60β,β ABC=100β,β ACB=20β and β DEC=80β, then the area of β³ABC plus twice the area of β³CDE equals
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