Individual Problems and Solutions
For problems and detailed solutions to each of the 1959 AHSME problems, please refer below:
Problem 1: Each edge of a cube is increased by . The percent of increase in the surface area of the cube is:
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Problem 2: Through a point inside the triangle a line is drawn parallel to the base , dividing the triangle into two equal areas. If the altitude to AB has a length of 1 , then the distance from P to AB is:
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Problem 3: If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:
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A. rhombus
B. rectangle
C. square
D. isosceles trapezoid
E. none of these
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Problem 4: If 78 is divided into three parts which are proportional to , the middle part is:
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Problem 5: The value of is:
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Problem 6: Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement,
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A. only the converse is true
B. only the inverse is true
C. both are true
D. neither is true
E. the inverse is true, but the converse is sometimes true
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Problem 7: The sides of a right triangle are , and , with and both positive. The ratio of to is:
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Problem 8: The value of can never be less than:
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Problem 9: A farmer divides his herd of cows among his four sons so that one son gets onehalf the herd, a second son, one-fourth, a third son one-fifth, and the fourth son, 7 cows. Then is:
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Problem 10: In triangle ABC , with , a point D is taken on AB at a distance 1.2 from . Point is joined to point in the prolongation of so that triangle is equal in area to triangle . Then equals:
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Problem 11: The logarithm of to the base is:
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Problem 12: By adding the same constant to each of a geometric progression results. The common ratio is:
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Problem 13: The arithmetic mean (average) of a set of 50 numbers is 38 . If two numbers, namely, 45 and 55, are discarded, the mean of the remaining set of numbers is:
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Problem 14: Given the set whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those operations that yield only elements of are:
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Problem 15: In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is:
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Problem 16: The expression , when simplified, is:
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Problem 17: If , where and are constants, and if when , and when , then equals:
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Problem 18: Let and be positive numbers with greater than . Let and . The difference in percent by which exceeds and exceeds is:
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Problem 19: With the use of three different weights, namely, ., and , how many objects of different weights can be weighed, if the objects to be weighed and the given weights may be placed in either pan of the scale?
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Problem 20: It is given that varies directly as and inversely as the square of , and that when and . Then, when and equals:
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Problem 21: If p is the perimeter of an equilateral triangle inscribed in a circle, the area of the circle is:
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Problem 22: The line joining the midpoints of the diagonals of a trapezoid has length 3. If the longer base is 97, then the shorter base is:
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Problem 23: The set of solutions for the equation consists of:
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A. two integers
B. one integer and one fraction
C. two irrational numbers
D. two non-real numbers
E. no numbers, that is, the set is empty.
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Problem 24: A chemist has ounces of salt water that is salt. How many ounces of salt must he add to make a solution that is salt?
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Problem 25: The symbol means if is greater than or equal to zero, and if is less than or equal to zero; the symbol means "less than"; the symbol means "greater than."
The set of values satisfying the inequality consists of all such that:
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Problem 26: The base of an isosceles triangle is . The medians to the legs intersect each other at right angles. The area of the triangle is:
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Problem 27: Which one of the following statements is not true for the equation where ?
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A. The sum of the roots is 2
B. The discriminant is 9
C. The roots are imaginary
D. The roots can be found by using the quadratic formula
E. The roots can be found by factoring, using imaginary numbers.
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Problem 28: In triangle bisects angle and bisects angle . Points and are on and , respectively. The sides of triangle are , and . Then where is:
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Problem 29: On an examination of questions a student answers correctly 15 of the first 20. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is , how many different values of can there be?
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Problem 30: A can run around a circular track in 40 seconds. B, running in the opposite direction, meets every 15 seconds. What is B's time to run around the track, expressed in seconds?
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Problem 31: A square, with an area of 40, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is:
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Problem 32: The length of a tangent, drawn from a point A to a circle, is of the radius . The (shortest) distance from to the circle is:
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Problem 33: A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let represent the sum of the first n terms of the harmonic progression; for example, represents the sum of the first three terms. If the first three terms of a harmonic progression are , then:
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Problem 34: Let the roots of be and . Then the expression is:
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A. a positive integer
B. a positive fraction greater than 1
C. a positive fraction less than 1
D. an irrational number
E. an imaginary number
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Problem 35: The symbol means "greater than or equal to"; the symbol means "less than or equal to". In the equation is a fixed positive number, and is a fixed negative number. The set of values satisfying the equation is:
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D. the set of all real numbers
E. none of these
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Problem 36: The base of a triangle is 80, and one of the base angles is . The sum of the other two sides is 90. The shortest side of the triangle is:
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Problem 37: When simplified the product becomes:
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Problem 38: If , then :
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A. is an integer
B. is fractional
C. is irrational
D. is imaginary
E. may have two different values
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Problem 39: Let be the sum of the first nine terms of the sequence, . Then equals:
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Problem 40: In triangle is a median. intersects at so that . Point is on AB. Then, if equals:
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Problem 41: On the same side of a straight line three circles are drawn as follows: a circle with a radius of 4 inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is:
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Problem 42: Given three positive integers , and . Their greatest common divisor is ; their least common multiple is . Then, which two of the following statements are true?
(1) The product MD cannot be less than abc.
(2) The product MD cannot be greater than abc.
(3) MD equals abc if and only if a,b,c are each prime.
(4) MD equals abc if and only if a,b,c are relatively prime in pairs,(This means: no two have a common factor greater than 1)
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Problem 43: The sides of a triangle are 25,39, and 40. The diameter of the circumscribed circle is:
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Problem 44: The roots of are both real and greater than 1. Let . Then :
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A. may be less than zero
B. may be equal to zero
C. must be greater than zero
D. must be less than zero
E. must be between -1 and +1
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Problem 45: If , then equals:
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Problem 46: A student on vacation for days observed that (1) it rained 7 times, morning or afternoon (2) when it rained in the afternoon, it was clear in the morning (3) there were five clear afternoons (4) there were six clear mornings. Then equals:
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Problem 47: Assume that the following three statements are true:
I. All freshmen are human.
II. All students are human.
III. Some students think
Given the following four statements:
(1) All freshmen are students.
(2) Some humans think.
(3) No freshmen think.
(4) Some humans who think are not students.
Those which are logical consequences of I, II, and III are:
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Problem 48: Given the polynomial , where is a positive integer or zero, and is a positive integer. The remaining 's are integers or zero. Set . The number of polynomials with is:
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Problem 49: For the infinite series let be the (limiting) sum. Then equals:
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Problem 50: A club with members is organized into four committees in accordance with these two rules: (1) Each member belongs to two and only two committees. (2) Each pair of committees has one and only one member in common. Then :
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A. cannot be determined
B. has a single value between and
C. has two values between and
D. has a single value between and
E. has two values between and
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions