1965 AHSME Problems
1965 AHSC (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also
Problem 1
The number of real values of satisfying the equation is:
Problem 2
A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side, is:
Problem 3
The expression has the same value as:
Problem 4
Line intersects line and line is parallel to . The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:
Problem 5
When the repeating decimal is written in simplest fractional form, the sum of the numerator and denominator is:
Problem 6
If then equals:
Problem 7
The sum of the reciprocals of the roots of the equation is:
Problem 8
One side of a given triangle is 18 inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is:
Problem 9
The vertex of the parabola will be a point on the -axis if the value of is:
Problem 10
The statement is equivalent to the statement:
Problem 11
Consider the statements: Of these the following are incorrect.
Problem 12
A rhombus is inscribed in in such a way that one of its vertices is and two of its sides lie along and . If inches, inches, and inches, the side of the rhombus, in inches, is:
Problem 13
Let be the number of number-pairs which satisfy and . Then is:
Problem 14
The sum of the numerical coefficients in the complete expansion of is:
Problem 15
The symbol represents a two-digit number in the base . If the number is double the number , then is:
Problem 16
Let line be perpendicular to line . Connect to , the midpoint of , and connect to , the midpoint of . If and intersect in point , and inches, then the area of triangle , in square inches, is:
Problem 17
Given the true statement: The picnic on Sunday will not be held only if the weather is not fair. We can then conclude that:
Problem 18
If is used as an approximation to the value of , the ratio of the error made to the correct value is:
Problem 19
If is exactly divisible by , the value of is:
Problem 20
For every the sum of n terms of an arithmetic progression is . The th term is:
Problem 21
It is possible to choose in such a way that the value of is
Problem 22
If and and are the roots of , then the equality holds:
Problem 23
If we write for all such that , the smallest value we can use for is:
Problem 24
Given the sequence , the smallest value of n such that the product of the first members of this sequence exceeds is:
Problem 25
Let be a quadrilateral with extended to so that . Lines and are drawn to form . For this angle to be a right angle it is necessary that quadrilateral have:
Problem 26
For the numbers define to be the arithmetic mean of all five numbers; to be the arithmetic mean of and ; to be the arithmetic mean of , and ; and to be the arithmetic mean of and . Then, no matter how , and are chosen, we shall always have:
Problem 27
When is divided by the quotient is and the remainder is . When is divided by the quotient is and the remainder is . If then is:
Problem 28
An escalator (moving staircase) of uniform steps visible at all times descends at constant speed. Two boys, and , walk down the escalator steadily as it moves, A negotiating twice as many escalator steps per minute as . reaches the bottom after taking steps while reaches the bottom after taking steps. Then is:
Problem 29
Of students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but not English. The number taking English and History only is five times the number taking all three subjects. If the number taking all three subjects is even and non-zero, the number taking English and Mathematics only is:
Problem 30
Let of right triangle be the diameter of a circle intersecting hypotenuse in . At a tangent is drawn cutting leg in . This information is not sufficient to prove that
Problem 31
The number of real values of satisfying the equality , where , is:
Problem 32
An article costing dollars is sold for $100 at a loss of percent of the selling price. It is then resold at a profit of percent of the new selling price . If the difference between and is dollars, then x is:
Problem 33
If the number , that is, , ends with zeros when given to the base and ends with zeros when given to the base , then equals:
Problem 34
For the smallest value of is:
Problem 35
The length of a rectangle is inches and its width is less than inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is , then the width is:
Problem 36
Given distinct straight lines and . From a point in a perpendicular is drawn to ; from the foot of this perpendicular a line is drawn perpendicular to . From the foot of this second perpendicular a line is drawn perpendicular to ; and so on indefinitely. The lengths of the first and second perpendiculars are and , respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:
Problem 37
Point is selected on side of in such a way that and point is selected on side such that . The point of intersection of and is . Then is:
Problem 38
A takes times as long to do a piece of work as and together; takes times as long as and together; and takes times as long as and together. Then , in terms of and , is:
Problem 39
A foreman noticed an inspector checking a "-hole with a "-plug and a "-plug and suggested that two more gauges be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter, , of each, to the nearest hundredth of an inch, is:
Problem 40
Let be the number of integer values of such that is the square of an integer. Then is:
See also
1965 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1964 AHSC |
Followed by 1966 AHSC | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.