1963 AHSME Problems
1963 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
Which one of the following points is not on the graph of ?
Problem 2
let . Find when and .
Problem 3
If the reciprocal of is , then equals:
Problem 4
For what value(s) of does the pair of equations and have two identical solutions?
Problem 5
If and are real numbers and , then:
Problem 6
is right-angled at . On there is a point for which and . The magnitude of is:
Problem 7
Given the four equations:
The pair representing the perpendicular lines is:
Problem 8
The smallest positive integer for which , where is an integer, is:
Problem 9
In the expansion of the coefficient of is:
Problem 10
Point is taken interior to a square with side-length and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If represents the common distance, then equals:
Problem 11
The arithmetic mean of a set of numbers is . If two numbers of the set, namely and , are discarded, the arithmetic mean of the remaining set of numbers is:
Problem 12
Three vertices of parallelogram are with and diagonally opposite. The sum of the coordinates of vertex is:
Problem 13
If , the number of integers which can possibly be negative, is, at most:
Problem 14
Given the equations and . If, when the roots of the equation are suitably listed, each root of the second equation is more than the corresponding root of the first equation, then equals:
Problem 15
A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:
Problem 16
Three numbers , none zero, form an arithmetic progression. Increasing by or increasing by results in a geometric progression. Then equals:
Problem 17
The expression , real, , has the value for:
Problem 18
Chord is the perpendicular bisector of chord , intersecting it in . Between and point is taken, and extended meets the circle in . Then, for any selection of , as described, is similar to:
Problem 19
In counting colored balls, some red and some black, it was found that of the first counted were red. Thereafter, out of every counted were red. If, in all, % or more of the balls counted were red, the maximum value of is:
Problem 20
Two men at points and , miles apart, set out at the same time to walk towards each other. The man at walks uniformly at the rate of miles per hour; the man at walks at the constant rate of miles per hour for the first hour, at miles per hour for the second hour, and so on, in arithmetic progression. If the men meet miles nearer than in an integral number of hours, then is:
Problem 21
The expression has:
Problem 22
Acute-angled is inscribed in a circle with center at ; and . point is taken in minor arc such that is perpendicular to . Then the ratio of the magnitudes of and is:
Problem 23
A gives as many cents as has and as many cents as has. Similarly, then gives and as many cents as each then has. , similarly, then gives and as many cents as each then has. If each finally has cents, with how many cents does start?
Problem 24
Consider equations of the form . How many such equations have real roots and have coefficients and selected from the set of integers ?
Problem 25
Point is taken in side of square . At a perpendicular is drawn to , meeting extended at . The area of is square inches and the area of is square inches. Then the number of inches in is:
Problem 26
Version 1 Consider the statements:
where , and are propositions. How many of these imply the truth of ?
Version 2 Consider the statements (1) and are true and is false (2) is true and and are false (3) is true and and are false (4) and are true and is false. How many of these imply the truth of the statement " is implied by the statement that implies "?
Problem 27
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
Problem 28
Given the equation with real roots. The value of for which the product of the roots of the equation is a maximum is:
Problem 29
A particle projected vertically upward reaches, at the end of seconds, an elevation of feet where . The highest elevation is:
Problem 30
Let . Find a new function by replacing each in by , and simplify. The simplified expression is equal to:
Problem 31
The number of solutions in positive integers of is:
Problem 32
The dimensions of a rectangle are and , . It is required to obtain a rectangle with dimensions and , , so that its perimeter is one-third that of , and its area is one-third that of . The number of such (different) rectangles is:
Problem 33
Given the line and a line parallel to the given line and units from it. A possible equation for is:
Problem 34
In , side , side , and side . Let be the largest number such that the magnitude, in degrees, of the angle opposite side exceeds . Then equals:
Problem 35
The lengths of the sides of a triangle are integers, and its area is also an integer. One side is and the perimeter is . The shortest side is:
Problem 36
A person starting with and making bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:
Problem 37
Given points on a straight line, in the order stated (not necessarily evenly spaced). Let be an arbitrarily selected point on the line and let be the sum of the undirected lengths . Then is smallest if and only if the point is:
Problem 38
Point is taken on the extension of side of parallelogram . intersects diagonal at and side at . If and , then equals:
Problem 39
In lines and are drawn so that and . Let where is the intersection point of and . Then equals:
Problem 40
If is a number satisfying the equation , then is between:
See also
1963 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1962 AHSC |
Followed by 1964 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.