Difference between revisions of "1950 AHSME Problems"
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== Problem 14 == | == Problem 14 == | ||
− | For the simultaneous equations <cmath>2x-3y=8 | + | For the simultaneous equations <cmath>2x-3y=8</cmath><cmath>6y-4x=9</cmath> |
<math> \textbf{(A)}\ x=4,y=0\qquad\textbf{(B)}\ x=0,y=\frac{3}{2}\qquad\textbf{(C)}\ x=0,y=0\qquad\\ \textbf{(D)}\ \text{There is no solution}\qquad\textbf{(E)}\ \text{There are an infinite number of solutions} </math> | <math> \textbf{(A)}\ x=4,y=0\qquad\textbf{(B)}\ x=0,y=\frac{3}{2}\qquad\textbf{(C)}\ x=0,y=0\qquad\\ \textbf{(D)}\ \text{There is no solution}\qquad\textbf{(E)}\ \text{There are an infinite number of solutions} </math> |
Revision as of 16:16, 21 December 2015
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 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 See also
Problem 1
If is divided into three parts proportional to , , and , the smallest part is:
Problem 2
Let . When , . When , is equal to:
Problem 3
The sum of the roots of the equation is equal to:
Problem 4
Reduced to lowest terms, is equal to:
Problem 5
If five geometric means are inserted between and , the fifth term in the geometric series is:
Problem 6
The values of which will satisfy the equations may be found by solving:
Problem 7
If the digit is placed after a two digit number whose tens' digit is , and units' digit is , the new number is:
Problem 8
If the radius of a circle is increased , the area is increased:
Problem 9
The largest area of a triangle that can be inscribed in a semi-circle whose radius is is:
Problem 10
After rationalizing the numerator of , the denominator in simplest form is:
Problem 11
If in the formula , is increased while , and are kept constant, then :
Problem 12
As the number of sides of a polygon increases from to , the sum of the exterior angles formed by extending each side in succession:
Problem 13
The roots of are:
Problem 14
For the simultaneous equations
Problem 15
The real factors of are:
Problem 16
The number of terms in the expansion of when simplified is:
Problem 17
The formula which expresses the relationship between and as shown in the accompanying table is:
Problem 18
Of the following:
(1)
(2)
(3)
(4)
(5)
Problem 19
If men can do a job in days, then men can do the job in:
Problem 20
When is divided by , the remainder is:
Problem 21
The volume of a rectangular solid each of whose side, front, and bottom faces are , , and respectively is:
Problem 22
Successive discounts of and are equivalent to a single discount of:
Problem 23
A man buys a house for dollars and rents it. He puts of each month's rent aside for repairs and upkeep; pays dollars a year taxes and realizes on his investment. The monthly rent is:
Problem 24
The equation has:
Problem 25
The value of is equal to:
Problem 26
if , then
Problem 27
A car travels miles from to at miles per hour but returns the same distance at miles per hour. The average speed for the round trip is closest to:
Problem 28
Two boys and start at the same time to ride from Port Jervis to Poughkeepsie, miles away. travels miles an hour slower than . reaches Poughkeepsie and at once turns back meeting miles from Poughkeepsie. The rate of was:
Problem 29
A manufacturer built a machine which will address envelopes in minutes. He wishes to build another machine so that when both are operating together they will address envelopes in minutes. The equation used to find how many minutes it would require the second machine to address envelopes alone is:
Problem 30
From a group of boys and girls, girls leave. There are then left two boys for each girl. After this boys leave. There are then girls for each boy. The number of girls in the beginning was:
Problem 31
John ordered pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by . The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:
Problem 32
A foot ladder is placed against a vertical wall of a building. The foot of the ladder is feet from the base of the building. If the top of the ladder slips feet, then the foot of the ladder will slide:
Problem 33
The number of circular pipes with an inside diameter of inch which will carry the same amount of water as a pipe with an inside diameter of inches is:
Problem 34
When the circumference of a toy balloon is increased from inches to inches, the radius is increased by:
Problem 35
In triangle , inches, inches, inches. The radius of the inscribed circle is:
Problem 36
A merchant buys goods at off the list price. He desires to mark the goods so that he can give a discount of on the marked price and still clear a profit of on the selling price. What percent of the list price must he mark the goods?
Problem 37
If , , which of the following statements is incorrect?
Problem 38
If the expression has the value for all values of and , then the equation :
Problem 39
Given the series and the following five statements: (1) the sum increases without limit (2) the sum decreases without limit (3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small (4) the difference between the sum and 4 can be made less than any positive quantity no matter how small (5) the sum approaches a limit Of these statements, the correct ones are:
Problem 40
The limit of as approaches as a limit is:
Problem 41
The least value of the function with is:
Problem 42
The equation is satisfied when is equal to:
Problem 43
The sum to infinity of is:
Problem 44
The graph of
Problem 45
The number of diagonals that can be drawn in a polygon of sides is:
Problem 46
In triangle , , , and . If sides and are doubled while remains the same, then:
Problem 47
A rectangle inscribed in a triangle has its base coinciding with the base of the triangle. If the altitude of the triangle is , and the altitude of the rectangle is half the base of the rectangle, then:
Problem 48
A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
Problem 49
A triangle has a fixed base that is inches long. The median from to side is inches long and can have any position emanating from . The locus of the vertex of the triangle is:
Problem 50
A privateer discovers a merchantman miles to leeward at and with a good breeze bears down upon her at mph, while the merchantman can only make mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only miles while the merchantman makes . The privateer will overtake the merchantman at:
See also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by First AHSME |
Followed by 1951 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.