MathDB

1950 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(50)
50
1

AHSME 1950- part 3

A privateer discovers a merchantman 1010 miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at 1111 mph, while the merchantman can only make 88 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 1717 miles while the merchantman makes 1515. The privateer will overtake the merchantman at:
<spanclass=′latex−bold′>(A)</span> 3:45 p.m.<spanclass=′latex−bold′>(B)</span> 3:30 p.m.<spanclass=′latex−bold′>(C)</span> 5:00 p.m.<spanclass=′latex−bold′>(D)</span> 2:45 p.m.<spanclass=′latex−bold′>(E)</span> 5:30 p.m.<span class='latex-bold'>(A)</span>\ 3\text{:}45\text{ p.m.} \qquad <span class='latex-bold'>(B)</span>\ 3\text{:}30\text{ p.m.} \qquad <span class='latex-bold'>(C)</span>\ 5\text{:}00\text{ p.m.} \qquad <span class='latex-bold'>(D)</span>\ 2\text{:}45\text{ p.m.} \qquad <span class='latex-bold'>(E)</span>\ 5\text{:}30\text{ p.m.}
49
1

AHSME 1950- part 3

A triangle has a fixed base ABAB that is 22 inches long. The median from AA to side BCBC is 112 1\frac{1}{2} inches long and can have any position emanating from AA. The locus of the vertex CC of the triangle is:
<spanclass=′latex−bold′>(A)</span> A straight line AB,112 inches from A<spanclass=′latex−bold′>(B)</span> A circle with A as center and radius 2 inches<spanclass=′latex−bold′>(C)</span> A circle with A as center and radius 3 inches<spanclass=′latex−bold′>(D)</span> A circle with radius 3 inches and center 4 inches from B along BA<spanclass=′latex−bold′>(E)</span> An ellipse with A as focus<span class='latex-bold'>(A)</span>\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ <span class='latex-bold'>(B)</span>\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ <span class='latex-bold'>(C)</span>\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ <span class='latex-bold'>(D)</span>\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ <span class='latex-bold'>(E)</span>\ \text{An ellipse with }A\text{ as focus}
48
1

AHSME 1950- part 3

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:
<spanclass=′latex−bold′>(A)</span> Least when the point is the center of gravity of the triangle<spanclass=′latex−bold′>(B)</span> Greater than the altitude of the triangle<spanclass=′latex−bold′>(C)</span> Equal to the altitude of the triangle<spanclass=′latex−bold′>(D)</span> One-half the sum of the sides of the triangle<spanclass=′latex−bold′>(E)</span> Greatest when the point is the center of gravity<span class='latex-bold'>(A)</span>\ \text{Least when the point is the center of gravity of the triangle}\qquad\\ <span class='latex-bold'>(B)</span>\ \text{Greater than the altitude of the triangle} \qquad\\ <span class='latex-bold'>(C)</span>\ \text{Equal to the altitude of the triangle}\qquad\\ <span class='latex-bold'>(D)</span>\ \text{One-half the sum of the sides of the triangle} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Greatest when the point is the center of gravity}
47
1

AHSME 1950- part 3

A rectangle inscribed in a triangle has its base coinciding with the base bb of the triangle. If the altitude of the triangle is hh, and the altitude xx of the rectangle is half the base of the rectangle, then:
<spanclass=′latex−bold′>(A)</span> x=12h<spanclass=′latex−bold′>(B)</span> x=bhb+h<spanclass=′latex−bold′>(C)</span> x=bh2h+b<spanclass=′latex−bold′>(D)</span> x=hb2<spanclass=′latex−bold′>(E)</span> x=12b<span class='latex-bold'>(A)</span>\ x=\dfrac{1}{2}h \qquad <span class='latex-bold'>(B)</span>\ x=\dfrac{bh}{b+h} \qquad <span class='latex-bold'>(C)</span>\ x=\dfrac{bh}{2h+b} \qquad <span class='latex-bold'>(D)</span>\ x=\sqrt{\dfrac{hb}{2}} \qquad <span class='latex-bold'>(E)</span>\ x=\dfrac{1}{2}b
46
1

AHSME 1950- part 3

In triangle ABCABC, AB=12AB=12, AC=7AC=7, and BC=10BC=10. If sides ABAB and ACAC are doubled while BCBC remains the same, then:
<spanclass=′latex−bold′>(A)</span> The area is doubled<spanclass=′latex−bold′>(B)</span> The altitude is doubled<spanclass=′latex−bold′>(C)</span> The area is four times the original area<spanclass=′latex−bold′>(D)</span> The median is unchanged<spanclass=′latex−bold′>(E)</span> The area of the triangle is 0<span class='latex-bold'>(A)</span>\ \text{The area is doubled} \qquad\\ <span class='latex-bold'>(B)</span>\ \text{The altitude is doubled} \qquad\\ <span class='latex-bold'>(C)</span>\ \text{The area is four times the original area} \qquad\\ <span class='latex-bold'>(D)</span>\ \text{The median is unchanged} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{The area of the triangle is 0}
45
1
44
1

AHSME 1950- part 3

The graph of y\equal{}\log x
<spanclass=′latex−bold′>(A)</span>Cuts the y-axis<spanclass=′latex−bold′>(B)</span> Cuts all lines perpendicular to the x-axis<spanclass=′latex−bold′>(C)</span> Cuts the x-axis<spanclass=′latex−bold′>(D)</span> Cuts neither axis<spanclass=′latex−bold′>(E)</span> Cuts all circles whose center is at the origin<span class='latex-bold'>(A)</span>\text{Cuts the }y\text{-axis} \qquad\\ <span class='latex-bold'>(B)</span>\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ <span class='latex-bold'>(C)</span>\ \text{Cuts the }x\text{-axis} \qquad\\ <span class='latex-bold'>(D)</span>\ \text{Cuts neither axis} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Cuts all circles whose center is at the origin}
43
1

AHSME 1950- part 3

The sum to infinity of \frac{1}{7}\plus{}\frac {2}{7^2}\plus{}\frac{1}{7^3}\plus{}\frac{2}{7^4}\plus{}... is:
<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 124<spanclass=′latex−bold′>(C)</span> 548<spanclass=′latex−bold′>(D)</span> 116<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ \frac{1}{5} \qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{24} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{5}{48} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{1}{16} \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
42
1

AHSME 1950- part 3

The equation x^{x^{x}}...\equal{}2 is satisfied when xx is equal to:
<spanclass=′latex−bold′>(A)</span> ∞<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ \infty \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ \sqrt[4]{2} \qquad <span class='latex-bold'>(D)</span>\ \sqrt{2} \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
41
1

AHSME 1950- part 3

The least value of the function ax^2\plus{}bx\plus{}c with a>0a>0 is:
<spanclass=′latex−bold′>(A)</span> −ba<spanclass=′latex−bold′>(B)</span> −b2a<spanclass=′latex−bold′>(C)</span> b2−4ac<spanclass=′latex−bold′>(D)</span> 4ac−b24a<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ -\dfrac{b}{a} \qquad <span class='latex-bold'>(B)</span>\ -\dfrac{b}{2a} \qquad <span class='latex-bold'>(C)</span>\ b^2-4ac \qquad <span class='latex-bold'>(D)</span>\ \dfrac{4ac-b^2}{4a}\qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
40
1

AHSME 1950- part 3

The limit of \frac {x^2\minus{}1}{x\minus{}1} as xx approaches 11 as a limit is:
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> Indeterminate<spanclass=′latex−bold′>(C)</span> x−1<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 1<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ \text{Indeterminate} \qquad <span class='latex-bold'>(C)</span>\ x-1 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 1
39
1

AHSME 1950- part 3

Given the series 2\plus{}1\plus{}\frac {1}{2}\plus{}\frac {1}{4}\plus{}... 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 statments, the correct ones are:
<spanclass=′latex−bold′>(A)</span> Only 3 and 4<spanclass=′latex−bold′>(B)</span> Only 5<spanclass=′latex−bold′>(C)</span> Only 2 and 4<spanclass=′latex−bold′>(D)</span> Only 2,3 and 4<spanclass=′latex−bold′>(E)</span> Only 4 and 5<span class='latex-bold'>(A)</span>\ \text{Only }3 \text{ and }4\qquad <span class='latex-bold'>(B)</span>\ \text{Only }5 \qquad <span class='latex-bold'>(C)</span>\ \text{Only }2\text{ and }4 \qquad <span class='latex-bold'>(D)</span>\ \text{Only }2,3\text{ and }4 \qquad <span class='latex-bold'>(E)</span>\ \text{Only }4\text{ and }5
38
1

AHSME 1950- part 3

If the expression (acdb) \begin{pmatrix}a & c \\ d & b \end{pmatrix} has the value ab\minus{}cd for all values of a,b,ca, b, c and dd, then the equation (2x1xx)=3 \begin{pmatrix}2x & 1 \\ x & x \end{pmatrix} = 3:
<spanclass=′latex−bold′>(A)</span> Is satisfied for only 1 value of x<spanclass=′latex−bold′>(B)</span> Is satisified for only 2 values of x<spanclass=′latex−bold′>(C)</span> Is satisified for no values of x<spanclass=′latex−bold′>(D)</span> Is satisfied for an infinite number of values of x<spanclass=′latex−bold′>(E)</span> None of these.<span class='latex-bold'>(A)</span>\ \text{Is satisfied for only 1 value of }x \qquad\\ <span class='latex-bold'>(B)</span>\ \text{Is satisified for only 2 values of }x \qquad\\ <span class='latex-bold'>(C)</span>\ \text{Is satisified for no values of }x \qquad\\ <span class='latex-bold'>(D)</span>\ \text{Is satisfied for an infinite number of values of }x \qquad\\ <span class='latex-bold'>(E)</span>\ \text{None of these.}
37
1

AHSME 1950- part 3

If y \equal{} \log_{a}{x}, a>1 a > 1, which of the following statements is incorrect?
<spanclass=′latex−bold′>(A)</span> If x=1,y=0<spanclass=′latex−bold′>(B)</span> If x=a,y=1<spanclass=′latex−bold′>(C)</span> If x=−1,y is imaginary (complex)<spanclass=′latex−bold′>(D)</span> If 0<x<z,y is always less than 0 and decreases without limit as x approaches zero<spanclass=′latex−bold′>(E)</span> Only some of the above statements are correct<span class='latex-bold'>(A)</span>\ \text{If }x=1,y=0 \qquad\\ <span class='latex-bold'>(B)</span>\ \text{If }x=a,y=1 \qquad\\ <span class='latex-bold'>(C)</span>\ \text{If }x=-1,y\text{ is imaginary (complex)} \qquad\\ <span class='latex-bold'>(D)</span>\ \text{If }0<x<z,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Only some of the above statements are correct}
36
1

AHSME 1950- part 3

A merchant buys goods at 25% 25\% of the list price. He desires to mark the goods so that he can give a discount of 20% 20\% on the marked price and still clear a profit of 25% 25\% on the selling price. What percent of the list price must he mark the goods?
<spanclass=′latex−bold′>(A)</span> 125%<spanclass=′latex−bold′>(B)</span> 100%<spanclass=′latex−bold′>(C)</span> 120%<spanclass=′latex−bold′>(D)</span> 80%<spanclass=′latex−bold′>(E)</span> 75%<span class='latex-bold'>(A)</span>\ 125\% \qquad <span class='latex-bold'>(B)</span>\ 100\% \qquad <span class='latex-bold'>(C)</span>\ 120\% \qquad <span class='latex-bold'>(D)</span>\ 80\% \qquad <span class='latex-bold'>(E)</span>\ 75\%
35
1

AHSME 1950- part 2

In triangle ABCABC, AC=24AC=24 inches, BC=10BC=10 inches, AB=26AB=26 inches. The radius of the inscribed circle is:
<spanclass=′latex−bold′>(A)</span> 26 in<spanclass=′latex−bold′>(B)</span> 4 in<spanclass=′latex−bold′>(C)</span> 13 in<spanclass=′latex−bold′>(D)</span> 8 in<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 26\text{ in} \qquad <span class='latex-bold'>(B)</span>\ 4\text{ in} \qquad <span class='latex-bold'>(C)</span>\ 13\text{ in} \qquad <span class='latex-bold'>(D)</span>\ 8\text{ in} \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
34
1

AHSME 1950- part 2

When the circumference of a toy balloon is increased from 2020 inches to 2525 inches, the radius is increased by:
<spanclass=′latex−bold′>(A)</span> 5 in<spanclass=′latex−bold′>(B)</span> 212 in<spanclass=′latex−bold′>(C)</span> 5π in<spanclass=′latex−bold′>(D)</span> 52π in<spanclass=′latex−bold′>(E)</span> π5 in<span class='latex-bold'>(A)</span>\ 5\text{ in} \qquad <span class='latex-bold'>(B)</span>\ 2\dfrac{1}{2}\text{ in} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{5}{\pi}\text{ in} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{5}{2\pi}\text{ in} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{\pi}{5}\text{ in}
33
1

AHSME 1950- part 2

The number of circular pipes with an inside diameter of 11 inch which will carry the same amount of water as a pipe with an inside diameter of 66 inches is:
<spanclass=′latex−bold′>(A)</span> 6π<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 36<spanclass=′latex−bold′>(E)</span> 36π<span class='latex-bold'>(A)</span>\ 6\pi \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 36 \qquad <span class='latex-bold'>(E)</span>\ 36\pi
32
1

AHSME 1950- part 2

A 2525 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 77 feet from the base of the building. If the top of the ladder slips 44 feet, then the foot of the ladder will slide:
<spanclass=′latex−bold′>(A)</span> 9 ft<spanclass=′latex−bold′>(B)</span> 15 ft<spanclass=′latex−bold′>(C)</span> 5 ft<spanclass=′latex−bold′>(D)</span> 8 ft<spanclass=′latex−bold′>(E)</span> 4 ft<span class='latex-bold'>(A)</span>\ 9\text{ ft} \qquad <span class='latex-bold'>(B)</span>\ 15\text{ ft} \qquad <span class='latex-bold'>(C)</span>\ 5\text{ ft} \qquad <span class='latex-bold'>(D)</span>\ 8\text{ ft} \qquad <span class='latex-bold'>(E)</span>\ 4\text{ ft}
31
1

AHSME 1950- part 2

John ordered 44 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 50% 50\%. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:
<spanclass=′latex−bold′>(A)</span> 4:1<spanclass=′latex−bold′>(B)</span> 2:1<spanclass=′latex−bold′>(C)</span> 1:4<spanclass=′latex−bold′>(D)</span> 1:2<spanclass=′latex−bold′>(E)</span> 1:8<span class='latex-bold'>(A)</span>\ 4:1 \qquad <span class='latex-bold'>(B)</span>\ 2:1 \qquad <span class='latex-bold'>(C)</span>\ 1:4 \qquad <span class='latex-bold'>(D)</span>\ 1:2 \qquad <span class='latex-bold'>(E)</span>\ 1:8
30
1

AHSME 1950- part 2

From a group of boys and girls, 1515 girls leave. There are then left two boys for each girl. After this 4545 boys leave. There are then 55 girls for each boy. The number of girls in the beginning was:
<spanclass=′latex−bold′>(A)</span> 40<spanclass=′latex−bold′>(B)</span> 43<spanclass=′latex−bold′>(C)</span> 29<spanclass=′latex−bold′>(D)</span> 50<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 40 \qquad <span class='latex-bold'>(B)</span>\ 43 \qquad <span class='latex-bold'>(C)</span>\ 29 \qquad <span class='latex-bold'>(D)</span>\ 50 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
29
1

AHSME 1950- part 2

A manufacturer built a machine which will address 500500 envelopes in 88 minutes. He wishes to build another machine so that when both are operating together they will address 500500 envelopes in 22 minutes. The equation used to find how many minutes xx it would require the second machine to address 500500 envelopes alone is:
<spanclass=′latex−bold′>(A)</span> 8−x=2<spanclass=′latex−bold′>(B)</span> 18+1x=12<spanclass=′latex−bold′>(C)</span> 5008+500x=500<spanclass=′latex−bold′>(D)</span> x2+x8=1<spanclass=′latex−bold′>(E)</span> None of these answers<span class='latex-bold'>(A)</span>\ 8-x=2 \qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{8}+\dfrac{1}{x}=\dfrac{1}{2} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{500}{8}+\dfrac{500}{x}=500 \qquad <span class='latex-bold'>(D)</span>\ \dfrac{x}{2}+\dfrac{x}{8}=1 \qquad\\ <span class='latex-bold'>(E)</span>\ \text{None of these answers}
28
1

AHSME 1950- part 2

Two boys AA and BB start at the same time to ride from Port Jervis to Poughkeepsie, 6060 miles away. AA travels 44 miles an hour slower than BB. BB reaches Poughkeepsie and at once turns back meeting AA 1212 miles from Poughkeepsie. The rate of AA was:
<spanclass=′latex−bold′>(A)</span> 4 mph<spanclass=′latex−bold′>(B)</span> 8 mph<spanclass=′latex−bold′>(C)</span> 12 mph<spanclass=′latex−bold′>(D)</span> 16 mph<spanclass=′latex−bold′>(E)</span> 20 mph<span class='latex-bold'>(A)</span>\ 4\text{ mph}\qquad <span class='latex-bold'>(B)</span>\ 8\text{ mph} \qquad <span class='latex-bold'>(C)</span>\ 12\text{ mph} \qquad <span class='latex-bold'>(D)</span>\ 16\text{ mph} \qquad <span class='latex-bold'>(E)</span>\ 20\text{ mph}
27
1

AHSME 1950- part 2

A car travels 120120 miles from AA to BB at 3030 miles per hour but returns the same distance at 4040 miles per hour. The average speed for the round trip is closest to:
<spanclass=′latex−bold′>(A)</span> 33 mph<spanclass=′latex−bold′>(B)</span> 34 mph<spanclass=′latex−bold′>(C)</span> 35 mph<spanclass=′latex−bold′>(D)</span> 36 mph<spanclass=′latex−bold′>(E)</span> 37 mph<span class='latex-bold'>(A)</span>\ 33\text{ mph} \qquad <span class='latex-bold'>(B)</span>\ 34\text{ mph} \qquad <span class='latex-bold'>(C)</span>\ 35\text{ mph} \qquad <span class='latex-bold'>(D)</span>\ 36\text{ mph} \qquad <span class='latex-bold'>(E)</span>\ 37\text{ mph}
26
1

AHSME 1950- part 2

If \log_{10}{m} \equal{} b \minus{} \log_{10}{n}, then m m=
<spanclass=′latex−bold′>(A)</span> bn<spanclass=′latex−bold′>(B)</span> bn<spanclass=′latex−bold′>(C)</span> 10bn<spanclass=′latex−bold′>(D)</span> b−10n<spanclass=′latex−bold′>(E)</span> 10bn<span class='latex-bold'>(A)</span>\ \dfrac{b}{n} \qquad <span class='latex-bold'>(B)</span>\ bn \qquad <span class='latex-bold'>(C)</span>\ 10^b n\qquad <span class='latex-bold'>(D)</span>\ b-10^n \qquad <span class='latex-bold'>(E)</span>\ \dfrac{10^b}{n}
25
1

AHSME 1950- part 2

The value of log⁡5(125)(625)25 \log_5 \frac {(125)(625)}{25} is equal to:
<spanclass=′latex−bold′>(A)</span> 725<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 3125<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 725 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 3125 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
24
1

AHSME 1950- part 2

The equation x\plus{}\sqrt{x\minus{}2}\equal{}4 has:
<spanclass=′latex−bold′>(A)</span> 2 real roots<spanclass=′latex−bold′>(B)</span> 1 real and 1 imaginary root<spanclass=′latex−bold′>(C)</span> 2 imaginary roots<spanclass=′latex−bold′>(D)</span> No roots<spanclass=′latex−bold′>(E)</span> 1 real root<span class='latex-bold'>(A)</span>\ \text{2 real roots} \qquad <span class='latex-bold'>(B)</span>\ \text{1 real and 1 imaginary root} \qquad <span class='latex-bold'>(C)</span>\ \text{2 imaginary roots} \qquad <span class='latex-bold'>(D)</span>\ \text{No roots} \qquad <span class='latex-bold'>(E)</span>\ \text{1 real root}
23
1

AHSME 1950- part 2

A man buys a house for 1000010000 dollars and rents it. He puts 1212% 12\frac{1}{2}\% of each month's rent aside for repairs and upkeep; pays 325325 dollars a year taxes and realizes 512% 5\frac{1}{2}\% on his investment. The monthly rent is:
<spanclass=′latex−bold′>(A)</span> 64.82 dollars<spanclass=′latex−bold′>(B)</span> 83.33 dollars<spanclass=′latex−bold′>(C)</span> 72.08 dollars<spanclass=′latex−bold′>(D)</span> 45.83 dollars<spanclass=′latex−bold′>(E)</span> 177.08 dollars<span class='latex-bold'>(A)</span>\ 64.82\text{ dollars} \qquad <span class='latex-bold'>(B)</span>\ 83.33\text{ dollars} \qquad <span class='latex-bold'>(C)</span>\ 72.08\text{ dollars} \qquad <span class='latex-bold'>(D)</span>\ 45.83\text{ dollars} \qquad <span class='latex-bold'>(E)</span>\ 177.08\text{ dollars}
22
1

AHSME 1950- part 2

Successive discounts of 10%10\% and 20%20\% are equivalent to a single discount of:
<spanclass=′latex−bold′>(A)</span> 30%<spanclass=′latex−bold′>(B)</span> 15%<spanclass=′latex−bold′>(C)</span> 72%<spanclass=′latex−bold′>(D)</span> 28%<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 30\% \qquad <span class='latex-bold'>(B)</span>\ 15\% \qquad <span class='latex-bold'>(C)</span>\ 72\% \qquad <span class='latex-bold'>(D)</span>\ 28\% \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
21
1

AHSME 1950- part 2

The volume of a rectangular solid each of whose side, front, and bottom faces are 12 in212\text{ in}^2, 8 in28\text{ in}^2, and 6 in26\text{ in}^2 respectively is:
<spanclass=′latex−bold′>(A)</span> 576 in3<spanclass=′latex−bold′>(B)</span> 24 in3<spanclass=′latex−bold′>(C)</span> 9 in3<spanclass=′latex−bold′>(D)</span> 104 in3<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 576\text{ in}^3 \qquad <span class='latex-bold'>(B)</span>\ 24\text{ in}^3 \qquad <span class='latex-bold'>(C)</span>\ 9\text{ in}^3 \qquad <span class='latex-bold'>(D)</span>\ 104\text{ in}^3 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
20
1

AHSME 1950- part 2

When x^{13} \plus{} 1 is divided by x \minus{} 1, the remainder is:
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> −1<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> None of these answers<span class='latex-bold'>(A)</span>\ 1\qquad <span class='latex-bold'>(B)</span>\ -1 \qquad <span class='latex-bold'>(C)</span>\ 0 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these answers}
19
1

AHSME 1950- part 2

If m m men can do a job in d d days, then m\plus{}r men can do the job in:
<spanclass=′latex−bold′>(A)</span> d+r days<spanclass=′latex−bold′>(B)</span> d−r days<spanclass=′latex−bold′>(C)</span> mdm+r days<spanclass=′latex−bold′>(D)</span> dm+r days<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ d+r\text{ days} \qquad <span class='latex-bold'>(B)</span>\ d-r\text{ days} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{md}{m+r}\text{ days} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{d}{m+r}\text{ days} \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
18
1

AHSME 1950- part 2

Of the following (1) a(x\minus{}y)\equal{}ax\minus{}ay (2) a^{x\minus{}y}\equal{}a^x\minus{}a^y (3) \log (x\minus{}y)\equal{}\log x\minus{}\log y (4) \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y} (5) a(xy)\equal{}ax\times ay
<spanclass=′latex−bold′>(A)</span> Only 1 and 4 are true<spanclass=′latex−bold′>(B)</span> Only 1 and 5 are true<spanclass=′latex−bold′>(C)</span> Only 1 and 3 are true<spanclass=′latex−bold′>(D)</span> Only 1 and 2 are true<spanclass=′latex−bold′>(E)</span> Only 1 is true<span class='latex-bold'>(A)</span>\ \text{Only 1 and 4 are true} \qquad\\ <span class='latex-bold'>(B)</span>\ \text{Only 1 and 5 are true} \qquad\\ <span class='latex-bold'>(C)</span>\ \text{Only 1 and 3 are true} \qquad\\ <span class='latex-bold'>(D)</span>\ \text{Only 1 and 2 are true} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Only 1 is true}
17
1

AHSME 1950- part 2

The formula which expresses the relationship between xx and yy as shown in the accompanying table is: \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular} <spanclass=′latex−bold′>(A)</span> y=100−10x<spanclass=′latex−bold′>(B)</span> y=100−5x2<spanclass=′latex−bold′>(C)</span> y=100−5x−5x2<spanclass=′latex−bold′>(D)</span> y=20−x−x2<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ y=100-10x \qquad <span class='latex-bold'>(B)</span>\ y=100-5x^2 \qquad <span class='latex-bold'>(C)</span>\ y=100-5x-5x^2 \qquad\\ <span class='latex-bold'>(D)</span>\ y=20-x-x^2 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
16
1
15
1

AHSME 1950- part 1

The real factors of x^2\plus{}4 are:
<spanclass=′latex−bold′>(A)</span> (x2+2)(x2+2)<spanclass=′latex−bold′>(B)</span> (x2+2)(x2−2)<spanclass=′latex−bold′>(C)</span> x2(x2+4)<spanclass=′latex−bold′>(D)</span> (x2−2x+2)(x2+2x+2)<spanclass=′latex−bold′>(E)</span> Non-existent<span class='latex-bold'>(A)</span>\ (x^2+2)(x^2+2) \qquad <span class='latex-bold'>(B)</span>\ (x^2+2)(x^2-2) \qquad <span class='latex-bold'>(C)</span>\ x^2(x^2+4) \qquad\\ <span class='latex-bold'>(D)</span>\ (x^2-2x+2)(x^2+2x+2) \qquad <span class='latex-bold'>(E)</span>\ \text{Non-existent}
14
1

AHSME 1950- part 1

For the simultaneous equations 2x\minus{}3y\equal{}8 6y\minus{}4x\equal{}9
<spanclass=′latex−bold′>(A)</span> x=4,y=0<spanclass=′latex−bold′>(B)</span> x=0,y=32<spanclass=′latex−bold′>(C)</span> x=0,y=0<spanclass=′latex−bold′>(D)</span> There is no solution<spanclass=′latex−bold′>(E)</span> There are an infinite number of solutions<span class='latex-bold'>(A)</span>\ x=4,y=0 \qquad <span class='latex-bold'>(B)</span>\ x=0,y=\dfrac{3}{2}\qquad <span class='latex-bold'>(C)</span>\ x=0,y=0 \qquad\\ <span class='latex-bold'>(D)</span>\ \text{There is no solution} \qquad <span class='latex-bold'>(E)</span>\ \text{There are an infinite number of solutions}
13
1

AHSME 1950- part 1

The roots of (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0 are:
<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 0 and 4<spanclass=′latex−bold′>(C)</span> 1 and 2<spanclass=′latex−bold′>(D)</span> 0,1,2 and 4<spanclass=′latex−bold′>(E)</span> 1,2 and 4<span class='latex-bold'>(A)</span>\ 4\qquad <span class='latex-bold'>(B)</span>\ 0\text{ and }4 \qquad <span class='latex-bold'>(C)</span>\ 1\text{ and }2 \qquad <span class='latex-bold'>(D)</span>\ 0,1,2\text{ and }4\qquad <span class='latex-bold'>(E)</span>\ 1,2\text{ and }4
12
1

AHSME 1950- part 1

As the number of sides of a polygon increases from 33 to n n, the sum of the exterior formed by extending each side in succession:
<spanclass=′latex−bold′>(A)</span> Increases<spanclass=′latex−bold′>(B)</span> Decreases<spanclass=′latex−bold′>(C)</span> Remains constant<spanclass=′latex−bold′>(D)</span> Cannot be predicted<spanclass=′latex−bold′>(E)</span> Becomes (n−3) straight angles<span class='latex-bold'>(A)</span>\ \text{Increases} \qquad <span class='latex-bold'>(B)</span>\ \text{Decreases} \qquad <span class='latex-bold'>(C)</span>\ \text{Remains constant} \qquad <span class='latex-bold'>(D)</span>\ \text{Cannot be predicted} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Becomes }(n-3)\text{ straight angles}
11
1

AHSME 1950- part 1

If in the formula C \equal{} \frac {en}{R\plus{}nr}, nn is increased while e e, RR and rr are kept constant, then CC:
<spanclass=′latex−bold′>(A)</span> Increases<spanclass=′latex−bold′>(B)</span> Decreases<spanclass=′latex−bold′>(C)</span> Remains constant<spanclass=′latex−bold′>(D)</span> Increases and then decreases<spanclass=′latex−bold′>(E)</span> Decreases and then increases<span class='latex-bold'>(A)</span>\ \text{Increases} \qquad <span class='latex-bold'>(B)</span>\ \text{Decreases} \qquad <span class='latex-bold'>(C)</span>\ \text{Remains constant} \qquad <span class='latex-bold'>(D)</span>\ \text{Increases and then decreases} \qquad\\ <span class='latex-bold'>(E)</span>\ \text{Decreases and then increases}
10
1

AHSME 1950- the original problems

After rationalizing the numerator of \frac {\sqrt{3}\minus{}\sqrt{2}}{\sqrt{3}}, the denominator in simplest form is:
<spanclass=′latex−bold′>(A)</span> 3(3+2)<spanclass=′latex−bold′>(B)</span> 3(3−2)<spanclass=′latex−bold′>(C)</span> 3−32<spanclass=′latex−bold′>(D)</span> 3+6<spanclass=′latex−bold′>(E)</span> None of these answers<span class='latex-bold'>(A)</span>\ \sqrt{3}(\sqrt{3}+\sqrt{2}) \qquad <span class='latex-bold'>(B)</span>\ \sqrt{3}(\sqrt{3}-\sqrt{2}) \qquad <span class='latex-bold'>(C)</span>\ 3-\sqrt{3}\sqrt{2} \qquad\\ <span class='latex-bold'>(D)</span>\ 3+\sqrt6 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these answers}
9
1

AHSME 1950- the original problems

The area of the largest triangle that can be inscribed in a semi-circle whose radius is rr is:
<spanclass=′latex−bold′>(A)</span> r2<spanclass=′latex−bold′>(B)</span> r3<spanclass=′latex−bold′>(C)</span> 2r2<spanclass=′latex−bold′>(D)</span> 2r3<spanclass=′latex−bold′>(E)</span> 12r2<span class='latex-bold'>(A)</span>\ r^2 \qquad <span class='latex-bold'>(B)</span>\ r^3 \qquad <span class='latex-bold'>(C)</span>\ 2r^2 \qquad <span class='latex-bold'>(D)</span>\ 2r^3 \qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{2}r^2
8
1

AHSME 1950- the original problems

If the radius of a circle is increased 100%100\%, the area is increased:
<spanclass=′latex−bold′>(A)</span> 100%<spanclass=′latex−bold′>(B)</span> 200%<spanclass=′latex−bold′>(C)</span> 300%<spanclass=′latex−bold′>(D)</span> 400%<spanclass=′latex−bold′>(E)</span> By none of these<span class='latex-bold'>(A)</span>\ 100\% \qquad <span class='latex-bold'>(B)</span>\ 200\% \qquad <span class='latex-bold'>(C)</span>\ 300\% \qquad <span class='latex-bold'>(D)</span>\ 400\% \qquad <span class='latex-bold'>(E)</span>\ \text{By none of these}
7
1

AHSME 1950- the original problems

If the digit 11 is placed after a two digit number whose tens' digit is tt, and units' digit is uu, the new number is:
<spanclass=′latex−bold′>(A)</span> 10t+u+1<spanclass=′latex−bold′>(B)</span> 100t+10u+1<spanclass=′latex−bold′>(C)</span> 100t+10u+1<spanclass=′latex−bold′>(D)</span> t+u+1<spanclass=′latex−bold′>(E)</span> None of these answers<span class='latex-bold'>(A)</span>\ 10t+u+1 \qquad <span class='latex-bold'>(B)</span>\ 100t+10u+1 \qquad <span class='latex-bold'>(C)</span>\ 100t+10u+1\qquad <span class='latex-bold'>(D)</span>\ t+u+1 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these answers}
6
1

AHSME 1950- the original problems

The values of y which will satisfy the equations 2x^2\plus{}6x\plus{}5y\plus{}1\equal{}0, 2x\plus{}y\plus{}3\equal{}0 may be found by solving:
<spanclass=′latex−bold′>(A)</span> y2+14y−7=0<spanclass=′latex−bold′>(B)</span> y2+8y+1=0<spanclass=′latex−bold′>(C)</span> y2+10y−7=0<spanclass=′latex−bold′>(D)</span> y2+y−12=0<spanclass=′latex−bold′>(E)</span> None of these equations<span class='latex-bold'>(A)</span>\ y^2+14y-7=0 \qquad <span class='latex-bold'>(B)</span>\ y^2+8y+1=0 \qquad <span class='latex-bold'>(C)</span>\ y^2+10y-7=0 \qquad <span class='latex-bold'>(D)</span>\ y^2+y-12=0 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these equations}
5
1

AHSME 1950- the original problems

If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series:
<spanclass=′latex−bold′>(A)</span> 648<spanclass=′latex−bold′>(B)</span> 832<spanclass=′latex−bold′>(C)</span> 1168<spanclass=′latex−bold′>(D)</span> 1944<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 648 \qquad <span class='latex-bold'>(B)</span>\ 832 \qquad <span class='latex-bold'>(C)</span>\ 1168 \qquad <span class='latex-bold'>(D)</span>\ 1944 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
4
1

AHSME 1950- the original problems

Reduced to lowest terms, \frac {a^2\minus{}b^2}{ab}\minus{} \frac {ab\minus{}b^2}{ab\minus{}a^2} is equal to:
<spanclass=′latex−bold′>(A)</span> ab<spanclass=′latex−bold′>(B)</span> a2−2b2ab<spanclass=′latex−bold′>(C)</span> a2<spanclass=′latex−bold′>(D)</span> a−2b<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ \dfrac{a}{b} \qquad <span class='latex-bold'>(B)</span>\ \dfrac{a^2-2b^2}{ab} \qquad <span class='latex-bold'>(C)</span>\ a^2 \qquad <span class='latex-bold'>(D)</span>\ a-2b \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
3
1

AHSME 1950- the original problems

The sum of the roots of the equation 4x^2\plus{}5\minus{}8x\equal{}0 is equal to:
<spanclass=′latex−bold′>(A)</span> 8<spanclass=′latex−bold′>(B)</span> −5<spanclass=′latex−bold′>(C)</span> −54<spanclass=′latex−bold′>(D)</span> −2<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 8 \qquad <span class='latex-bold'>(B)</span>\ -5 \qquad <span class='latex-bold'>(C)</span>\ -\dfrac{5}{4} \qquad <span class='latex-bold'>(D)</span>\ -2 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
2
1

AHSME 1950- the original problems

Let R=gS−4R=gS-4. When S=8S=8, R=16R=16. When S=10S=10, RR is equal to:
<spanclass=′latex−bold′>(A)</span> 11<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> None of these<span class='latex-bold'>(A)</span>\ 11 \qquad <span class='latex-bold'>(B)</span>\ 14 \qquad <span class='latex-bold'>(C)</span>\ 20\qquad <span class='latex-bold'>(D)</span>\ 21 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these}
1
1

AHSME 1950-the original problems

If 64 is divided into three parts proportional to 2, 4, and 6, the smallest part is:
<spanclass=′latex−bold′>(A)</span> 513<spanclass=′latex−bold′>(B)</span> 11<spanclass=′latex−bold′>(C)</span> 1023<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> None of these answers<span class='latex-bold'>(A)</span>\ 5\dfrac{1}{3} \qquad <span class='latex-bold'>(B)</span>\ 11 \qquad <span class='latex-bold'>(C)</span>\ 10\dfrac{2}{3} \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ \text{None of these answers}