MathDB

1959 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(50)
50
1

Making committees

A club with xx members is organized into four committees in accordance with these two rules: (1) Each member belongs to two and only two committees \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad (2) Each pair of committees has one and only one member in common\text{(2)}\ \text{Each pair of committees has one and only one member in common} Then xx: <spanclass=′latex−bold′>(A)</span> cannont be determined<span class='latex-bold'>(A)</span> \ \text{cannont be determined} \qquad <spanclass=′latex−bold′>(B)</span> has a single value between 8 and 16<span class='latex-bold'>(B)</span> \ \text{has a single value between 8 and 16} \qquad <spanclass=′latex−bold′>(C)</span> has two values between 8 and 16<span class='latex-bold'>(C)</span> \ \text{has two values between 8 and 16} \qquad <spanclass=′latex−bold′>(D)</span> has a single value between 4 and 8<span class='latex-bold'>(D)</span> \ \text{has a single value between 4 and 8} \qquad <spanclass=′latex−bold′>(E)</span> has two values between 4 and 8<span class='latex-bold'>(E)</span> \ \text{has two values between 4 and 8} \qquad
49
1

Finding an infinite geometric series (sorta)

For the infinite series 1−12−14+18−116−132+164−1128−⋯1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots let SS be the (limiting) sum. Then SS equals: <spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 27<spanclass=′latex−bold′>(C)</span> 67<spanclass=′latex−bold′>(D)</span> 932<spanclass=′latex−bold′>(E)</span> 2732 <span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ \frac27\qquad<span class='latex-bold'>(C)</span>\ \frac67\qquad<span class='latex-bold'>(D)</span>\ \frac{9}{32}\qquad<span class='latex-bold'>(E)</span>\ \frac{27}{32}
48
1

Counting number of polynomials

Given the polynomial a0xn+a1xn−1+⋯+an−1x+ana_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n, where nn is a positive integer or zero, and a0a_0 is a positive integer. The remaining aa's are integers or zero. Set h=n+a0+∣a1∣+∣a2∣+⋯+∣an∣h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|. [See example 25 for the meaning of ∣x∣|x|.] The number of polynomials with h=3h=3 is: <spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 6<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> 9 <span class='latex-bold'>(A)</span>\ 3\qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 6\qquad<span class='latex-bold'>(D)</span>\ 7\qquad<span class='latex-bold'>(E)</span>\ 9
47
1

Logical statements

Assume that the following three statements are true: II. All freshmen are human. IIII. All students are human. IIIIII. Some students think. Given the following four statements: <spanclass=′latex−bold′>(1)</span> All freshmen are students. <span class='latex-bold'>(1)</span>\ \text{All freshmen are students.}\qquad <spanclass=′latex−bold′>(2)</span> Some humans think.<span class='latex-bold'>(2)</span>\ \text{Some humans think.}\qquad <spanclass=′latex−bold′>(3)</span> No freshmen think.<span class='latex-bold'>(3)</span>\ \text{No freshmen think.}\qquad <spanclass=′latex−bold′>(4)</span> Some humans who think are not students.<span class='latex-bold'>(4)</span>\ \text{Some humans who think are not students.} Those which are logical consequences of I,II, and III are: <spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 2,3<spanclass=′latex−bold′>(D)</span> 2,4<spanclass=′latex−bold′>(E)</span> 1,2 <span class='latex-bold'>(A)</span>\ 2\qquad<span class='latex-bold'>(B)</span>\ 4\qquad<span class='latex-bold'>(C)</span>\ 2,3\qquad<span class='latex-bold'>(D)</span>\ 2,4\qquad<span class='latex-bold'>(E)</span>\ 1,2
46
1

Logic puzzle

A student on vacation for dd days observed that (1)(1) it rained 77 times, morning or afternoon (2)(2) when it rained in the afternoon, it was clear in the morning (3)(3) there were five clear afternoons (4)(4) there were six clear mornings. Then dd equals: <spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 7\qquad<span class='latex-bold'>(B)</span>\ 9\qquad<span class='latex-bold'>(C)</span>\ 10\qquad<span class='latex-bold'>(D)</span>\ 11\qquad<span class='latex-bold'>(E)</span>\ 12
45
1

Logarithmic equation

If (log⁡3x)(log⁡x2x)(log⁡2xy)=log⁡xx2\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2, then yy equals: <spanclass=′latex−bold′>(A)</span> 92<spanclass=′latex−bold′>(B)</span> 9<spanclass=′latex−bold′>(C)</span> 18<spanclass=′latex−bold′>(D)</span> 27<spanclass=′latex−bold′>(E)</span> 81 <span class='latex-bold'>(A)</span>\ \frac92\qquad<span class='latex-bold'>(B)</span>\ 9\qquad<span class='latex-bold'>(C)</span>\ 18\qquad<span class='latex-bold'>(D)</span>\ 27\qquad<span class='latex-bold'>(E)</span>\ 81
44
1

Roots of equation

The roots of x2+bx+c=0x^2+bx+c=0 are both real and greater than 11. Let s=b+c+1s=b+c+1. Then s:s: <spanclass=′latex−bold′>(A)</span> may be less than zero<spanclass=′latex−bold′>(B)</span> may be equal to zero <span class='latex-bold'>(A)</span>\ \text{may be less than zero}\qquad<span class='latex-bold'>(B)</span>\ \text{may be equal to zero}\qquad <spanclass=′latex−bold′>(C)</span> must be greater than zero<spanclass=′latex−bold′>(D)</span> must be less than zero<span class='latex-bold'>(C)</span>\ \text{must be greater than zero}\qquad<span class='latex-bold'>(D)</span>\ \text{must be less than zero}\qquad <spanclass=′latex−bold′>(E)</span> must be between -1 and +1<span class='latex-bold'>(E)</span>\text{ must be between -1 and +1}
43
1

Circumradius

The sides of a triangle are 25,39,25,39, and 4040. The diameter of the circumscribed circle is: <spanclass=′latex−bold′>(A)</span> 1333<spanclass=′latex−bold′>(B)</span> 1253<spanclass=′latex−bold′>(C)</span> 42<spanclass=′latex−bold′>(D)</span> 41<spanclass=′latex−bold′>(E)</span> 40 <span class='latex-bold'>(A)</span>\ \frac{133}{3}\qquad<span class='latex-bold'>(B)</span>\ \frac{125}{3}\qquad<span class='latex-bold'>(C)</span>\ 42\qquad<span class='latex-bold'>(D)</span>\ 41\qquad<span class='latex-bold'>(E)</span>\ 40
42
1

gcd(a,b,c)lcm(a,b,c)

Given three positive integers a,b,a,b, and cc. Their greatest common divisor is DD; their least common multiple is mm. Then, which two of the following statements are true? (1) the product MD cannot be less than abc \text{(1)}\ \text{the product MD cannot be less than abc} \qquad (2) the product MD cannot be greater than abc\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad (3) MD equals abc if and only if a,b,c are each prime\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad (4) MD equals abc if and only if a,b,c are each relatively prime in pairs\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}  (This means: no two have a common factor greater than 1.)\text{ (This means: no two have a common factor greater than 1.)} <spanclass=′latex−bold′>(A)</span> 1,2<spanclass=′latex−bold′>(B)</span> 1,3<spanclass=′latex−bold′>(C)</span> 1,4<spanclass=′latex−bold′>(D)</span> 2,3<spanclass=′latex−bold′>(E)</span> 2,4 <span class='latex-bold'>(A)</span>\ 1,2 \qquad<span class='latex-bold'>(B)</span>\ 1,3\qquad<span class='latex-bold'>(C)</span>\ 1,4\qquad<span class='latex-bold'>(D)</span>\ 2,3\qquad<span class='latex-bold'>(E)</span>\ 2,4
41
1

Constructing circles

On the same side of a straight line three circles are drawn as follows: a circle with a radius of 44 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: <spanclass=′latex−bold′>(A)</span> 24<spanclass=′latex−bold′>(B)</span> 20<spanclass=′latex−bold′>(C)</span> 18<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 24 \qquad<span class='latex-bold'>(B)</span>\ 20\qquad<span class='latex-bold'>(C)</span>\ 18\qquad<span class='latex-bold'>(D)</span>\ 16\qquad<span class='latex-bold'>(E)</span>\ 12
40
1

Triangle with medians

In triangle ABCABC, BDBD is a median. CFCF intersects BDBD at EE so that BE‾=ED‾\overline{BE}=\overline{ED}. Point FF is on ABAB. Then, if BF‾=5\overline{BF}=5, BA‾\overline{BA} equals: <spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 20<spanclass=′latex−bold′>(E)</span> none of these <span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 12\qquad<span class='latex-bold'>(C)</span>\ 15\qquad<span class='latex-bold'>(D)</span>\ 20\qquad<span class='latex-bold'>(E)</span>\ \text{none of these}
39
1

Sum of terms of a sequence

Let SS be the sum of the first nine terms of the sequence x+a,x2+2a,x3+3a,⋯ .x+a, x^2+2a, x^3+3a, \cdots. Then SS equals: <spanclass=′latex−bold′>(A)</span> 50a+x+x8x+1<spanclass=′latex−bold′>(B)</span> 50a−x+x10x−1<spanclass=′latex−bold′>(C)</span> x9−1x+1+45a <span class='latex-bold'>(A)</span>\ \frac{50a+x+x^8}{x+1} \qquad<span class='latex-bold'>(B)</span>\ 50a-\frac{x+x^{10}}{x-1}\qquad<span class='latex-bold'>(C)</span>\ \frac{x^9-1}{x+1}+45a\qquad<spanclass=′latex−bold′>(D)</span> x10−xx−1+45a<spanclass=′latex−bold′>(E)</span> x11−xx−1+45a<span class='latex-bold'>(D)</span>\ \frac{x^{10}-x}{x-1}+45a\qquad<span class='latex-bold'>(E)</span>\ \frac{x^{11}-x}{x-1}+45a
38
1

4x+sqrt(2x)=1

If 4x+2x=14x+\sqrt{2x}=1, then xx: <spanclass=′latex−bold′>(A)</span> is an integer<spanclass=′latex−bold′>(B)</span> is fractional<spanclass=′latex−bold′>(C)</span> is irrational<spanclass=′latex−bold′>(D)</span> is imaginary<spanclass=′latex−bold′>(E)</span> may have two different values <span class='latex-bold'>(A)</span>\ \text{is an integer} \qquad<span class='latex-bold'>(B)</span>\ \text{is fractional}\qquad<span class='latex-bold'>(C)</span>\ \text{is irrational}\qquad<span class='latex-bold'>(D)</span>\ \text{is imaginary}\qquad<span class='latex-bold'>(E)</span>\ \text{may have two different values}
37
1

Telescoping product

When simplified the product (1−13)(1−14)(1−15)⋯(1−1n)\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right) becomes: <spanclass=′latex−bold′>(A)</span> 1n<spanclass=′latex−bold′>(B)</span> 2n<spanclass=′latex−bold′>(C)</span> 2(n−1)n<spanclass=′latex−bold′>(D)</span> 2n(n+1)<spanclass=′latex−bold′>(E)</span> 3n(n+1) <span class='latex-bold'>(A)</span>\ \frac1n \qquad<span class='latex-bold'>(B)</span>\ \frac2n\qquad<span class='latex-bold'>(C)</span>\ \frac{2(n-1)}{n}\qquad<span class='latex-bold'>(D)</span>\ \frac{2}{n(n+1)}\qquad<span class='latex-bold'>(E)</span>\ \frac{3}{n(n+1)}
36
1

Law of cosines?

The base of a triangle is 8080, and one side of the base angle is 60∘60^\circ. The sum of the lengths of the other two sides is 9090. The shortest side is: <spanclass=′latex−bold′>(A)</span> 45<spanclass=′latex−bold′>(B)</span> 40<spanclass=′latex−bold′>(C)</span> 36<spanclass=′latex−bold′>(D)</span> 17<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 45 \qquad<span class='latex-bold'>(B)</span>\ 40\qquad<span class='latex-bold'>(C)</span>\ 36\qquad<span class='latex-bold'>(D)</span>\ 17\qquad<span class='latex-bold'>(E)</span>\ 12
35
1

Set of numbers satisfying equation

The symbol ≥\ge means "greater than or equal to"; the symbol ≤\le means "less than or equal to". In the equation (x−m)2−(x−n)2=(m−n)2(x-m)^2-(x-n)^2=(m-n)^2; m is a fixed positive number, and nn is a fixed negative number. The set of values xx satisfying the equation is: <spanclass=′latex−bold′>(A)</span> x≥0<spanclass=′latex−bold′>(B)</span> x≤n<spanclass=′latex−bold′>(C)</span> x=0<spanclass=′latex−bold′>(D)</span> the set of all real numbers<spanclass=′latex−bold′>(E)</span> none of these <span class='latex-bold'>(A)</span>\ x\ge 0 \qquad<span class='latex-bold'>(B)</span>\ x\le n\qquad<span class='latex-bold'>(C)</span>\ x=0\qquad<span class='latex-bold'>(D)</span>\ \text{the set of all real numbers}\qquad<span class='latex-bold'>(E)</span>\ \text{none of these}
34
1

Sum of squares of roots

Let the roots of x2−3x+1=0x^2-3x+1=0 be rr and ss. Then the expression r2+s2r^2+s^2 is:
<spanclass=′latex−bold′>(A)</span> a positive integer<spanclass=′latex−bold′>(B)</span> a positive fraction greater than 1<spanclass=′latex−bold′>(C)</span> a positive fraction less than 1 <span class='latex-bold'>(A)</span>\ \text{a positive integer} \qquad<span class='latex-bold'>(B)</span>\ \text{a positive fraction greater than 1}\qquad<span class='latex-bold'>(C)</span>\ \text{a positive fraction less than 1} <spanclass=′latex−bold′>(D)</span> an irrational number<spanclass=′latex−bold′>(E)</span> an imaginary number<span class='latex-bold'>(D)</span>\ \text{an irrational number}\qquad<span class='latex-bold'>(E)</span>\ \text{an imaginary number}
33
1

Harmonic progression

A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let SnS_n represent the sum of the first nn terms of the harmonic progression; for example S3S_3 represents the sum of the first three terms. If the first three terms of a harmonic progression are 3,4,63,4,6, then: <spanclass=′latex−bold′>(A)</span> S4=20<spanclass=′latex−bold′>(B)</span> S4=25<spanclass=′latex−bold′>(C)</span> S5=49<spanclass=′latex−bold′>(D)</span> S6=49<spanclass=′latex−bold′>(E)</span> S2=12S4 <span class='latex-bold'>(A)</span>\ S_4=20 \qquad<span class='latex-bold'>(B)</span>\ S_4=25\qquad<span class='latex-bold'>(C)</span>\ S_5=49\qquad<span class='latex-bold'>(D)</span>\ S_6=49\qquad<span class='latex-bold'>(E)</span>\ S_2=\frac12 S_4
32
1

LPower of a point time

The length ll of a tangent, drawn from a point AA to a circle, is 43\frac43 of the radius rr. The (shortest) distance from AA to the circle is: <spanclass=′latex−bold′>(A)</span> 12r<spanclass=′latex−bold′>(B)</span> r<spanclass=′latex−bold′>(C)</span> 12l<spanclass=′latex−bold′>(D)</span> 23l<spanclass=′latex−bold′>(E)</span> a value between r and l. <span class='latex-bold'>(A)</span>\ \frac{1}{2}r \qquad<span class='latex-bold'>(B)</span>\ r\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}l\qquad<span class='latex-bold'>(D)</span>\ \frac23l \qquad<span class='latex-bold'>(E)</span>\ \text{a value between r and l.}
31
1

Square inscribed in semicircle

A square, with an area of 4040, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is: <spanclass=′latex−bold′>(A)</span> 80<spanclass=′latex−bold′>(B)</span> 100<spanclass=′latex−bold′>(C)</span> 120<spanclass=′latex−bold′>(D)</span> 160<spanclass=′latex−bold′>(E)</span> 200 <span class='latex-bold'>(A)</span>\ 80 \qquad<span class='latex-bold'>(B)</span>\ 100\qquad<span class='latex-bold'>(C)</span>\ 120\qquad<span class='latex-bold'>(D)</span>\ 160\qquad<span class='latex-bold'>(E)</span>\ 200
30
1

A and B running around a track

AA can run around a circular track in 4040 seconds. BB, running in the opposite direction, meets AA every 1515 seconds. What is BB's time to run around the track, expressed in seconds? <spanclass=′latex−bold′>(A)</span> 1212<spanclass=′latex−bold′>(B)</span> 24<spanclass=′latex−bold′>(C)</span> 25<spanclass=′latex−bold′>(D)</span> 2712<spanclass=′latex−bold′>(E)</span> 55 <span class='latex-bold'>(A)</span>\ 12\frac12 \qquad<span class='latex-bold'>(B)</span>\ 24\qquad<span class='latex-bold'>(C)</span>\ 25\qquad<span class='latex-bold'>(D)</span>\ 27\frac12\qquad<span class='latex-bold'>(E)</span>\ 55
29
1

Solving a word question

On a examination of nn questions a student answers correctly 1515 of the first 2020. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is 50%50\%, how many different values of nn can there be? <spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> the problem cannot be solved <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3\qquad<span class='latex-bold'>(C)</span>\ 2\qquad<span class='latex-bold'>(D)</span>\ 1\qquad<span class='latex-bold'>(E)</span>\ \text{the problem cannot be solved}
28
1

Angle bisectors

In triangle ABCABC, ALAL bisects angle AA and CMCM bisects angle CC. Points LL and MM are on BCBC and ABAB, respectively. The sides of triangle ABCABC are a,b,a,b, and cc. Then AM‾MB‾=kCL‾LB‾\frac{\overline{AM}}{\overline{MB}}=k\frac{\overline{CL}}{\overline{LB}} where kk is: <spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> bca2<spanclass=′latex−bold′>(C)</span> a2bc<spanclass=′latex−bold′>(D)</span> cb<spanclass=′latex−bold′>(E)</span> ca <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ \frac{bc}{a^2}\qquad<span class='latex-bold'>(C)</span>\ \frac{a^2}{bc}\qquad<span class='latex-bold'>(D)</span>\ \frac{c}{b}\qquad<span class='latex-bold'>(E)</span>\ \frac{c}{a}
27
1

Which statement is not true

Which one of the following is not true for the equation ix2−x+2i=0,ix^2-x+2i=0, where i=−1i=\sqrt{-1}? <spanclass=′latex−bold′>(A)</span> The sum of the roots is 2 <span class='latex-bold'>(A)</span>\ \text{The sum of the roots is 2} \qquad <spanclass=′latex−bold′>(B)</span> The discriminant is 9<span class='latex-bold'>(B)</span>\ \text{The discriminant is 9}\qquad <spanclass=′latex−bold′>(C)</span> The roots are imaginary<span class='latex-bold'>(C)</span>\ \text{The roots are imaginary}\qquad <spanclass=′latex−bold′>(D)</span> The roots can be found using the quadratic formula<span class='latex-bold'>(D)</span>\ \text{The roots can be found using the quadratic formula}\qquad <spanclass=′latex−bold′>(E)</span> The roots can be found by factoring, using imaginary numbers<span class='latex-bold'>(E)</span>\ \text{The roots can be found by factoring, using imaginary numbers}
26
1

Triangle geometry again

The base of an isosceles triangle is 2\sqrt 2. The medians to the leg intersect each other at right angles. The area of the triangle is: <spanclass=′latex−bold′>(A)</span> 1.5<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 2.5<spanclass=′latex−bold′>(D)</span> 3.5<spanclass=′latex−bold′>(E)</span> 4 <span class='latex-bold'>(A)</span>\ 1.5 \qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 2.5\qquad<span class='latex-bold'>(D)</span>\ 3.5\qquad<span class='latex-bold'>(E)</span>\ 4
25
1

Absolute value equation

The symbol ∣a∣|a| means +a+a if aa is greater than or equal to zero, and −a-a if aa is less than or equal to zero; the symbol << means "less than"; the symbol >> means "greater than." The set of values xx satisfying the inequality ∣3−x∣<4|3-x|<4 consists of all xx such that: <spanclass=′latex−bold′>(A)</span> x2<49<spanclass=′latex−bold′>(B)</span> x2>1<spanclass=′latex−bold′>(C)</span> 1<x2<49<spanclass=′latex−bold′>(D)</span> −1<x<7<spanclass=′latex−bold′>(E)</span> −7<x<1 <span class='latex-bold'>(A)</span>\ x^2<49 \qquad<span class='latex-bold'>(B)</span>\ x^2>1 \qquad<span class='latex-bold'>(C)</span>\ 1<x^2<49\qquad<span class='latex-bold'>(D)</span>\ -1<x<7\qquad<span class='latex-bold'>(E)</span>\ -7<x<1
24
1

m ounces of salt that is m% salt

A chemist has mm ounces of salt that is m%m\% salt. How many ounces of salt must he add to make a solution that is 2m%2m\% salt?
<spanclass=′latex−bold′>(A)</span> m100+m<spanclass=′latex−bold′>(B)</span> 2m100−2m<spanclass=′latex−bold′>(C)</span> m2100−2m<spanclass=′latex−bold′>(D)</span> m2100+2m<spanclass=′latex−bold′>(E)</span> 2m100+2m <span class='latex-bold'>(A)</span>\ \frac{m}{100+m} \qquad<span class='latex-bold'>(B)</span>\ \frac{2m}{100-2m}\qquad<span class='latex-bold'>(C)</span>\ \frac{m^2}{100-2m}\qquad<span class='latex-bold'>(D)</span>\ \frac{m^2}{100+2m}\qquad<span class='latex-bold'>(E)</span>\ \frac{2m}{100+2m}
23
1

Solution set of log equation

The set of solutions of the equation log⁡10(a2−15a)=2\log_{10}\left( a^2-15a\right)=2 consists of <spanclass=′latex−bold′>(A)</span> two integers <spanclass=′latex−bold′>(B)</span> one integer and one fraction <span class='latex-bold'>(A)</span>\ \text{two integers } \qquad<span class='latex-bold'>(B)</span>\ \text{one integer and one fraction}\qquad <spanclass=′latex−bold′>(C)</span> two irrational numbers <spanclass=′latex−bold′>(D)</span> two non-real numbers<spanclass=′latex−bold′>(E)</span> no numbers, that is, the empty set<span class='latex-bold'>(C)</span>\ \text{two irrational numbers }\qquad<span class='latex-bold'>(D)</span>\ \text{two non-real numbers} \qquad<span class='latex-bold'>(E)</span>\ \text{no numbers, that is, the empty set}
22
1

Trapezoid problem

The line joining the midpoints of the diagonals of a trapezoid has length 33. If the longer base is 9797, then the shorter base is: <spanclass=′latex−bold′>(A)</span> 94<spanclass=′latex−bold′>(B)</span> 92<spanclass=′latex−bold′>(C)</span> 91<spanclass=′latex−bold′>(D)</span> 90<spanclass=′latex−bold′>(E)</span> 89 <span class='latex-bold'>(A)</span>\ 94 \qquad<span class='latex-bold'>(B)</span>\ 92\qquad<span class='latex-bold'>(C)</span>\ 91\qquad<span class='latex-bold'>(D)</span>\ 90\qquad<span class='latex-bold'>(E)</span>\ 89
21
1

Equilateral triangle inscribed in a circle

If pp is the perimeter of an equilateral triangle inscribed in a circle, the area of the circle is: <spanclass=′latex−bold′>(A)</span> πp23<spanclass=′latex−bold′>(B)</span> πp29<spanclass=′latex−bold′>(C)</span> πp227<spanclass=′latex−bold′>(D)</span> πp281<spanclass=′latex−bold′>(E)</span> πp2327 <span class='latex-bold'>(A)</span>\ \frac{\pi p^2}{3} \qquad<span class='latex-bold'>(B)</span>\ \frac{\pi p^2}{9}\qquad<span class='latex-bold'>(C)</span>\ \frac{\pi p^2}{27}\qquad<span class='latex-bold'>(D)</span>\ \frac{\pi p^2}{81} \qquad<span class='latex-bold'>(E)</span>\ \frac{\pi p^2 \sqrt3}{27}
20
1

Direct and inverse proportions

It is given that xx varies directly as yy and inversely as the square of zz, and that x=10x=10 when y=4y=4 and z=14z=14. Then, when y=16y=16 and z=7z=7, xx equals: <spanclass=′latex−bold′>(A)</span> 180<spanclass=′latex−bold′>(B)</span> 160<spanclass=′latex−bold′>(C)</span> 154<spanclass=′latex−bold′>(D)</span> 140<spanclass=′latex−bold′>(E)</span> 120 <span class='latex-bold'>(A)</span>\ 180\qquad<span class='latex-bold'>(B)</span>\ 160\qquad<span class='latex-bold'>(C)</span>\ 154\qquad<span class='latex-bold'>(D)</span>\ 140\qquad<span class='latex-bold'>(E)</span>\ 120
19
1

How many objects can be weighed

With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale? <spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 13<spanclass=′latex−bold′>(C)</span> 11<spanclass=′latex−bold′>(D)</span> 9<spanclass=′latex−bold′>(E)</span> 7 <span class='latex-bold'>(A)</span>\ 15 \qquad<span class='latex-bold'>(B)</span>\ 13\qquad<span class='latex-bold'>(C)</span>\ 11\qquad<span class='latex-bold'>(D)</span>\ 9\qquad<span class='latex-bold'>(E)</span>\ 7
18
1

Arithmetic mean of first n positive integers

The arithmetic mean (average) of the first nn positive integers is: <spanclass=′latex−bold′>(A)</span> n2<spanclass=′latex−bold′>(B)</span> n22<spanclass=′latex−bold′>(C)</span> n<spanclass=′latex−bold′>(D)</span> n−12<spanclass=′latex−bold′>(E)</span> n+12 <span class='latex-bold'>(A)</span>\ \frac{n}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac{n^2}{2}\qquad<span class='latex-bold'>(C)</span>\ n\qquad<span class='latex-bold'>(D)</span>\ \frac{n-1}{2}\qquad<span class='latex-bold'>(E)</span>\ \frac{n+1}{2}
17
1

System of linear equations

If y=a+bxy=a+\frac{b}{x}, where aa and bb are constants, and if y=1y=1 when x=−1x=-1, and y=5y=5 when x=−5x=-5, then a+ba+b equals: <spanclass=′latex−bold′>(A)</span> −1<spanclass=′latex−bold′>(B)</span> 0<spanclass=′latex−bold′>(C)</span> 1<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11 <span class='latex-bold'>(A)</span>\ -1 \qquad<span class='latex-bold'>(B)</span>\ 0\qquad<span class='latex-bold'>(C)</span>\ 1\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 11
16
1

Oh boy time to factor some quadratics

The expression x2−3x+2x2−5x+6÷x2−5x+4x2−7x+12\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}, when simplified is: <spanclass=′latex−bold′>(A)</span> (x−1)(x−6)(x−3)(x−4)<spanclass=′latex−bold′>(B)</span> x+3x−3<spanclass=′latex−bold′>(C)</span> x+1x−1<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> 2 <span class='latex-bold'>(A)</span>\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad<span class='latex-bold'>(B)</span>\ \frac{x+3}{x-3}\qquad<span class='latex-bold'>(C)</span>\ \frac{x+1}{x-1}\qquad<span class='latex-bold'>(D)</span>\ 1\qquad<span class='latex-bold'>(E)</span>\ 2
15
1

Pythagorean Theorem time

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: <spanclass=′latex−bold′>(A)</span> 15∘<spanclass=′latex−bold′>(B)</span> 30∘<spanclass=′latex−bold′>(C)</span> 45∘<spanclass=′latex−bold′>(D)</span> 60∘<spanclass=′latex−bold′>(E)</span> 75∘ <span class='latex-bold'>(A)</span>\ 15^{\circ} \qquad<span class='latex-bold'>(B)</span>\ 30^{\circ} \qquad<span class='latex-bold'>(C)</span>\ 45^{\circ} \qquad<span class='latex-bold'>(D)</span>\ 60^{\circ} \qquad<span class='latex-bold'>(E)</span>\ 75^{\circ}
14
1

A brief introduction to groups

Given the set SS 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 elements that only yield elements of SS are: <spanclass=′latex−bold′>(A)</span> all<spanclass=′latex−bold′>(B)</span> 1,2,3,4<spanclass=′latex−bold′>(C)</span> 1,2,3,5<spanclass=′latex−bold′>(D)</span> 1,2,3<spanclass=′latex−bold′>(E)</span> 1,3,5 <span class='latex-bold'>(A)</span>\ \text{all} \qquad<span class='latex-bold'>(B)</span>\ 1,2,3,4\qquad<span class='latex-bold'>(C)</span>\ 1,2,3,5\qquad<span class='latex-bold'>(D)</span>\ 1,2,3\qquad<span class='latex-bold'>(E)</span>\ 1,3,5
13
1

The classical arithmetic mean problem

The arithmetic mean (average) of a set of 5050 numbers is 3838. If two numbers, namely, 4545 and 5555, are discarded, the mean of the remaining set of numbers is: <spanclass=′latex−bold′>(A)</span> 36.5<spanclass=′latex−bold′>(B)</span> 37<spanclass=′latex−bold′>(C)</span> 37.2<spanclass=′latex−bold′>(D)</span> 37.5<spanclass=′latex−bold′>(E)</span> 37.52 <span class='latex-bold'>(A)</span>\ 36.5 \qquad<span class='latex-bold'>(B)</span>\ 37\qquad<span class='latex-bold'>(C)</span>\ 37.2\qquad<span class='latex-bold'>(D)</span>\ 37.5\qquad<span class='latex-bold'>(E)</span>\ 37.52
12
1

Adding constant to geometric series

By adding the same constant to 20,50,10020,50,100 a geometric progression results. The common ratio is: <spanclass=′latex−bold′>(A)</span> 53<spanclass=′latex−bold′>(B)</span> 43<spanclass=′latex−bold′>(C)</span> 32<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 13 <span class='latex-bold'>(A)</span>\ \frac53 \qquad<span class='latex-bold'>(B)</span>\ \frac43\qquad<span class='latex-bold'>(C)</span>\ \frac32\qquad<span class='latex-bold'>(D)</span>\ \frac12\qquad<span class='latex-bold'>(E)</span>\ \frac13
11
1
10
1

Equal areas again

In triangle ABCABC with AB‾=AC‾=3.6\overline{AB}=\overline{AC}=3.6, a point DD is taken on ABAB at a distance 1.21.2 from AA. Point DD is joined to EE in the prolongation of ACAC so that triangle AEDAED is equal in area to ABCABC. Then AE‾\overline{AE} is: <spanclass=′latex−bold′>(A)</span> 4.8<spanclass=′latex−bold′>(B)</span> 5.4<spanclass=′latex−bold′>(C)</span> 7.2<spanclass=′latex−bold′>(D)</span> 10.8<spanclass=′latex−bold′>(E)</span> 12.6 <span class='latex-bold'>(A)</span>\ 4.8 \qquad<span class='latex-bold'>(B)</span>\ 5.4\qquad<span class='latex-bold'>(C)</span>\ 7.2\qquad<span class='latex-bold'>(D)</span>\ 10.8\qquad<span class='latex-bold'>(E)</span>\ 12.6
9
1

The dreaded word problem

A farmer divides his herd of nn cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then nn is: <spanclass=′latex−bold′>(A)</span> 80<spanclass=′latex−bold′>(B)</span> 100<spanclass=′latex−bold′>(C)</span> 140<spanclass=′latex−bold′>(D)</span> 180<spanclass=′latex−bold′>(E)</span> 240 <span class='latex-bold'>(A)</span>\ 80 \qquad<span class='latex-bold'>(B)</span>\ 100\qquad<span class='latex-bold'>(C)</span>\ 140\qquad<span class='latex-bold'>(D)</span>\ 180\qquad<span class='latex-bold'>(E)</span>\ 240
8
1
7
1

Right triangle sides in arithmetic progression

The sides of a right triangle are a,a+d,a, a+d, and a+2da+2d, with aa and dd both positive. The ratio of aa to dd is: <spanclass=′latex−bold′>(A)</span> 1:3<spanclass=′latex−bold′>(B)</span> 1:4<spanclass=′latex−bold′>(C)</span> 2:1<spanclass=′latex−bold′>(D)</span> 3:1<spanclass=′latex−bold′>(E)</span> 3:4 <span class='latex-bold'>(A)</span>\ 1:3 \qquad<span class='latex-bold'>(B)</span>\ 1:4 \qquad<span class='latex-bold'>(C)</span>\ 2:1\qquad<span class='latex-bold'>(D)</span>\ 3:1\qquad<span class='latex-bold'>(E)</span>\ 3:4
6
1

Converse and Inverse of Statement

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 is: <spanclass=′latex−bold′>(A)</span> only the converse is true<spanclass=′latex−bold′>(B)</span> only the inverse is true <spanclass=′latex−bold′>(C)</span> both are true <span class='latex-bold'>(A)</span>\ \text{only the converse is true} \qquad<span class='latex-bold'>(B)</span>\ \text{only the inverse is true }\qquad <span class='latex-bold'>(C)</span>\ \text{both are true} \qquad <spanclass=′latex−bold′>(D)</span> neither is true<spanclass=′latex−bold′>(E)</span> the inverse is true, but the converse is sometimes true<span class='latex-bold'>(D)</span>\ \text{neither is true} \qquad<span class='latex-bold'>(E)</span>\ \text{the inverse is true, but the converse is sometimes true}
5
1

Calculating exponents to fractions

The value of (256).16(256).09\left(256\right)^{.16}\left(256\right)^{.09} is: <spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 64<spanclass=′latex−bold′>(D)</span> 256.25<spanclass=′latex−bold′>(E)</span> −16 <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 16\qquad<span class='latex-bold'>(C)</span>\ 64\qquad<span class='latex-bold'>(D)</span>\ 256.25\qquad<span class='latex-bold'>(E)</span>\ -16
4
1

78 divided into three parts

If 7878 is divided into three parts which are proportional to 1,13,161, \frac13, \frac16, the middle part is: <spanclass=′latex−bold′>(A)</span> 913<spanclass=′latex−bold′>(B)</span> 13<spanclass=′latex−bold′>(C)</span> 1713<spanclass=′latex−bold′>(D)</span> 1813<spanclass=′latex−bold′>(E)</span> 26 <span class='latex-bold'>(A)</span>\ 9\frac13 \qquad<span class='latex-bold'>(B)</span>\ 13\qquad<span class='latex-bold'>(C)</span>\ 17\frac13 \qquad<span class='latex-bold'>(D)</span>\ 18\frac13\qquad<span class='latex-bold'>(E)</span>\ 26
3
1

Two diagonals perpendicular

If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification: <spanclass=′latex−bold′>(A)</span> rhombus<spanclass=′latex−bold′>(B)</span> rectangles<spanclass=′latex−bold′>(C)</span> square<spanclass=′latex−bold′>(D)</span> isosceles trapezoid<spanclass=′latex−bold′>(E)</span> none of these <span class='latex-bold'>(A)</span>\ \text{rhombus} \qquad<span class='latex-bold'>(B)</span>\ \text{rectangles} \qquad<span class='latex-bold'>(C)</span>\ \text{square} \qquad<span class='latex-bold'>(D)</span>\ \text{isosceles trapezoid}\qquad<span class='latex-bold'>(E)</span>\ \text{none of these}
2
1

Distance from line parallel to base

Through a point PP inside the triangle ABCABC a line is drawn parallel to the base ABAB, dividing the triangle into two equal areas. If the altitude to ABAB has a length of 11, then the distance from PP to ABAB is: <spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 2−2<spanclass=′latex−bold′>(D)</span> 2−22<spanclass=′latex−bold′>(E)</span> 2+28 <span class='latex-bold'>(A)</span>\ \frac12 \qquad<span class='latex-bold'>(B)</span>\ \frac14\qquad<span class='latex-bold'>(C)</span>\ 2-\sqrt2\qquad<span class='latex-bold'>(D)</span>\ \frac{2-\sqrt2}{2}\qquad<span class='latex-bold'>(E)</span>\ \frac{2+\sqrt2}{8}
1
1

Increasing the side lengths of a cube

Each edge of a cube is increased by 50%50 \%. The percent of increase of the surface area of the cube is: <spanclass=′latex−bold′>(A)</span> 50<spanclass=′latex−bold′>(B)</span> 125<spanclass=′latex−bold′>(C)</span> 150<spanclass=′latex−bold′>(D)</span> 300<spanclass=′latex−bold′>(E)</span> 750 <span class='latex-bold'>(A)</span>\ 50 \qquad<span class='latex-bold'>(B)</span>\ 125\qquad<span class='latex-bold'>(C)</span>\ 150\qquad<span class='latex-bold'>(D)</span>\ 300\qquad<span class='latex-bold'>(E)</span>\ 750