MathDB

1964 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(40)
40
1

Broken Watch

A watch loses 2122\frac{1}{2} minutes per day. It is set right at 11 P.M. on March 15. Let nn be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows 99 A.M. on March 21, nn equals:
<spanclass=′latex−bold′>(A)</span>141423<spanclass=′latex−bold′>(B)</span>14114<spanclass=′latex−bold′>(C)</span>13101115<spanclass=′latex−bold′>(D)</span>1383115<spanclass=′latex−bold′>(E)</span>131323<span class='latex-bold'>(A) </span>14\frac{14}{23}\qquad<span class='latex-bold'>(B) </span>14\frac{1}{14}\qquad<span class='latex-bold'>(C) </span>13\frac{101}{115}\qquad<span class='latex-bold'>(D) </span>13\frac{83}{115}\qquad <span class='latex-bold'>(E) </span>13\frac{13}{23}
39
1

Magnitudes of triangle ABC

The magnitudes of the sides of triangle ABCABC are aa, bb, and cc, as shown, with c≤b≤ac\le b\le a. Through interior point PP and the vertices AA, BB, CC, lines are drawn meeting the opposite sides in A′A', B′B', C′C', respectively. Let s=AA′+BB′+CC′s=AA'+BB'+CC'. Then, for all positions of point PP, ss is less than:
<spanclass=′latex−bold′>(A)</span>2a+b<spanclass=′latex−bold′>(B)</span>2a+c<spanclass=′latex−bold′>(C)</span>2b+c<spanclass=′latex−bold′>(D)</span>a+2b<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(A) </span>2a+b\qquad<span class='latex-bold'>(B) </span>2a+c\qquad<span class='latex-bold'>(C) </span>2b+c\qquad<span class='latex-bold'>(D) </span>a+2b\qquad <span class='latex-bold'>(E) </span> a+b+ca+b+c
[asy] import math; defaultpen(fontsize(11pt)); pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1);
pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P);
draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("AA",A,dir(210)); label("BB",B,dir(90)); label("CC",C,dir(-30)); label("A′A'",X,dir(-100)); label("B′B'",Y,dir(65)); label("C′C'",Z,dir(20)); label("PP",P,dir(70)); label("aa",X,dir(80)); label("bb",Y,dir(-90)); label("cc",Z,dir(110)); //Credit to bobthesmartypants for the diagram [/asy]
38
1

Missing Side Length of a Triangle

The sides PQPQ and PRPR of triangle PQRPQR are respectively of lengths 44 inches, and 77 inches. The median PMPM is 3123\frac{1}{2} inches. Then QRQR, in inches, is:
<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>7<spanclass=′latex−bold′>(C)</span>8<spanclass=′latex−bold′>(D)</span>9<spanclass=′latex−bold′>(E)</span>10<span class='latex-bold'>(A) </span>6\qquad<span class='latex-bold'>(B) </span>7\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>9\qquad <span class='latex-bold'>(E) </span>10
37
1

Difference between AM and GM

Given two positive number aa, bb such that a<ba<b. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
<spanclass=′latex−bold′>(A)</span>(b+a)2ab<spanclass=′latex−bold′>(B)</span>(b+a)28b<spanclass=′latex−bold′>(C)</span>(b−a)2ab<span class='latex-bold'>(A) </span>\dfrac{(b+a)^2}{ab}\qquad<span class='latex-bold'>(B) </span>\dfrac{(b+a)^2}{8b}\qquad<span class='latex-bold'>(C) </span>\dfrac{(b-a)^2}{ab}
<spanclass=′latex−bold′>(D)</span>(b−a)28a<spanclass=′latex−bold′>(E)</span>(b−a)28b<span class='latex-bold'>(D) </span>\dfrac{(b-a)^2}{8a}\qquad <span class='latex-bold'>(E) </span>\dfrac{(b-a)^2}{8b}
36
1

Permissible positions of a circle

In this figure the radius of the circle is equal to the altitude of the equilateral triangle ABCABC. The circle is made to roll along the side ABAB, remaining tangent to it at a variable point TT and intersecting lines ACAC and BCBC in variable points MM and NN, respectively. Let nn be the number of degrees in arc MTNMTN. Then nn, for all permissible positions of the circle:
<spanclass=′latex−bold′>(A)</span>varies from 30∘ to 90∘<span class='latex-bold'>(A) </span>\text{varies from }30^{\circ}\text{ to }90^{\circ}
<spanclass=′latex−bold′>(B)</span>varies from 30∘ to 60∘<span class='latex-bold'>(B) </span>\text{varies from }30^{\circ}\text{ to }60^{\circ}
<spanclass=′latex−bold′>(C)</span>varies from 60∘ to 90∘<span class='latex-bold'>(C) </span>\text{varies from }60^{\circ}\text{ to }90^{\circ}
<spanclass=′latex−bold′>(D)</span>remains constant at 30∘<span class='latex-bold'>(D) </span>\text{remains constant at }30^{\circ}
<spanclass=′latex−bold′>(E)</span>remains constant at 60∘<span class='latex-bold'>(E) </span>\text{remains constant at }60^{\circ}
[asy] pair A = (0,0), B = (1,0), C = dir(60), T = (2/3,0); pair M = intersectionpoint(A--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)), N = intersectionpoint(B--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2));
draw((0,0)--(1,0)--dir(60)--cycle); draw(Circle((2/3,sqrt(3)/2),sqrt(3)/2)); label("AA",A,dir(210)); label("BB",B,dir(-30)); label("CC",C,dir(90)); label("MM",M,dir(190)); label("NN",N,dir(75)); label("TT",T,dir(-90)); //Credit to bobthesmartypants for the diagram [/asy]
33
1

Find the missing length, PB

PP is a point interior to rectangle ABCDABCD and such that PA=3PA=3 inches, PD=4PD=4 inches, and PC=5PC=5 inches. Then PBPB, in inches, equals:
<spanclass=′latex−bold′>(A)</span>23<spanclass=′latex−bold′>(B)</span>32<spanclass=′latex−bold′>(C)</span>33<spanclass=′latex−bold′>(D)</span>42<spanclass=′latex−bold′>(E)</span>2<span class='latex-bold'>(A) </span>2\sqrt{3}\qquad<span class='latex-bold'>(B) </span>3\sqrt{2}\qquad<span class='latex-bold'>(C) </span>3\sqrt{3}\qquad<span class='latex-bold'>(D) </span>4\sqrt{2}\qquad <span class='latex-bold'>(E) </span>2
[asy] draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle); draw((2.5,1.5)--(0,0)); draw((2.5,1.5)--(0,4.5)); draw((2.5,1.5)--(6.5,4.5)); draw((2.5,1.5)--(6.5,0),linetype("8 8")); label("AA",(0,0),dir(-135)); label("BB",(6.5,0),dir(-45)); label("CC",(6.5,4.5),dir(45)); label("DD",(0,4.5),dir(135)); label("PP",(2.5,1.5),dir(-90)); label("33",(1.25,0.75),dir(120)); label("44",(1.25,3),dir(35)); label("55",(4.5,3),dir(120)); //Credit to bobthesmartypants for the diagram [/asy]
32
1

Deduction from a fraction

If a+bb+c=c+dd+a\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}, then:
<spanclass=′latex−bold′>(A)</span> a must equal c<spanclass=′latex−bold′>(B)</span> a+b+c+d must equal zero <span class='latex-bold'>(A)</span>\ a\text{ must equal }c \qquad <span class='latex-bold'>(B)</span>\ a+b+c+d\text{ must equal zero }\qquad
<spanclass=′latex−bold′>(C)</span> either a=c or a+b+c+d=0, or both <span class='latex-bold'>(C)</span>\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad
<spanclass=′latex−bold′>(D)</span> a+b+c+d≠0 if a=c<spanclass=′latex−bold′>(E)</span> a(b+c+d)=c(a+b+d) <span class='latex-bold'>(D)</span>\ a+b+c+d\neq 0\text{ if }a=c \qquad <span class='latex-bold'>(E)</span>\ a(b+c+d)=c(a+b+d)
31
1

Function f(n)

Let f(n)=5+3510(1+52)n+5−3510(1−52)n.f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n. Then f(n+1)−f(n−1)f(n+1)-f(n-1), expressed in terms of f(n)f(n), equals:
<spanclass=′latex−bold′>(A)</span> 12f(n)<spanclass=′latex−bold′>(B)</span> f(n)<spanclass=′latex−bold′>(C)</span> 2f(n)+1<spanclass=′latex−bold′>(D)</span> f2(n)<spanclass=′latex−bold′>(E)</span> 12(f2(n)−1)<span class='latex-bold'>(A)</span>\ \dfrac{1}{2}f(n) \qquad <span class='latex-bold'>(B)</span>\ f(n)\qquad <span class='latex-bold'>(C)</span>\ 2f(n)+1 \qquad <span class='latex-bold'>(D)</span>\ f^2(n) \qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{2}(f^2(n)-1)
29
1

Application of SAS Similarity

In this figure ∠RFS=∠FDR\angle RFS = \angle FDR, FD=4FD = 4 inches, DR=6DR = 6 inches, FR=5FR = 5 inches, FS=712FS = 7\dfrac{1}{2} inches. The length of RSRS, in inches, is:
[asy] import olympiad; pair F,R,S,D; F=origin; R=5*dir(aCos(9/16)); S=(7.5,0); D=4*dir(aCos(9/16)+aCos(1/8)); label("FF",F,SW);label("RR",R,N); label("SS",S,SE); label("DD",D,W); label("7127\frac{1}{2}",(F+S)/2.5,SE); label("44",midpoint(F--D),SW); label("55",midpoint(F--R),W); label("66",midpoint(D--R),N); draw(F--D--R--F--S--R); markscalefactor=0.1; draw(anglemark(S,F,R)); draw(anglemark(F,D,R)); //Credit to throwaway1489 for the diagram[/asy]

<spanclass=′latex−bold′>(A)</span> undetermined<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 512<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 614<span class='latex-bold'>(A)</span>\ \text{undetermined} \qquad <span class='latex-bold'>(B)</span>\ 4\qquad <span class='latex-bold'>(C)</span>\ 5\dfrac{1}{2} \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 6\dfrac{1}{4}
34
1

Imaginary i

If nn is a multiple of 44, the sum s=1+2i+3i2+...+(n+1)ins=1+2i+3i^2+ ... +(n+1)i^{n}, where i=−1i=\sqrt{-1}, equals:
<spanclass=′latex−bold′>(A)</span> 1+i<spanclass=′latex−bold′>(B)</span> 12(n+2)<spanclass=′latex−bold′>(C)</span> 12(n+2−ni) <span class='latex-bold'>(A)</span>\ 1+i\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2}(n+2) \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}(n+2-ni) \qquad
<spanclass=′latex−bold′>(D)</span> 12[(n+1)(1−i)+2]<spanclass=′latex−bold′>(E)</span> 18(n2+8−4ni) <span class='latex-bold'>(D)</span>\ \frac{1}{2}[(n+1)(1-i)+2]\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{8}(n^2+8-4ni)
30
1

Dealing with roots

If (7+43)x2+(2+3)x−2=0(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0, the larger root minus the smaller root is:
<spanclass=′latex−bold′>(A)</span> −2+33<spanclass=′latex−bold′>(B)</span> 2−3<spanclass=′latex−bold′>(C)</span> 6+33<spanclass=′latex−bold′>(D)</span> 6−33<spanclass=′latex−bold′>(E)</span> 33+2 <span class='latex-bold'>(A)</span>\ -2+3\sqrt{3}\qquad<span class='latex-bold'>(B)</span>\ 2-\sqrt{3}\qquad<span class='latex-bold'>(C)</span>\ 6+3\sqrt{3}\qquad<span class='latex-bold'>(D)</span>\ 6-3\sqrt{3}\qquad<span class='latex-bold'>(E)</span>\ 3\sqrt{3}+2
28
1

Sum Arithmetic Progression

The sum of nn terms of an arithmetic progression is 153153, and the common difference is 22. If the first interm is an integer, and n>1n>1, then the number of possible values for nn is:
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 <span class='latex-bold'>(A)</span>\ 2\qquad<span class='latex-bold'>(B)</span>\ 3\qquad<span class='latex-bold'>(C)</span>\ 4\qquad<span class='latex-bold'>(D)</span>\ 5\qquad<span class='latex-bold'>(E)</span>\ 6
27
1

Absolute Values

If xx is a real number and ∣x−4∣+∣x−3∣<a|x-4|+|x-3|<a where a>0a>0, then:
<spanclass=′latex−bold′>(A)</span> 0<a<.01<spanclass=′latex−bold′>(B)</span> .01<a<1<spanclass=′latex−bold′>(C)</span> 0<a<1 <span class='latex-bold'>(A)</span>\ 0<a<.01\qquad<span class='latex-bold'>(B)</span>\ .01<a<1 \qquad<span class='latex-bold'>(C)</span>\ 0<a<1\qquad
<spanclass=′latex−bold′>(D)</span> 0<a≤1<spanclass=′latex−bold′>(E)</span> a>1<span class='latex-bold'>(D)</span>\ 0<a \le 1\qquad<span class='latex-bold'>(E)</span>\ a>1
26
1

A Race

In a ten-mile race First beats Second by 22 miles and First beats Third by 44 miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 214<spanclass=′latex−bold′>(C)</span> 212<spanclass=′latex−bold′>(D)</span> 234<spanclass=′latex−bold′>(E)</span> 3 <span class='latex-bold'>(A)</span>\ 2\qquad<span class='latex-bold'>(B)</span>\ 2\frac{1}{4}\qquad<span class='latex-bold'>(C)</span>\ 2\frac{1}{2}\qquad<span class='latex-bold'>(D)</span>\ 2\frac{3}{4}\qquad<span class='latex-bold'>(E)</span>\ 3
25
1

Set values of m

The set of values of mm for which x2+3xy+x+my−mx^2+3xy+x+my-m has two factors, with integer coefficients, which are linear in xx and yy, is precisely:
<spanclass=′latex−bold′>(A)</span> 0,12,−12<spanclass=′latex−bold′>(B)</span> 0,12<spanclass=′latex−bold′>(C)</span> 12,−12<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 0 <span class='latex-bold'>(A)</span>\ 0, 12, -12\qquad<span class='latex-bold'>(B)</span>\ 0, 12\qquad<span class='latex-bold'>(C)</span>\ 12, -12\qquad<span class='latex-bold'>(D)</span>\ 12\qquad<span class='latex-bold'>(E)</span>\ 0
24
1

Minimizing y

Let y=(x−a)2+(x−b)2,a,by=(x-a)^2+(x-b)^2, a, b constants. For what value of xx is yy a minimum?
<spanclass=′latex−bold′>(A)</span> a+b2<spanclass=′latex−bold′>(B)</span> a+b<spanclass=′latex−bold′>(C)</span> ab<spanclass=′latex−bold′>(D)</span> a2+b22<spanclass=′latex−bold′>(E)</span> a+b2ab <span class='latex-bold'>(A)</span>\ \frac{a+b}{2} \qquad<span class='latex-bold'>(B)</span>\ a+b \qquad<span class='latex-bold'>(C)</span>\ \sqrt{ab} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{\frac{a^2+b^2}{2}}\qquad<span class='latex-bold'>(E)</span>\ \frac{a+b}{2ab}
23
1

Difference, Sum, Product

Two numbers are such that their difference, their sum, and their product are to one another as 1:7:241:7:24. The product of the two numbers is:
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 48<spanclass=′latex−bold′>(E)</span> 96 <span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 12\qquad<span class='latex-bold'>(C)</span>\ 24\qquad<span class='latex-bold'>(D)</span>\ 48\qquad<span class='latex-bold'>(E)</span>\ 96
22
1

Parrallelogram =&gt; Quadrilateral

Given parallelogram ABCDABCD with EE the midpoint of diagonal BDBD. Point EE is connected to a point FF in DADA so that DF=13DADF=\frac{1}{3}DA. What is the ratio of the area of triangle DFEDFE to the area of quadrilateral ABEFABEF?
<spanclass=′latex−bold′>(A)</span> 1:2<spanclass=′latex−bold′>(B)</span> 1:3<spanclass=′latex−bold′>(C)</span> 1:5<spanclass=′latex−bold′>(D)</span> 1:6<spanclass=′latex−bold′>(E)</span> 1:7 <span class='latex-bold'>(A)</span>\ 1:2 \qquad<span class='latex-bold'>(B)</span>\ 1:3 \qquad<span class='latex-bold'>(C)</span>\ 1:5 \qquad<span class='latex-bold'>(D)</span>\ 1:6 \qquad<span class='latex-bold'>(E)</span>\ 1:7
21
1

Logs

If log⁡b2x+log⁡x2b=1,b>0,b≠1,x≠1\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1, then xx equals:
<spanclass=′latex−bold′>(A)</span> 1/b2<spanclass=′latex−bold′>(B)</span> 1/b<spanclass=′latex−bold′>(C)</span> b2<spanclass=′latex−bold′>(D)</span> b<spanclass=′latex−bold′>(E)</span> b <span class='latex-bold'>(A)</span>\ 1/b^2 \qquad<span class='latex-bold'>(B)</span>\ 1/b \qquad<span class='latex-bold'>(C)</span>\ b^2 \qquad<span class='latex-bold'>(D)</span>\ b \qquad<span class='latex-bold'>(E)</span>\ \sqrt{b}
20
1

Sum of coefficients

The sum of the numerical coefficients of all the terms in the expansion of (x−2y)18(x-2y)^{18} is:
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 19<spanclass=′latex−bold′>(D)</span> −1<spanclass=′latex−bold′>(E)</span> −19 <span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 1\qquad<span class='latex-bold'>(C)</span>\ 19\qquad<span class='latex-bold'>(D)</span>\ -1\qquad<span class='latex-bold'>(E)</span>\ -19
19
1

Value of fraction

If 2x−3y−z=02x-3y-z=0 and x+3y−14z=0x+3y-14z=0, z≠0z \neq 0, the numerical value of x2+3xyy2+z2\frac{x^2+3xy}{y^2+z^2} is:
<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> −20/17<spanclass=′latex−bold′>(E)</span> −2 <span class='latex-bold'>(A)</span>\ 7\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 0\qquad<span class='latex-bold'>(D)</span>\ -20/17\qquad<span class='latex-bold'>(E)</span>\ -2
18
1

Same Graph

Let nn be the number of pairs of values of bb and cc such that 3x+by+c=03x+by+c=0 and cx−2y+12=0cx-2y+12=0 have the same graph. Then nn is:
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> finite but more than 2<spanclass=′latex−bold′>(E)</span> greater than any finite number <span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 1\qquad<span class='latex-bold'>(C)</span>\ 2\qquad<span class='latex-bold'>(D)</span>\ \text{finite but more than 2}\qquad<span class='latex-bold'>(E)</span>\ \text{greater than any finite number}
17
1

Points P, Q, R

Given the distinct points P(x1,y1)P(x_1, y_1), Q(x2,y2)Q(x_2, y_2) and R(x1+x2,y1+y2)R(x_1+x_2, y_1+y_2). Line segments are drawn connecting these points to each other and to the origin 00. Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure OPRQOPRQ, depending upon the location of the points P,Q,P, Q, and RR, can be:
<spanclass=′latex−bold′>(A)</span> (1) only<spanclass=′latex−bold′>(B)</span> (2) only<spanclass=′latex−bold′>(C)</span> (3) only<spanclass=′latex−bold′>(D)</span> (1) or (2) only<spanclass=′latex−bold′>(E)</span> all three <span class='latex-bold'>(A)</span>\ \text{(1) only}\qquad<span class='latex-bold'>(B)</span>\ \text{(2) only}\qquad<span class='latex-bold'>(C)</span>\ \text{(3) only}\qquad<span class='latex-bold'>(D)</span>\ \text{(1) or (2) only}\qquad<span class='latex-bold'>(E)</span>\ \text{all three}
16
1

Set of Integers

Let f(x)=x2+3x+2f(x)=x^2+3x+2 and let SS be the set of integers {0,1,2,…,25}\{0, 1, 2, \dots , 25 \}. The number of members ss of SS such that f(s)f(s) has remainder zero when divided by 6 is:
<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 22<spanclass=′latex−bold′>(C)</span> 21<spanclass=′latex−bold′>(D)</span> 18<spanclass=′latex−bold′>(E)</span> 17{{ <span class='latex-bold'>(A)</span>\ 25\qquad<span class='latex-bold'>(B)</span>\ 22\qquad<span class='latex-bold'>(C)</span>\ 21\qquad<span class='latex-bold'>(D)</span>\ 18 }\qquad<span class='latex-bold'>(E)</span>\ 17 }
15
1

Line Cutting From Second Quadrant

A line through the point (−a,0)(-a,0) cuts from the second quadrant a triangular region with area TT. The equation of the line is:
<spanclass=′latex−bold′>(A)</span> 2Tx+a2y+2aT=0<spanclass=′latex−bold′>(B)</span> 2Tx−a2y+2aT=0 <span class='latex-bold'>(A)</span>\ 2Tx+a^2y+2aT=0 \qquad<span class='latex-bold'>(B)</span>\ 2Tx-a^2y+2aT=0 \qquad <spanclass=′latex−bold′>(C)</span> 2Tx+a2y−2aT=0<spanclass=′latex−bold′>(D)</span> 2Tx−a2y−2aT=0<spanclass=′latex−bold′>(E)</span> none of these{{<span class='latex-bold'>(C)</span>\ 2Tx+a^2y-2aT=0 \qquad<span class='latex-bold'>(D)</span>\ 2Tx-a^2y-2aT=0 }\qquad<span class='latex-bold'>(E)</span>\ \text{none of these} }
14
1

SHEEP!!!

A farmer bought 749749 sheeps. He sold 700700 of them for the price paid for the 749749 sheep. The remaining 4949 sheep were sold at the same price per head as the other 700700. Based on the cost, the percent gain on the entire transaction is:
<spanclass=′latex−bold′>(A)</span> 6.5<spanclass=′latex−bold′>(B)</span> 6.75<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 7.5<spanclass=′latex−bold′>(E)</span> 8{{ <span class='latex-bold'>(A)</span>\ 6.5 \qquad<span class='latex-bold'>(B)</span>\ 6.75 \qquad<span class='latex-bold'>(C)</span>\ 7 \qquad<span class='latex-bold'>(D)</span>\ 7.5 }\qquad<span class='latex-bold'>(E)</span>\ 8 }
13
1

Circle Inscribed In Circle

A circle is inscribed in a triangle with side lengths 88, 1313, and 1717. Let the segments of the side of length 88, made by a point of tangency, be rr and ss, with r<sr<s. What is the ratio r:sr:s?
<spanclass=′latex−bold′>(A)</span> 1:3<spanclass=′latex−bold′>(B)</span> 2:5<spanclass=′latex−bold′>(C)</span> 1:2<spanclass=′latex−bold′>(D)</span> 2:3<spanclass=′latex−bold′>(E)</span> 3:4{{ <span class='latex-bold'>(A)</span>\ 1:3 \qquad<span class='latex-bold'>(B)</span>\ 2:5 \qquad<span class='latex-bold'>(C)</span>\ 1:2 \qquad<span class='latex-bold'>(D)</span>\ 2:3 }\qquad<span class='latex-bold'>(E)</span>\ 3:4 }
12
1

Negation of Statement

Which of the following is the negation of the statement: For all xx of a certain set, x2>0x^2>0?
<spanclass=′latex−bold′>(A)</span> For all x,x2<0 <span class='latex-bold'>(A)</span>\ \text{For all x}, x^2 < 0\qquad <spanclass=′latex−bold′>(B)</span> For all x,x2≤0<span class='latex-bold'>(B)</span>\ \text{For all x}, x^2 \le 0\qquad <spanclass=′latex−bold′>(C)</span> For no x,x2>0<span class='latex-bold'>(C)</span>\ \text{For no x}, x^2>0\qquad <spanclass=′latex−bold′>(D)</span> For some x,x2>0{<span class='latex-bold'>(D)</span>\ \text{For some x}, x^2>0 }\qquad <spanclass=′latex−bold′>(E)</span> For some x,x2≤0{{<span class='latex-bold'>(E)</span>\ \text{For some x}, x^2 \le 0}}
11
1
10
1

Square and Triangle

Given a square side of length ss. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:
<spanclass=′latex−bold′>(A)</span> s2<spanclass=′latex−bold′>(B)</span> s/2<spanclass=′latex−bold′>(C)</span> 2s<spanclass=′latex−bold′>(D)</span> 2s<spanclass=′latex−bold′>(E)</span> 2/s{{ <span class='latex-bold'>(A)</span>\ s\sqrt{2} \qquad<span class='latex-bold'>(B)</span>\ s/\sqrt{2} \qquad<span class='latex-bold'>(C)</span>\ 2s \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt{s} }\qquad<span class='latex-bold'>(E)</span>\ 2/ \sqrt{s} }
9
1

A Jobber

A jobber buys an article at $24\$24 less 1212%12\frac{1}{2}\%. He then wishes to sell the article at a gain of 3313%33\frac{1}{3}\% of his cost after allowing a 20%20\% discount on his marked price. At what price, in dollars, should the article be marked?
<spanclass=′latex−bold′>(A)</span> 25.20<spanclass=′latex−bold′>(B)</span> 30.00<spanclass=′latex−bold′>(C)</span> 33.60<spanclass=′latex−bold′>(D)</span> 40.00<spanclass=′latex−bold′>(E)</span> none of these{{ <span class='latex-bold'>(A)</span>\ 25.20 \qquad<span class='latex-bold'>(B)</span>\ 30.00 \qquad<span class='latex-bold'>(C)</span>\ 33.60 \qquad<span class='latex-bold'>(D)</span>\ 40.00 }\qquad<span class='latex-bold'>(E)</span>\ \text{none of these} }
8
1

Smaller Root of Equation

The smaller root of the equation (x−34)(x−34)+(x−34)(x−12)=0 \left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0 is:
<spanclass=′latex−bold′>(A)</span> −34<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 58<spanclass=′latex−bold′>(D)</span> 34<spanclass=′latex−bold′>(E)</span> 1{{ <span class='latex-bold'>(A)</span>\ -\frac{3}{4}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2}\qquad<span class='latex-bold'>(C)</span>\ \frac{5}{8}\qquad<span class='latex-bold'>(D)</span>\ \frac{3}{4} }\qquad<span class='latex-bold'>(E)</span>\ 1 }
7
1

Find the value of n

Let nn be the number of real values of pp for which the roots of x2−px+p=0 x^2-px+p=0 are equal. Then nn equals:
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> a finite number greater than 2<spanclass=′latex−bold′>(E)</span> an infinitely large number{{ <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ \text{a finite number greater than 2} }\qquad<span class='latex-bold'>(E)</span>\ \text{an infinitely large number} }
6
1

Geometric Progression

If x,2x+2,3x+3,…x, 2x+2, 3x+3, \dots are in geometric progression, the fourth term is:
<spanclass=′latex−bold′>(A)</span> −27<spanclass=′latex−bold′>(B)</span> −1312<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 1312<spanclass=′latex−bold′>(E)</span> 27{{ <span class='latex-bold'>(A)</span>\ -27 \qquad<span class='latex-bold'>(B)</span>\ -13\frac{1}{2} \qquad<span class='latex-bold'>(C)</span>\ 12\qquad<span class='latex-bold'>(D)</span>\ 13\frac{1}{2} }\qquad<span class='latex-bold'>(E)</span>\ 27 }
5
1

Varies Directly...

If yy varies directly as xx, and if y=8y=8 when x=4x=4, the value of yy when x=−8x=-8 is:
<spanclass=′latex−bold′>(A)</span> −16<spanclass=′latex−bold′>(B)</span> −4<spanclass=′latex−bold′>(C)</span> −2<spanclass=′latex−bold′>(D)</span> 4k,k=±1,±2,…{{ <span class='latex-bold'>(A)</span>\ -16} \qquad<span class='latex-bold'>(B)</span>\ -4 \qquad<span class='latex-bold'>(C)</span>\ -2 \qquad<span class='latex-bold'>(D)</span>\ 4k, k= \pm1, \pm2, \dots} <spanclass=′latex−bold′>(E)</span> 16k,k=±1,±2,…{\qquad<span class='latex-bold'>(E)</span>\ 16k, k=\pm1,\pm2,\dots }
4
1

Expression is equivalent to...

The expression
P+QP−Q−P−QP+Q \frac{P+Q}{P-Q}-\frac{P-Q}{P+Q}
where P=x+yP=x+y and Q=x−yQ=x-y, is equivalent to:
<spanclass=′latex−bold′>(A)</span> x2−y2xy<spanclass=′latex−bold′>(B)</span> x2−y22xy<spanclass=′latex−bold′>(C)</span> 1<spanclass=′latex−bold′>(D)</span> x2+y2xy<spanclass=′latex−bold′>(E)</span> x2+y22xy{ <span class='latex-bold'>(A)</span>\ \frac{x^2-y^2}{xy}\qquad<span class='latex-bold'>(B)</span>\ \frac{x^2-y^2}{2xy}\qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ \frac{x^2+y^2}{xy} \qquad<span class='latex-bold'>(E)</span>\ \frac{x^2+y^2}{2xy} }
3
1

Quotient Remainder

When a positive integer xx is divided by a positive integer yy, the quotient is uu and the remainder is vv, where uu and vv are integers. What is the remainder when x+2uyx+2uy is divided by yy?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 2u<spanclass=′latex−bold′>(C)</span> 3u<spanclass=′latex−bold′>(D)</span> v<spanclass=′latex−bold′>(E)</span> 2v{{ <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2u \qquad<span class='latex-bold'>(C)</span>\ 3u \qquad<span class='latex-bold'>(D)</span>\ v }\qquad<span class='latex-bold'>(E)</span>\ 2v }
2
1

What is this Graph?

The graph of x2−4y2=0x^2-4y^2=0 is:
<spanclass=′latex−bold′>(A)</span> a parabola<spanclass=′latex−bold′>(B)</span> an ellipse<spanclass=′latex−bold′>(C)</span> a pair of straight lines<spanclass=′latex−bold′>(D)</span> a point<spanclass=′latex−bold′>(E)</span> none of these{{ <span class='latex-bold'>(A)</span>\ \text{a parabola} \qquad<span class='latex-bold'>(B)</span>\ \text{an ellipse} \qquad<span class='latex-bold'>(C)</span>\ \text{a pair of straight lines} \qquad<span class='latex-bold'>(D)</span>\ \text{a point} }\qquad<span class='latex-bold'>(E)</span>\ \text{none of these} }
1
1

Log Value

What is the value of [log⁡10(5log⁡10100)]2[\log_{10}(5\log_{10}100)]^2?
<spanclass=′latex−bold′>(A)</span> log⁡1050<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 1{{ <span class='latex-bold'>(A)</span>\ \log_{10}50 \qquad<span class='latex-bold'>(B)</span>\ 25\qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 2}\qquad<span class='latex-bold'>(E)</span>\ 1 }
35
1

Ratio of segments divided by orthocenter

The sides of a triangle are of lengths 1313, 1414, and 1515. The altitudes of the triangle meet at point HH. If ADAD is the altitude to the side length 1414, what is the ratio HD:HAHD:HA?
<spanclass=′latex−bold′>(A)</span>3:11<spanclass=′latex−bold′>(B)</span>5:11<spanclass=′latex−bold′>(C)</span>1:2<spanclass=′latex−bold′>(D)</span>2:3<spanclass=′latex−bold′>(E)</span>25:33<span class='latex-bold'>(A) </span> 3 : 11\qquad <span class='latex-bold'>(B) </span> 5 : 11\qquad <span class='latex-bold'>(C) </span> 1 : 2\qquad <span class='latex-bold'>(D) </span>2 : 3\qquad <span class='latex-bold'>(E) </span>25 : 33