MathDB

2002 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(21)
-1
1
11
1

Sum of Reciprocals of Triangular Numbers

Let tn=n(n+1)2t_n=\dfrac{n(n+1)}2 be the nnth triangular number. Find 1t1+1t2+1t3+⋯+1t2002.\dfrac1{t_1}+\dfrac1{t_2}+\dfrac1{t_3}+\cdots+\dfrac1{t_{2002}}. <spanclass=′latex−bold′>(A)</span>40032003<spanclass=′latex−bold′>(B)</span>20011001<spanclass=′latex−bold′>(C)</span>40042003<spanclass=′latex−bold′>(D)</span>40012001<spanclass=′latex−bold′>(E)</span>2<span class='latex-bold'>(A) </span>\dfrac{4003}{2003}\qquad<span class='latex-bold'>(B) </span>\dfrac{2001}{1001}\qquad<span class='latex-bold'>(C) </span>\dfrac{4004}{2003}\qquad<span class='latex-bold'>(D) </span>\dfrac{4001}{2001}\qquad<span class='latex-bold'>(E) </span>2
8
1

Points on Semicircles

Let ABAB be a segment of length 2626, and let points CC and DD be located on ABAB such that AC=1AC=1 and AD=8AD=8. Let EE and FF be points on one of the semicircles with diameter ABAB for which ECEC and FDFD are perpendicular to ABAB. Find EFEF.
<spanclass=′latex−bold′>(A)</span>5<spanclass=′latex−bold′>(B)</span>52<spanclass=′latex−bold′>(C)</span>7<spanclass=′latex−bold′>(D)</span>72<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span>5\qquad<span class='latex-bold'>(B) </span>5\sqrt2\qquad<span class='latex-bold'>(C) </span>7\qquad<span class='latex-bold'>(D) </span>7\sqrt2\qquad<span class='latex-bold'>(E) </span>12
25
3

Nonzero Coefficient of Polynomials

The nonzero coefficients of a polynomial PP with real coefficients are all replaced by their mean to form a polynomial QQ. Which of the following could be a graph of y=P(x)y = P(x) and y=Q(x)y = Q(x) over the interval −4≤x≤4-4\leq x \leq 4?
[asy]//Choice A size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125; } real g(real x) { return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(A)</span><span class='latex-bold'>(A)</span>", (-5,4.5)); [/asy] [asy]//Choice B size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2; } real g(real x) { return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(B)</span><span class='latex-bold'>(B)</span>", (-5,4.5)); [/asy] [asy]//Choice C size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.21875 x^2+0.28125 x+0.5; } real g(real x) { return -0.375 x^2-0.75 x+0.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(C)</span><span class='latex-bold'>(C)</span>", (-5,4.5)); [/asy] [asy]//Choice D size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875; } real g(real x) { return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625; } real z=3.14; draw(graph(f,-z, z), heavygray); draw(graph(g,-z, z), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(D)</span><span class='latex-bold'>(D)</span>", (-5,4.5)); [/asy] [asy]//Choice E size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("xx", (4,0), N); label("yy", (0,4), W); label("−4-4", (-4,-ticks), S); label("−1-1", (-1,-ticks), S); label("11", (1,-ticks), S); label("44", (4,-ticks), S);
real f(real x) { return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598; } real g(real x) { return -0.166667 x^3+0.125 x^2+0.479167 x-0.375; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(E)</span>", (-5,4.5)); [/asy]

Parabola Inequalities

Let f(x)\equal{}x^2\plus{}6x\plus{}1, and let R R denote the set of points (x,y) (x,y) in the coordinate plane such that f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0 The area of R R is closest to <spanclass=′latex−bold′>(A)</span> 21<spanclass=′latex−bold′>(B)</span> 22<spanclass=′latex−bold′>(C)</span> 23<spanclass=′latex−bold′>(D)</span> 24<spanclass=′latex−bold′>(E)</span> 25 <span class='latex-bold'>(A)</span>\ 21 \qquad <span class='latex-bold'>(B)</span>\ 22 \qquad <span class='latex-bold'>(C)</span>\ 23 \qquad <span class='latex-bold'>(D)</span>\ 24 \qquad <span class='latex-bold'>(E)</span>\ 25

Trigonometric System

Let aa and bb be real numbers such that sin⁡a+sin⁡b=22\sin a+\sin b=\dfrac{\sqrt2}2 and cos⁡a+cos⁡b=62\cos a+\cos b=\dfrac{\sqrt6}2. Find sin⁡(a+b)\sin(a+b).
<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>22<spanclass=′latex−bold′>(C)</span>32<spanclass=′latex−bold′>(D)</span>62<spanclass=′latex−bold′>(E)</span>1<span class='latex-bold'>(A) </span>\dfrac12\qquad<span class='latex-bold'>(B) </span>\dfrac{\sqrt2}2\qquad<span class='latex-bold'>(C) </span>\dfrac{\sqrt3}2\qquad<span class='latex-bold'>(D) </span>\dfrac{\sqrt6}2\qquad<span class='latex-bold'>(E) </span>1
16
2

Juan and Amal's Dice

Juan rolls a fair regular octahedral die marked with the numbers 1 1 through 8 8. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3 3? <spanclass=′latex−bold′>(A)</span> 112<spanclass=′latex−bold′>(B)</span> 13<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 712<spanclass=′latex−bold′>(E)</span> 23 <span class='latex-bold'>(A)</span>\ \frac{1}{12} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(D)</span>\ \frac{7}{12} \qquad <span class='latex-bold'>(E)</span>\ \frac{2}{3}

Lengths of Altitudes of a Triangle

The altitudes of a triangles are 1212, 1515, and 2020. The largest angle in this triangle is
<spanclass=′latex−bold′>(A)</span>72∘<spanclass=′latex−bold′>(B)</span>75∘<spanclass=′latex−bold′>(C)</span>90∘<spanclass=′latex−bold′>(D)</span>108∘<spanclass=′latex−bold′>(E)</span>120∘<span class='latex-bold'>(A) </span>72^\circ\qquad<span class='latex-bold'>(B) </span>75^\circ\qquad<span class='latex-bold'>(C) </span>90^\circ\qquad<span class='latex-bold'>(D) </span>108^\circ\qquad<span class='latex-bold'>(E) </span>120^\circ
15
1

Removing Leftmost Digit

How many four-digit numbers N N have the property that the three-digit number obtained by removing the leftmost digit is one ninth of N N? <spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 6<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> 8 <span class='latex-bold'>(A)</span>\ 4 \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>\ 8
10
2

Function in Sine and Cosine

Let fn(x)=sin⁡nx+cos⁡nxf_n(x)=\sin^n x + \cos^n x. For how many xx in [0,π][0,\pi] is it true that 6f4(x)−4f6(x)=2f2(x)?6f_4(x)-4f_6(x)=2f_2(x)? <spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>8<spanclass=′latex−bold′>(E)</span>more than 8<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>8\qquad<span class='latex-bold'>(E) </span>\text{more than 8}

Sums in a Set

How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19} \{1, 4, 7, 10, 13, 16, 19\}? <spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 30<spanclass=′latex−bold′>(E)</span> 35 <span class='latex-bold'>(A)</span>\ 13 \qquad <span class='latex-bold'>(B)</span>\ 16 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 30 \qquad <span class='latex-bold'>(E)</span>\ 35
9
1

Two Sequences

If a a, b b, c c, and d d are positive real numbers such that a a, b b, c c, d d form an increasing arithmetic sequence and a a, b b, d d form a geometric sequence, then ad \frac{a}{d} is <spanclass=′latex−bold′>(A)</span> 112<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ \frac{1}{12} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{2}
5
2

Pentagon Angles

Let v v, w w, x x, y y, and z z be the degree measures of the five angles of a pentagon. Suppose v<w<x<y<z v < w < x < y < z and v v, w w, x x, y y, and z z form an arithmetic sequence. Find the value of x x. <spanclass=′latex−bold′>(A)</span> 72<spanclass=′latex−bold′>(B)</span> 84<spanclass=′latex−bold′>(C)</span> 90<spanclass=′latex−bold′>(D)</span> 108<spanclass=′latex−bold′>(E)</span> 120 <span class='latex-bold'>(A)</span>\ 72 \qquad <span class='latex-bold'>(B)</span>\ 84 \qquad <span class='latex-bold'>(C)</span>\ 90 \qquad <span class='latex-bold'>(D)</span>\ 108 \qquad <span class='latex-bold'>(E)</span>\ 120

Expression an Integer

For how many positive integers mm is 2002m2−2\dfrac{2002}{m^2-2} a positive integer?
<spanclass=′latex−bold′>(A)</span>one<spanclass=′latex−bold′>(B)</span>two<spanclass=′latex−bold′>(C)</span>three<spanclass=′latex−bold′>(D)</span>four<spanclass=′latex−bold′>(E)</span>more than four<span class='latex-bold'>(A) </span>\text{one}\qquad<span class='latex-bold'>(B) </span>\text{two}\qquad<span class='latex-bold'>(C) </span>\text{three}\qquad<span class='latex-bold'>(D) </span>\text{four}\qquad<span class='latex-bold'>(E) </span>\text{more than four}
1
1

Lots of 9s

The arithmetic mean of the nine numbers in the set {9,99,999,9999,...,999999999} \{9,99,999,9999, . . . ,999999999\} is a 9 9-digit number M M, all of whose digits are distinct. The number M M does not contain the digit <spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 8 <span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 8
22
2

Segment Length Probability

Triangle ABC ABC is a right triangle with ∠ACB \angle ACB as its right angle, m\angle ABC \equal{} 60^\circ, and AB \equal{} 10. Let P P be randomly chosen inside △ABC \triangle ABC, and extend BP‾ \overline{BP} to meet AC‾ \overline{AC} at D D. What is the probability that BD>52 BD > 5\sqrt2? [asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P);
draw(A--C--B--cycle); draw(B--D); dot(P); label("A",A,S); label("D",D,S); label("C",C,S); label("P",P,NE); label("B",B,N);[/asy] (A)\ \frac {2 \minus{} \sqrt2}{2} \qquad (B)\ \frac {1}{3} \qquad (C)\ \frac {3 \minus{} \sqrt3}{3} \qquad (D)\ \frac {1}{2} \qquad (E)\ \frac {5 \minus{} \sqrt5}{5}

Log Sum

For all integers n n greater than 1 1, define a_n \equal{} \frac {1}{\log_n 2002}. Let b \equal{} a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 and c \equal{} a_{10} \plus{} a_{11} \plus{} a_{12} \plus{} a_{13} \plus{} a_{14}. Then b \minus{} c equals (A)\ \minus{} 2 \qquad (B)\ \minus{} 1 \qquad (C)\ \frac {1}{2002} \qquad (D)\ \frac {1}{1001} \qquad (E)\ \frac {1}{2}
20
1

Repeating Decimal

Suppose that a a and b b are digits, not both nine and not both zero, and the repeating decimal 0.ab‾ 0.\overline{ab} is expressed as a fraction in lowest terms. How many different denominators are possible? <spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 9 <span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 9
19
3

f(f(x))=6

The graph of the function f f is shown below. How many solutions does the equation f(f(x)) \equal{} 6 have? [asy]size(220); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4;
pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6); real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}; real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
draw(P1--P2--P3--P4--P5);
dot("(-7, -4)",P1); dot("(-2, 6)",P2,LeftSide); dot("(1, 6)",P4); dot("(5, -6)",P5);
xaxis("xx",-7.5,7,Ticks(xticks),EndArrow(6)); yaxis("yy",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7 <span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 7

System

If a a, b b, and c c are positive real numbers such that a(b \plus{} c) \equal{} 152, b(c \plus{} a) \equal{} 162, and c(a \plus{} b) \equal{} 170, then abc is <spanclass=′latex−bold′>(A)</span> 672<spanclass=′latex−bold′>(B)</span> 688<spanclass=′latex−bold′>(C)</span> 704<spanclass=′latex−bold′>(D)</span> 720<spanclass=′latex−bold′>(E)</span> 750 <span class='latex-bold'>(A)</span>\ 672 \qquad <span class='latex-bold'>(B)</span>\ 688 \qquad <span class='latex-bold'>(C)</span>\ 704 \qquad <span class='latex-bold'>(D)</span>\ 720 \qquad <span class='latex-bold'>(E)</span>\ 750

Area of Quadrilateral with 120 Degree Angles

In quadrilateral ABCDABCD, m∠B=m∠C=120∘m\angle B=m\angle C=120^\circ, AB=3AB=3, BC=4BC=4, and CD=5CD=5. Find the area of ABCDABCD.
<spanclass=′latex−bold′>(A)</span>15<spanclass=′latex−bold′>(B)</span>93<spanclass=′latex−bold′>(C)</span>4534<spanclass=′latex−bold′>(D)</span>4734<spanclass=′latex−bold′>(E)</span>153<span class='latex-bold'>(A) </span>15\qquad<span class='latex-bold'>(B) </span>9\sqrt3\qquad<span class='latex-bold'>(C) </span>\dfrac{45\sqrt3}4\qquad<span class='latex-bold'>(D) </span>\dfrac{47\sqrt3}4\qquad<span class='latex-bold'>(E) </span>15\sqrt3
18
2

Common Tangent

Let C1 C_1 and C2 C_2 be circles defined by (x \minus{} 10)^2 \plus{} y^2 \equal{} 36and (x \plus{} 15)^2 \plus{} y^2 \equal{} 81,respectively. What is the length of the shortest line segment PQ‾ \overline{PQ} that is tangent to C1 C_1 at P P and to C2 C_2 at Q Q? <spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 18 \qquad <span class='latex-bold'>(C)</span>\ 20 \qquad <span class='latex-bold'>(D)</span>\ 21 \qquad <span class='latex-bold'>(E)</span>\ 24

Random Point

A point P P is randomly selected from the rectangular region with vertices (0,0) (0, 0), (2,0) (2, 0), (2,1) (2, 1), (0,1) (0, 1). What is the probability that P P is closer to the origin than it is to the point (3,1) (3, 1)? <spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 23<spanclass=′latex−bold′>(C)</span> 34<spanclass=′latex−bold′>(D)</span> 45<spanclass=′latex−bold′>(E)</span> 1 <span class='latex-bold'>(A)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(B)</span>\ \frac{2}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{3}{4} \qquad <span class='latex-bold'>(D)</span>\ \frac{4}{5} \qquad <span class='latex-bold'>(E)</span>\ 1
14
2

Base-2002 Log

For all positive integers n n, let f(n) \equal{} \log_{2002} n^2. Let N \equal{} f(11) \plus{} f(13) \plus{} f(14) Which of the following relations is true? (A)\ N < 1 \qquad (B)\ N \equal{} 1 \qquad (C)\ 1 < N < 2 \qquad (D)\ N \equal{} 2 \qquad (E)\ N > 2

Sum of ni^n from 1 to 2002

Find i+2i2+3i3+…+2002i2002i+2i^2+3i^3+\ldots+2002i^{2002}.
<spanclass=′latex−bold′>(A)</span>−999+1002i<spanclass=′latex−bold′>(B)</span>−1002+999i2<spanclass=′latex−bold′>(C)</span>−1001+1000i<span class='latex-bold'>(A) </span>-999+1002i\qquad<span class='latex-bold'>(B) </span>-1002+999i\sqrt2\qquad<span class='latex-bold'>(C) </span>-1001+1000i <spanclass=′latex−bold′>(D)</span>−1002+1001i<spanclass=′latex−bold′>(E)</span>i<span class='latex-bold'>(D) </span>-1002+1001i\qquad<span class='latex-bold'>(E) </span>i
4
1

Complement + Supplement Relation

Find the degree measure of an angle whose complement is 25% 25\% of its supplement. <spanclass=′latex−bold′>(A)</span> 48<spanclass=′latex−bold′>(B)</span> 60<spanclass=′latex−bold′>(C)</span> 75<spanclass=′latex−bold′>(D)</span> 120<spanclass=′latex−bold′>(E)</span> 150 <span class='latex-bold'>(A)</span>\ 48 \qquad <span class='latex-bold'>(B)</span>\ 60 \qquad <span class='latex-bold'>(C)</span>\ 75 \qquad <span class='latex-bold'>(D)</span>\ 120 \qquad <span class='latex-bold'>(E)</span>\ 150
23
3

Perpendicular Bisectors and Angle Bisectors

In triangle ABC ABC, side AC AC and the perpendicular bisector of BC BC meet in point D D, and BD BD bisects ∠ABC \angle ABC. If AD \equal{} 9 and DC \equal{} 7, what is the area of triangle ABD ABD? <spanclass=′latex−bold′>(A)</span> 14<spanclass=′latex−bold′>(B)</span> 21<spanclass=′latex−bold′>(C)</span> 28<spanclass=′latex−bold′>(D)</span> 145<spanclass=′latex−bold′>(E)</span> 285 <span class='latex-bold'>(A)</span>\ 14 \qquad <span class='latex-bold'>(B)</span>\ 21 \qquad <span class='latex-bold'>(C)</span>\ 28 \qquad <span class='latex-bold'>(D)</span>\ 14\sqrt5 \qquad <span class='latex-bold'>(E)</span>\ 28\sqrt5

Equal Median and Side

In △ABC \triangle{ABC}, we have AB\equal{}1 and AC\equal{}2. Side BC BC and the median from A A to BC BC have the same length. What is BC BC? (A)\ \frac{1\plus{}\sqrt2}{2} \qquad (B)\ \frac{1\plus{}\sqrt3}{2} \qquad (C)\ \sqrt2 \qquad (D)\ \frac{3}{2} \qquad (E)\ \sqrt3

Contrived Equation in Complex Numbers

The equation z(z+i)(z+3i)=2002iz(z+i)(z+3i)=2002i has a zero of the form a+bia+bi, where aa and bb are positive real numbers. Find aa.
<spanclass=′latex−bold′>(A)</span>118<spanclass=′latex−bold′>(B)</span>210<spanclass=′latex−bold′>(C)</span>2210<spanclass=′latex−bold′>(D)</span>2002<spanclass=′latex−bold′>(E)</span>1002<span class='latex-bold'>(A) </span>\sqrt{118}\qquad<span class='latex-bold'>(B) </span>\sqrt{210}\qquad<span class='latex-bold'>(C) </span>2\sqrt{210}\qquad<span class='latex-bold'>(D) </span>\sqrt{2002}\qquad<span class='latex-bold'>(E) </span>100\sqrt2
24
3

Binomial raised to a large power

Find the number of ordered pairs of real numbers (a,b) (a,b) such that (a \plus{} bi)^{2002} \equal{} a \minus{} bi. <spanclass=′latex−bold′>(A)</span> 1001<spanclass=′latex−bold′>(B)</span> 1002<spanclass=′latex−bold′>(C)</span> 2001<spanclass=′latex−bold′>(D)</span> 2002<spanclass=′latex−bold′>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 1001\qquad <span class='latex-bold'>(B)</span>\ 1002\qquad <span class='latex-bold'>(C)</span>\ 2001\qquad <span class='latex-bold'>(D)</span>\ 2002\qquad <span class='latex-bold'>(E)</span>\ 2004

quadrilateral area

A convex quadrilateral ABCD ABCD with area 2002 2002 contains a point P P in its interior such that PA \equal{} 24, PB \equal{} 32, PC \equal{} 28, and PD \equal{} 45. FInd the perimeter of ABCD ABCD. (A)\ 4\sqrt {2002}\qquad (B)\ 2\sqrt {8465}\qquad (C)\ 2\left(48 \plus{} \sqrt {2002}\right) (D)\ 2\sqrt {8633}\qquad (E)\ 4\left(36 \plus{} \sqrt {113}\right)

Distances from a point inside a Tetrahedron

Let ABCDABCD be a regular tetrahedron and let EE be a point inside the face ABCABC. Denote by ss the sum of the distances from EE to the faces DABDAB, DBCDBC, DCADCA, and by SS the sum of the distances from EE to the edges ABAB, BCBC, CACA. Then sS\dfrac sS equals
<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>223<spanclass=′latex−bold′>(C)</span>62<spanclass=′latex−bold′>(D)</span>2<spanclass=′latex−bold′>(E)</span>3<span class='latex-bold'>(A) </span>\sqrt2\qquad<span class='latex-bold'>(B) </span>\dfrac{2\sqrt2}3\qquad<span class='latex-bold'>(C) </span>\dfrac{\sqrt6}2\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>3
17
2

Minimal sum of a set of primes

Several sets of prime numbers, such as {7,83,421,659} \{ 7, 83, 421, 659\} use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? <spanclass=′latex−bold′>(A)</span> 193<spanclass=′latex−bold′>(B)</span> 207<spanclass=′latex−bold′>(C)</span> 225<spanclass=′latex−bold′>(D)</span> 252<spanclass=′latex−bold′>(E)</span> 447 <span class='latex-bold'>(A)</span>\ 193\qquad<span class='latex-bold'>(B)</span>\ 207\qquad<span class='latex-bold'>(C)</span>\ 225\qquad<span class='latex-bold'>(D)</span>\ 252\qquad<span class='latex-bold'>(E)</span>\ 447

Equivalent Form of Trigonometric Function

Let f(x)=sin⁡4x+4cos⁡2x−cos⁡4x+4sin⁡2xf(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}. An equivalent form of f(x)f(x) is
<spanclass=′latex−bold′>(A)</span>1−2sin⁡x<spanclass=′latex−bold′>(B)</span>−1+2cos⁡x<spanclass=′latex−bold′>(C)</span>cos⁡x2−sin⁡x2<span class='latex-bold'>(A) </span>1-\sqrt2\sin x\qquad<span class='latex-bold'>(B) </span>-1+\sqrt2\cos x\qquad<span class='latex-bold'>(C) </span>\cos\dfrac x2-\sin\dfrac x2 <spanclass=′latex−bold′>(D)</span>cos⁡x−sin⁡x<spanclass=′latex−bold′>(E)</span>cos⁡2x<span class='latex-bold'>(D) </span>\cos x-\sin x\qquad<span class='latex-bold'>(E) </span>\cos2x
21
3

Digit Sequence

Consider the sequence of numbers: 4,7,1,8,9,7,6,…. 4, 7, 1, 8, 9, 7, 6, \ldots . For n>2 n > 2, the n nth term of the sequence is the units digit of the sum of the two previous terms. Let Sn S_n denote the sum of the first n n terms of this sequence. The smallest value of n n for which Sn>10,000 S_n > 10,000 is: <spanclass=′latex−bold′>(A)</span> 1992<spanclass=′latex−bold′>(B)</span> 1999<spanclass=′latex−bold′>(C)</span> 2001<spanclass=′latex−bold′>(D)</span> 2002<spanclass=′latex−bold′>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 1992 \qquad <span class='latex-bold'>(B)</span>\ 1999 \qquad <span class='latex-bold'>(C)</span>\ 2001 \qquad <span class='latex-bold'>(D)</span>\ 2002 \qquad <span class='latex-bold'>(E)</span>\ 2004

Divisibility Series

For all positive integers n n less than 2002 2002, let a_n \equal{} \begin{cases} 11 & \text{if }n\text{ is divisible by }13\text{ and }14 \\ 13 & \text{if }n\text{ is divisible by }11\text{ and }14 \\ 14 & \text{if }n\text{ is divisible by }11\text{ and }13 \\ 0 & \text{otherwise} \end{cases} Calculate \sum_{n \equal{} 1}^{2001} a_n. <spanclass=′latex−bold′>(A)</span> 448<spanclass=′latex−bold′>(B)</span> 486<spanclass=′latex−bold′>(C)</span> 1560<spanclass=′latex−bold′>(D)</span> 2001<spanclass=′latex−bold′>(E)</span> 2002 <span class='latex-bold'>(A)</span>\ 448 \qquad <span class='latex-bold'>(B)</span>\ 486 \qquad <span class='latex-bold'>(C)</span>\ 1560 \qquad <span class='latex-bold'>(D)</span>\ 2001 \qquad <span class='latex-bold'>(E)</span>\ 2002

Maximizing a Log

Let aa and bb be real numbers greater than 11 for which there exists a positive real number cc, different from 11, such that 2(log⁡ac+log⁡bc)=9log⁡abc.2(\log_ac+\log_bc)=9\log_{ab}c. Find the largest possible value of log⁡ab\log_ab.
<spanclass=′latex−bold′>(A)</span>2<spanclass=′latex−bold′>(B)</span>3<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>3<span class='latex-bold'>(A) </span>\sqrt2\qquad<span class='latex-bold'>(B) </span>\sqrt3\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>\sqrt6\qquad<span class='latex-bold'>(E) </span>3
13
2

Differs from reciprocal by 1

Two different positive numbers a a and b b each differ from their reciprocals by 1. What is a \plus{} b? <spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>2<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>3 <span class='latex-bold'>(A) </span> 1 \qquad <span class='latex-bold'>(B) </span> 2 \qquad <span class='latex-bold'>(C) </span> \sqrt {5} \qquad <span class='latex-bold'>(D) </span> \sqrt {6} \qquad <span class='latex-bold'>(E) </span> 3

Perfect Square Sums

The sum of 18 18 consecutive positive integers is a perfect square. The smallest possible value of this sum is <spanclass=′latex−bold′>(A)</span> 169<spanclass=′latex−bold′>(B)</span> 225<spanclass=′latex−bold′>(C)</span> 289<spanclass=′latex−bold′>(D)</span> 361<spanclass=′latex−bold′>(E)</span> 441 <span class='latex-bold'>(A)</span>\ 169 \qquad <span class='latex-bold'>(B)</span>\ 225 \qquad <span class='latex-bold'>(C)</span>\ 289 \qquad <span class='latex-bold'>(D)</span>\ 361 \qquad <span class='latex-bold'>(E)</span>\ 441