Difference between revisions of "2000 AMC 8 Problems"
(New page: ==Problem 1== Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? <math> \mathrm{(A)...) |
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− | ==Problem 1== | + | {{AMC8 Problems|year=2000|}} |
+ | == Problem 1 == | ||
Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? | Aunt Anna is <math>42</math> years old. Caitlin is <math>5</math> years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? | ||
− | <math> | + | <math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qquad\mathrm{(E)}\ 37</math> |
− | \mathrm{(A)}\ 15 | ||
− | \qquad | ||
− | \mathrm{(B)}\ 16 | ||
− | \qquad | ||
− | \mathrm{(C)}\ 17 | ||
− | \qquad | ||
− | \mathrm{(D)}\ 21 | ||
− | \qquad | ||
− | \mathrm{(E)}\ 37 | ||
− | </math> | ||
[[2000 AMC 8 Problems/Problem 1|Solution]] | [[2000 AMC 8 Problems/Problem 1|Solution]] | ||
− | ==Problem 2== | + | == Problem 2 == |
+ | |||
Which of these numbers is less than its reciprocal? | Which of these numbers is less than its reciprocal? | ||
− | <math> | + | <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2</math> |
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | </math> | ||
[[2000 AMC 8 Problems/Problem 2|Solution]] | [[2000 AMC 8 Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
+ | |||
How many whole numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi?</math> | How many whole numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi?</math> | ||
− | + | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ \text{infinitely many}</math> | |
− | <math> | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
[[2000 AMC 8 Problems/Problem 3|Solution]] | [[2000 AMC 8 Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | |||
− | <math> | + | In <math>1960</math> only <math>5\%</math> of the working adults in Carlin City worked at home. By <math>1970</math> the "at-home" work force increased to <math>8\%</math>. In <math>1980</math> there were approximately <math>15\%</math> working at home, and in <math>1990</math> there were <math>30\%</math>. The graph that best illustrates this is |
− | </ | + | |
+ | <asy> | ||
+ | unitsize(18); | ||
+ | |||
+ | draw((0,4)--(0,0)--(7,0)); | ||
+ | draw((0,1)--(.2,1)); draw((0,2)--(.2,2)); draw((0,3)--(.2,3)); | ||
+ | draw((2,0)--(2,.2)); draw((4,0)--(4,.2)); draw((6,0)--(6,.2)); | ||
+ | for (int a = 1; a < 4; ++a) | ||
+ | { | ||
+ | for (int b = 1; b < 4; ++b) | ||
+ | { | ||
+ | draw((2*a,b-.1)--(2*a,b+.1)); | ||
+ | draw((2*a-.1,b)--(2*a+.1,b)); | ||
+ | } | ||
+ | } | ||
+ | label("1960",(0,0),S); label("1970",(2,0),S); label("1980",(4,0),S); label("1990",(6,0),S); | ||
+ | label("10",(0,1),W); label("20",(0,2),W); label("30",(0,3),W); | ||
+ | label("$\%$",(0,4),N); | ||
+ | |||
+ | draw((12,4)--(12,0)--(19,0)); | ||
+ | draw((12,1)--(12.2,1)); draw((12,2)--(12.2,2)); draw((12,3)--(12.2,3)); | ||
+ | draw((14,0)--(14,.2)); draw((16,0)--(16,.2)); draw((18,0)--(18,.2)); | ||
+ | for (int a = 1; a < 4; ++a) | ||
+ | { | ||
+ | for (int b = 1; b < 4; ++b) | ||
+ | { | ||
+ | draw((2*a+12,b-.1)--(2*a+12,b+.1)); | ||
+ | draw((2*a+11.9,b)--(2*a+12.1,b)); | ||
+ | } | ||
+ | } | ||
+ | label("1960",(12,0),S); label("1970",(14,0),S); label("1980",(16,0),S); label("1990",(18,0),S); | ||
+ | label("10",(12,1),W); label("20",(12,2),W); label("30",(12,3),W); | ||
+ | label("$\%$",(12,4),N); | ||
+ | |||
+ | draw((0,12)--(0,8)--(7,8)); | ||
+ | draw((0,9)--(.2,9)); draw((0,10)--(.2,10)); draw((0,11)--(.2,11)); | ||
+ | draw((2,8)--(2,8.2)); draw((4,8)--(4,8.2)); draw((6,8)--(6,8.2)); | ||
+ | for (int a = 1; a < 4; ++a) | ||
+ | { | ||
+ | for (int b = 1; b < 4; ++b) | ||
+ | { | ||
+ | draw((2*a,b+7.9)--(2*a,b+8.1)); | ||
+ | draw((2*a-.1,b+8)--(2*a+.1,b+8)); | ||
+ | } | ||
+ | } | ||
+ | label("1960",(0,8),S); label("1970",(2,8),S); label("1980",(4,8),S); label("1990",(6,8),S); | ||
+ | label("10",(0,9),W); label("20",(0,10),W); label("30",(0,11),W); | ||
+ | label("$\%$",(0,12),N); | ||
+ | |||
+ | draw((12,12)--(12,8)--(19,8)); | ||
+ | draw((12,9)--(12.2,9)); draw((12,10)--(12.2,10)); draw((12,11)--(12.2,11)); | ||
+ | draw((14,8)--(14,8.2)); draw((16,8)--(16,8.2)); draw((18,8)--(18,8.2)); | ||
+ | for (int a = 1; a < 4; ++a) | ||
+ | { | ||
+ | for (int b = 1; b < 4; ++b) | ||
+ | { | ||
+ | draw((2*a+12,b+7.9)--(2*a+12,b+8.1)); | ||
+ | draw((2*a+11.9,b+8)--(2*a+12.1,b+8)); | ||
+ | } | ||
+ | } | ||
+ | label("1960",(12,8),S); label("1970",(14,8),S); label("1980",(16,8),S); label("1990",(18,8),S); | ||
+ | label("10",(12,9),W); label("20",(12,10),W); label("30",(12,11),W); | ||
+ | label("$\%$",(12,12),N); | ||
+ | |||
+ | draw((24,12)--(24,8)--(31,8)); | ||
+ | draw((24,9)--(24.2,9)); draw((24,10)--(24.2,10)); draw((24,11)--(24.2,11)); | ||
+ | draw((26,8)--(26,8.2)); draw((28,8)--(28,8.2)); draw((30,8)--(30,8.2)); | ||
+ | for (int a = 1; a < 4; ++a) | ||
+ | { | ||
+ | for (int b = 1; b < 4; ++b) | ||
+ | { | ||
+ | draw((2*a+24,b+7.9)--(2*a+24,b+8.1)); | ||
+ | draw((2*a+23.9,b+8)--(2*a+24.1,b+8)); | ||
+ | } | ||
+ | } | ||
+ | label("1960",(24,8),S); label("1970",(26,8),S); label("1980",(28,8),S); label("1990",(30,8),S); | ||
+ | label("10",(24,9),W); label("20",(24,10),W); label("30",(24,11),W); | ||
+ | label("$\%$",(24,12),N); | ||
+ | |||
+ | draw((0,9)--(2,9.25)--(4,10)--(6,11)); | ||
+ | draw((12,8.5)--(14,9)--(16,10)--(18,10.5)); | ||
+ | draw((24,8.5)--(26,8.8)--(28,10.5)--(30,11)); | ||
+ | draw((0,0.5)--(2,1)--(4,2.8)--(6,3)); | ||
+ | draw((12,0.5)--(14,.8)--(16,1.5)--(18,3)); | ||
+ | |||
+ | label("(A)",(-1,12),W); | ||
+ | label("(B)",(11,12),W); | ||
+ | label("(C)",(23,12),W); | ||
+ | label("(D)",(-1,4),W); | ||
+ | label("(E)",(11,4),W); | ||
+ | </asy> | ||
[[2000 AMC 8 Problems/Problem 4|Solution]] | [[2000 AMC 8 Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | |||
− | <math> | + | Each principal of Lincoln High School serves exactly one <math>3</math>-year term. What is the maximum number of principals this school could have during an 8-year period? |
− | \ | + | |
− | \qquad | + | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 8</math> |
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | </math> | ||
[[2000 AMC 8 Problems/Problem 5|Solution]] | [[2000 AMC 8 Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | Figure <math>ABCD</math> is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is | + | Figure <math>ABCD</math> is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded <math>L</math>-shaped region is |
− | < | + | <asy> |
− | + | pair A,B,C,D; | |
− | + | A = (5,5); B = (5,0); C = (0,0); D = (0,5); | |
− | + | fill((0,0)--(0,4)--(1,4)--(1,1)--(4,1)--(4,0)--cycle,gray); | |
− | + | draw(A--B--C--D--cycle); | |
− | + | draw((4,0)--(4,4)--(0,4)); | |
− | + | draw((1,5)--(1,1)--(5,1)); | |
− | |||
− | |||
− | |||
− | |||
+ | label("$A$",A,NE); | ||
+ | label("$B$",B,SE); | ||
+ | label("$C$",C,SW); | ||
+ | label("$D$",D,NW); | ||
+ | label("$1$",(1,4.5),E); | ||
+ | label("$1$",(0.5,5),N); | ||
+ | label("$3$",(1,2.5),E); | ||
+ | label("$3$",(2.5,1),N); | ||
+ | label("$1$",(4,0.5),E); | ||
+ | label("$1$",(4.5,1),N); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 7 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15</math> | ||
[[2000 AMC 8 Problems/Problem 6|Solution]] | [[2000 AMC 8 Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | |||
− | <math> | + | What is the minimum possible product of three different numbers of the set <math>\{-8,-6,-4,0,3,5,7\}</math>? |
− | \ | + | |
− | \qquad | + | <math>\text{(A)}\ -336 \qquad \text{(B)}\ -280 \qquad \text{(C)}\ -210 \qquad \text{(D)}\ -192 \qquad \text{(E)}\ 0</math> |
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | \qquad | ||
− | \ | ||
− | </math> | ||
[[2000 AMC 8 Problems/Problem 7|Solution]] | [[2000 AMC 8 Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | |||
− | <math> | + | Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is |
− | \ | + | |
− | \qquad | + | <asy> |
− | \ | + | draw((0,0)--(2,0)--(3,1)--(3,7)--(1,7)--(0,6)--cycle); |
− | \qquad | + | draw((3,7)--(2,6)--(0,6)); |
− | \ | + | draw((3,5)--(2,4)--(0,4)); |
− | \qquad | + | draw((3,3)--(2,2)--(0,2)); |
− | \ | + | draw((2,0)--(2,6)); |
− | \qquad | + | |
− | \ | + | dot((1,1)); dot((.5,.5)); dot((1.5,.5)); dot((1.5,1.5)); dot((.5,1.5)); |
− | </math> | + | dot((2.5,1.5)); |
+ | dot((.5,2.5)); dot((1.5,2.5)); dot((1.5,3.5)); dot((.5,3.5)); | ||
+ | dot((2.25,2.75)); dot((2.5,3)); dot((2.75,3.25)); dot((2.25,3.75)); dot((2.5,4)); dot((2.75,4.25)); | ||
+ | dot((.5,5.5)); dot((1.5,4.5)); | ||
+ | dot((2.25,4.75)); dot((2.5,5.5)); dot((2.75,6.25)); | ||
+ | dot((1.5,6.5)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 21 \qquad \text{(B)}\ 22 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 41 \qquad \text{(E)}\ 53</math> | ||
− | [[2000 AMC 8 Problems/Problem | + | [[2000 AMC 8 Problems/Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
+ | |||
+ | Three-digit powers of <math>2</math> and <math>5</math> are used in this "cross-number" puzzle. What is the only possible digit for the outlined square? | ||
+ | <cmath>\begin{array}{lcl} | ||
+ | \textbf{ACROSS} & & \textbf{DOWN} \\ | ||
+ | \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n | ||
+ | \end{array}</cmath> | ||
+ | |||
+ | <asy> | ||
+ | draw((0,-1)--(1,-1)--(1,2)--(0,2)--cycle); | ||
+ | draw((0,1)--(3,1)--(3,0)--(0,0)); | ||
+ | draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(2)); | ||
+ | |||
+ | label("$1$",(0,2),SE); | ||
+ | label("$2$",(0,1),SE); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? | ||
+ | |||
+ | <math>\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | |||
+ | The number <math>64</math> has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property? | ||
+ | |||
+ | <math>\text{(A)}\ 15 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 20</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(6,0)--(6,1)--(5,1)--(5,2)--(0,2)--cycle); | ||
+ | draw((0,1)--(5,1)); | ||
+ | draw((1,1)--(1,2)); | ||
+ | draw((3,1)--(3,2)); | ||
+ | draw((2,0)--(2,1)); | ||
+ | draw((4,0)--(4,1)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 344 \qquad \text{(B)}\ 347 \qquad \text{(C)}\ 350 \qquad \text{(D)}\ 353 \qquad \text{(E)}\ 356</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 12|Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | In triangle <math>CAT</math>, we have <math>\angle ACT = \angle ATC</math> and <math>\angle CAT = 36^\circ</math>. If <math>\overline{TR}</math> bisects <math>\angle ATC</math>, then <math>\angle CRT =</math> | ||
+ | |||
+ | <asy> | ||
+ | pair A,C,T,R; | ||
+ | C = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570); | ||
+ | draw(C--A--T--cycle); | ||
+ | draw(T--R); | ||
+ | label("$A$",A,N); | ||
+ | label("$T$",T,SE); | ||
+ | label("$C$",C,SW); | ||
+ | label("$R$",R,NW); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 36^\circ \qquad \text{(B)}\ 54^\circ \qquad \text{(C)}\ 72^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 108^\circ</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | |||
+ | What is the units digit of <math>19^{19} + 99^{99}</math>? | ||
+ | |||
+ | <math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 14|Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | Triangles <math>ABC</math>, <math>ADE</math>, and <math>EFG</math> are all equilateral. Points <math>D</math> and <math>G</math> are midpoints of <math>\overline{AC}</math> and <math>\overline{AE}</math>, respectively. If <math>AB = 4</math>, what is the perimeter of figure <math>ABCDEFG</math>? | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,EE,F,G; | ||
+ | A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3)); | ||
+ | EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2); | ||
+ | draw(A--B--C--cycle); | ||
+ | draw(D--EE--A); | ||
+ | draw(EE--F--G); | ||
+ | |||
+ | label("$A$",A,S); | ||
+ | label("$B$",B,SW); | ||
+ | label("$C$",C,N); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",EE,NE); | ||
+ | label("$F$",F,SE); | ||
+ | label("$G$",G,SE); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 15|Solution]] | ||
+ | |||
+ | ==Problem 16== | ||
+ | |||
+ | In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters? | ||
+ | |||
+ | <math>\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 16|Solution]] | ||
+ | |||
+ | ==Problem 17== | ||
+ | |||
+ | The operation <math>\otimes</math> is defined for all nonzero numbers by <math>a\otimes b = \dfrac{a^2}{b}</math>. Determine <math>[(1\otimes 2)\otimes 3] - [1\otimes (2\otimes 3)]</math>. | ||
+ | |||
+ | <math>\text{(A)}\ -\dfrac{2}{3} \qquad \text{(B)}\ -\dfrac{1}{4} \qquad \text{(C)}\ 0 \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{2}{3}</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 17|Solution]] | ||
+ | |||
+ | ==Problem 18== | ||
+ | |||
+ | Consider these two geoboard quadrilaterals. Which of the following statements is true? | ||
+ | |||
+ | <asy> | ||
+ | for (int a = 0; a < 5; ++a) | ||
+ | { | ||
+ | for (int b = 0; b < 5; ++b) | ||
+ | { | ||
+ | dot((a,b)); | ||
+ | } | ||
+ | } | ||
+ | |||
+ | draw((0,3)--(0,4)--(1,3)--(1,2)--cycle); | ||
+ | draw((2,1)--(4,2)--(3,1)--(3,0)--cycle); | ||
+ | |||
+ | label("I",(0.4,3),E); | ||
+ | label("II",(2.9,1),W); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}</math> | ||
+ | |||
+ | <math>\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}</math> | ||
+ | |||
+ | <math>\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}</math> | ||
+ | |||
+ | <math>\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}</math> | ||
+ | |||
+ | <math>\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 18|Solution]] | ||
+ | |||
+ | ==Problem 19== | ||
+ | |||
+ | Three circular arcs of radius 5 units bound the region shown. Arcs <math>AB</math> and <math>AD</math> are quarter-circles, and arc <math>BCD</math> is a semicircle. What is the area, in square units, of the region? | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D; | ||
+ | A = (0,0); | ||
+ | B = (-5,5); | ||
+ | C = (0,10); | ||
+ | D = (5,5); | ||
+ | |||
+ | draw(arc((-5,0),A,B,CCW)); | ||
+ | draw(arc((0,5),B,D,CW)); | ||
+ | draw(arc((5,0),D,A,CCW)); | ||
+ | |||
+ | label("$A$",A,S); | ||
+ | label("$B$",B,W); | ||
+ | label("$C$",C,N); | ||
+ | label("$D$",D,E); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 25 \qquad \text{(B)}\ 10 + 5\pi \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 50 + 5\pi \qquad \text{(E)}\ 25\pi</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 19|Solution]] | ||
+ | |||
+ | ==Problem 20== | ||
+ | |||
+ | You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $<math>1.02</math>, with at least one coin of each type. How many dimes must you have? | ||
+ | |||
+ | <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 20|Solution]] | ||
+ | |||
+ | ==Problem 21== | ||
+ | |||
+ | Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is: | ||
+ | |||
+ | <math>\text{(A)}\ \dfrac{1}{4} \qquad \text{(B)}\ \dfrac{3}{8} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{3}{4}</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 21|Solution]] | ||
+ | |||
+ | ==Problem 22== | ||
+ | |||
+ | A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); | ||
+ | draw((2,0)--(2,2)); | ||
+ | draw((0,2)--(1,3)); | ||
+ | draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); | ||
+ | draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); | ||
+ | draw((2,10/3)--(5/2,23/6)); | ||
+ | draw((3,3)--(5/2,3)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 21 \qquad \text{(E)}\ 25</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 22|Solution]] | ||
+ | |||
+ | ==Problem 23== | ||
+ | |||
+ | There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is <math>6\frac{4}{7}</math>, then the number common to both sets of four numbers is | ||
+ | |||
+ | <math>\text{(A)}\ 5\frac{3}{7} \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 6\frac{4}{7} \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 7\frac{3}{7}</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 23|Solution]] | ||
+ | |||
+ | ==Problem 24== | ||
+ | |||
+ | If <math>\angle A = 20^\circ</math> and <math>\angle AFG = \angle AGF</math>, then <math>\angle B + \angle D = </math> | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,EE,F,G; | ||
+ | A = (0,0); | ||
+ | B = (9,4); | ||
+ | C = (21,0); | ||
+ | D = (13,-12); | ||
+ | EE = (4,-16); | ||
+ | F = (13/2,-6); | ||
+ | G = (8,0); | ||
+ | |||
+ | draw(A--C--EE--B--D--cycle); | ||
+ | |||
+ | label("$A$",A,W); | ||
+ | label("$B$",B,N); | ||
+ | label("$C$",C,E); | ||
+ | label("$D$",D,SE); | ||
+ | label("$E$",EE,SW); | ||
+ | label("$F$",F,WSW); | ||
+ | label("$G$",G,NW); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 48^\circ \qquad \text{(B)}\ 60^\circ \qquad \text{(C)}\ 72^\circ \qquad \text{(D)}\ 80^\circ \qquad \text{(E)}\ 90^\circ</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 24|Solution]] | ||
+ | |||
+ | ==Problem 25== | ||
+ | |||
+ | The area of rectangle <math>ABCD</math> is 72. If point <math>A</math> and the midpoints of <math>\overline{BC}</math> and <math>\overline{CD}</math> are joined to form a triangle, the area of that triangle is | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D; | ||
+ | A = (0,8); B = (9,8); C = (9,0); D = (0,0); | ||
+ | draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle); | ||
+ | |||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,SW); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40</math> | ||
+ | |||
+ | [[2000 AMC 8 Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2000|before=[[1999 AMC 8 Problems|1999 AMC 8]]|after=[[2001 AMC 8 Problems|2001 AMC 8]]}} | ||
+ | * [[AMC 8]] | ||
+ | * [[AMC 8 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 16:37, 11 September 2024
2000 AMC 8 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Aunt Anna is years old. Caitlin is years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
Problem 2
Which of these numbers is less than its reciprocal?
Problem 3
How many whole numbers lie in the interval between and
Problem 4
In only of the working adults in Carlin City worked at home. By the "at-home" work force increased to . In there were approximately working at home, and in there were . The graph that best illustrates this is
Problem 5
Each principal of Lincoln High School serves exactly one -year term. What is the maximum number of principals this school could have during an 8-year period?
Problem 6
Figure is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded -shaped region is
Problem 7
What is the minimum possible product of three different numbers of the set ?
Problem 8
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is
Problem 9
Three-digit powers of and are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
Problem 10
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?
Problem 11
The number has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property?
Problem 12
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
Problem 13
In triangle , we have and . If bisects , then
Problem 14
What is the units digit of ?
Problem 15
Triangles , , and are all equilateral. Points and are midpoints of and , respectively. If , what is the perimeter of figure ?
Problem 16
In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
Problem 17
The operation is defined for all nonzero numbers by . Determine .
Problem 18
Consider these two geoboard quadrilaterals. Which of the following statements is true?
Problem 19
Three circular arcs of radius 5 units bound the region shown. Arcs and are quarter-circles, and arc is a semicircle. What is the area, in square units, of the region?
Problem 20
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $, with at least one coin of each type. How many dimes must you have?
Problem 21
Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is:
Problem 22
A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to
Problem 23
There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is , then the number common to both sets of four numbers is
Problem 24
If and , then
Problem 25
The area of rectangle is 72. If point and the midpoints of and are joined to form a triangle, the area of that triangle is
See Also
2000 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 1999 AMC 8 |
Followed by 2001 AMC 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.