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2005 AMC 12A Problems/Problem 25
...ly possible side length (red triangle in diagram). Each of these triangles is determined by one vertex of the cube, so in one cube we have 8 equilateral currentprojection=perspective(1/3,-1,1/2);

4 KB (495 words) - 01:36, 26 May 2024
2005 AMC 12B Problems/Problem 2
...property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? ...h> means <math>0.01x</math>, the statement "<math>x\% \text{ of } x \text{ is 4}</math>" can be rewritten as "<math>0.01x \cdot x = 4</math>":

1 KB (145 words) - 13:56, 14 December 2021
2005 AMC 12B Problems/Problem 4
...>A</math> on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she e \textbf{(C) }\ 3 \qquad

1 KB (197 words) - 14:16, 14 December 2021
2005 AMC 12B Problems/Problem 6
...lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad

2 KB (299 words) - 15:29, 5 July 2022
2005 AMC 12B Problems/Problem 7
What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>? ...equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-b</math>):

2 KB (357 words) - 20:15, 27 December 2020
2005 AMC 12B Problems/Problem 9
...got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the [[mean]] and the [[median]] score on this exam? ...720}{20}=86</math>. The difference between the mean and median, therefore, is <math>\boxed{\textbf{(B)}\ 1}</math>.

2 KB (280 words) - 15:35, 16 December 2021
2005 AMC 12B Problems/Problem 10
...ding term is the sum of the cubes of the digits of the previous term. What is the <math>{2005}^{\text{th}}</math> term of the sequence? ...<math>250</math>. It just so happens that <math>2005\equiv 1\ (\text{mod}\ 3)</math>, which leads us to the answer of <math>\boxed{\textbf{(E) } 250}</m

1 KB (204 words) - 14:37, 15 December 2021
2005 AMC 12B Problems/Problem 11
...awn at random without replacement. What is the probability that their sum is $<math>20</math> or more? ...\qquad \textbf{(D) }\ {{{\frac{1}{2}}}} \qquad \textbf{(E) }\ {{{\frac{2}{3}}}}</math>

4 KB (607 words) - 15:16, 23 June 2024
2005 AMC 12B Problems/Problem 13
...math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>. What is <math>x_1x_2...x_{124}</math>? ...)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</mat

1 KB (203 words) - 19:57, 24 December 2020
2005 AMC 12B Problems/Problem 14
...o the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle? ...</math> and the diagonal is <math>k = R+6</math>. The diagonal of a square is <math>\sqrt{2}</math> times the side length. Therefore, <math>R+6 = R\sqrt{

2 KB (278 words) - 21:12, 24 December 2020
2005 AMC 12B Problems/Problem 15
...is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits? \mathrm{(C)}\ 3 \qquad

2 KB (411 words) - 21:02, 21 December 2020
2005 AMC 12B Problems/Problem 16
...radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains th \mathrm {(B)}\ \sqrt{3} \qquad

2 KB (364 words) - 04:54, 16 January 2023
2005 AMC 12B Problems/Problem 17
<cmath>a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?</cmath> <cmath>\log_{10}2^{a}+\log_{10}3^{b}+\log_{10}5^{c}+\log_{10}7^{d}=2005</cmath>

1 KB (159 words) - 21:18, 21 December 2020
2005 AMC 12B Problems/Problem 20
...g</math> and <math>h</math> be distinct elements in the set <math>\{-7,-5,-3,-2,2,4,6,13\}.</math> What is the minimum possible value of <math>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</math>

3 KB (584 words) - 01:17, 17 July 2024
2005 AMC 12B Problems/Problem 21
...60</math> divisors and <math>7n</math> has <math>80</math> divisors. What is the greatest integer <math>k</math> such that <math>7^k</math> divides <mat ...\mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}</math>

888 bytes (140 words) - 20:04, 24 December 2020
2005 AMC 12B Problems/Problem 22
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>.

4 KB (673 words) - 17:50, 27 June 2024
2005 AMC 12B Problems/Problem 23
...> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> Therefore, <math>x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})</math>.

5 KB (786 words) - 11:36, 19 May 2024
2005 AMC 12B Problems/Problem 24
...h>m</math> and <math>n</math> are relatively prime positive integers. What is the value of <math>m + n</math>? ...that the slope between the first two is <math>2</math>, and <math>A</math> is the point with the least <math>y</math>-coordinate.

4 KB (761 words) - 09:10, 1 August 2023
2005 AMC 12B Problems/Problem 25
...o one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex? \qquad\mathrm{(E)}\ \frac {3}{128}</math>

10 KB (1,840 words) - 15:01, 4 July 2024
2006 AMC 10A Problems
Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \textdollar 2 </math> each. How many dol Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?

13 KB (2,028 words) - 16:32, 22 March 2022

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