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• 2006 AMC 12A Problems/Problem 14
  ...debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? ...mon divisor]]) of <math>a</math> and <math>b</math>. Therefore, the answer is <math>gcd(300,210)=\boxed{\textbf{(C) }30}.</math>
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 15
  Suppose <math>\cos x=0</math> and <math>\cos (x+z)=1/2</math>. What is the smallest possible positive value of <math>z</math>? <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \
  919 bytes (138 words) - 12:45, 4 August 2017
• 2006 AMC 12A Problems/Problem 16
  ...d <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? dotfactor=3;
  2 KB (286 words) - 10:16, 19 December 2021
• 2006 AMC 12A Problems/Problem 17
  ...th> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? ...rac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}</math>
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 20
  ...s equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vert Therefore, starting at <math>A</math>, the bug has a <math>\frac{3}{3}</math> chance of finding a good path to the next vertex, and call it <math
  5 KB (908 words) - 19:23, 22 September 2022
• 2006 AMC 12A Problems/Problem 22
  ...sible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...quad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}</math>
  2 KB (343 words) - 15:39, 14 June 2023
• 2006 AMC 12A Problems/Problem 23
  ...,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? <cmath>A^2(S)=\left(\frac{1+2x+x^2}{2^2},\frac{x+2x^2+x^3}{2^2},...,\frac{x^{98}+2x^{99}+x^{100}}{2^2}\right)</cmath>
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  is simplified by expanding it and combining like terms. How many terms are in if the exponent of <math>y</math> is <math>1</math>, the exponent of <math>z</math> can be all even integers up
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties? ...k+1</math>, with no restriction on consecutive numbers. Since this process is easily reversible, we have a [[bijection]].
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  ...\geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> is <math>f^{[2005]}(x) = \frac {1}{2}</math>? ...<math>f(x)=2-2x,\frac{1}{2}\le x\le 1</math>,as long as <math>f(x)</math> is between <math>0</math> and <math>1</math>, <math>x</math> will be in the ri
  3 KB (437 words) - 23:49, 28 September 2022
• 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 (498 words) - 00:46, 4 August 2023
• 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 &#36;<math>20</math> or more? ...\qquad \textbf{(D) }\ {{{\frac{1}{2}}}} \qquad \textbf{(E) }\ {{{\frac{2}{3}}}}</math>
  4 KB (607 words) - 21:01, 20 May 2023
• 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 (463 words) - 19:28, 6 November 2022
• 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 (660 words) - 17:40, 24 January 2021
• 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) - 16:49, 31 January 2023
• 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) - 21:35, 7 September 2023
• 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
• 1995 AHSME Problems/Problem 6
  ...to the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x? path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
  1 KB (168 words) - 00:49, 14 October 2013
• 2006 AMC 10A Problems/Problem 8
  ...es through the points <math> (2,3) </math> and <math> (4,3) </math>. What is <math>c</math>? Substitute the points <math> (2,3) </math> and <math> (4,3) </math> into the given equation for <math> (x,y) </math>.
  2 KB (348 words) - 23:10, 16 December 2021
• 2006 AMC 12A Problems/Problem 12
  ...ove it. The bottom ring has an outside diameter of <math>3</math> cm. What is the distance, in cm, from the top of the top ring to the bottom of the bott D(CR((0,-39),3));
  2 KB (292 words) - 11:56, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  .../math> meters in the opposite direction and the circumference of his track is <math>100\pi</math>. ...will meet again in <math>k</math> minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=\boxed{\text
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 16
  ...h>\overline{AB}</math> and <math>\overline{AC}</math> are congruent. What is the area of <math>\triangle ABC</math>? MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);</asy>
  5 KB (732 words) - 23:19, 19 September 2023
• 2006 AMC 10A Problems/Problem 17
  ...HE}</math>. In addition, <math>AH=AC=2</math>, and <math>AD=3</math>. What is the area of quadrilateral <math>WXYZ</math> shown in the figure? A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 18
  ...quad\textbf{(D) } 10^2\times 26^4\qquad\textbf{(E) } 5\times 10^3\times 26^3\qquad</math> Therefore, the number of distinct license plates is <math> 5\times 10^4\times 26^2 \Longrightarrow \boxed{\mathrm{C}}</math>.
  2 KB (254 words) - 14:39, 5 April 2024
• 2006 AMC 10A Problems/Problem 19
  ...le value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>\boxed ==Solution 3 (Quick Summation)==
  2 KB (259 words) - 03:10, 22 June 2023
• 2006 AMC 10A Problems/Problem 20
  ...is the probability that some pair of these integers has a difference that is a multiple of <math>5</math>? ...) } \frac{1}{2}\qquad\textbf{(B) } \frac{3}{5}\qquad\textbf{(C) } \frac{2}{3}\qquad\textbf{(D) } \frac{4}{5}\qquad\textbf{(E) } 1\qquad</math>
  1 KB (187 words) - 08:21, 17 March 2023
• 2006 AMC 10A Problems/Problem 21
  ...itive integers have at least one digit that is a <math>2</math> or a <math>3</math>? ...s and subtracting off those which do not have any <math>2</math>s or <math>3</math>s as digits.
  3 KB (525 words) - 20:25, 30 April 2024
• 2006 AMC 10A Problems/Problem 24
  ...ent faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? ...) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{1}{3}\qquad\textbf{(E) } \frac{1}{2}\qquad</math>
  2 KB (292 words) - 10:19, 19 December 2021
• Set
  ...ames really do not define the meaning of the word ''set''; all they can do is replace it in various sentences. So, instead of defining what sets are, one ...uch as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] (which is well-defined by this formula for <math>\Re s>0</math>) admits an
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2006 USAMO Problems
  if and only if <math>s</math> is not a divisor of <math>p-1</math>. ...rms of <math>k</math>, the minimum value of <math>N</math> for which there is a set of <math>2k+1</math> distinct positive integers that has sum greater
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  == Problem 3 == [[1991 AJHSME Problems/Problem 3|Solution]]
  17 KB (2,246 words) - 13:37, 19 February 2020
• 1990 AJHSME Problems/Problem 1
  What is the smallest sum of two <math>3</math>-digit numbers that can be obtained by placing each of the six digits draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);
  1 KB (191 words) - 17:12, 29 October 2016
• 2022 USAJMO Problems/Problem 1
  <math>\bullet</math> <math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>; <math>\bullet</math> <math>a_2-a_1</math> is not divisible by <math>m</math>.
  4 KB (792 words) - 00:29, 13 April 2024
• 2006 AIME II Problems/Problem 2
  ...th>\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>. ...number of positive integer <math>n</math> which satisfies this requirement is <math>\boxed{893}</math>.
  1 KB (164 words) - 14:58, 14 April 2020
• 2005 AIME II Problems/Problem 1
  ...that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...ial coefficient]] <math>{n \choose 6} = \frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1}</math>.
  1 KB (239 words) - 11:54, 31 July 2023
• 2005 AIME II Problems/Problem 2
  ...uests. Given that the [[probability]] each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are [[ *Person 1: <math>\frac{9 \cdot 6 \cdot 3}{9 \cdot 8 \cdot 7} = \frac{9}{28}</math>
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 4
  <math>15^7 = 3^7\cdot5^7</math> so <math>15^7</math> has <math>8\cdot8 = 64</math> divisor <math>\gcd(15^7, 18^{11}) = 3^7 </math> which has 8 divisors.
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  ...th> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_ ...h>(x-17)(x-24)</math> to be a factor of <math>10</math>. Hence the answer is <math>19\cdot 22=\boxed{418}</math>.
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 5
  ...og b=3\log a </math> or <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>. ...e <math> b=a^3 </math>, note that <math> 12^3=1728 </math> while <math> 13^3=2197 </math>. Therefore, for this case, all values of <math>a</math> from <
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 6
  ...agical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in t ...s suggests that <math>n = 131 + 65 = 196</math>; the total number of cards is <math>196 \cdot 2 = \boxed{392}</math>.
  2 KB (384 words) - 00:31, 26 July 2018
• 2005 AIME II Problems
  ...that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math> ...he guests. Given that the probability each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  It follows that <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = \boxed{125}</math>. ...+1) = (y^{15}+y^{14}+y^{13}+y^{12}+y^{11}+y^{10}+y^9+y^8+y^7+y^6+y^5+y^4+y^3+y^2+y+1)=\frac{y^{16}-1}{y-1}</cmath>
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 8
  .../math> and <math> p </math> are [[relatively prime]], and <math> n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> ...= (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-acos(3/7));
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 9
  ...positive integers <math> n </math> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all rea ...t certainly hold for <math>t = \frac{\pi}2 - u</math>. Thus, the question is equivalent to asking for how many [[positive integer]]s <math>n \leq 1000</
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...h> and <math> r </math> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math ...- y</math> again, we know have <math>xy = (400 - y)y = 150^2</math>. This is a quadratic with roots <math>200 \pm 50\sqrt{7}</math>. Since <math>y < x</
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...at the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re ...,0,-3)--(0,-3,0)--(3,0,0)--(0,0,-3)--(0,3,0)--(0,0,3)--(3,0,0)--(0,3,0)--(-3,0,0));
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME II Problems/Problem 11
  ...>a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math>m. </math> <math>a_{k}a_{k+1} = a_{k-1}a_{k} - 3 </math>.
  3 KB (499 words) - 18:52, 21 November 2022
• Euler's identity
  ...er's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is named after the 18th-century mathematician [[Leonhard Euler]]. ...umbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties:
  3 KB (452 words) - 23:17, 4 January 2021
• 2006 AMC 12A Problems/Problem 4
  A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? 1+2&9&6&3\\
  2 KB (257 words) - 11:20, 2 January 2022
• 2005 AIME I Problems
  ...ose common difference is <math> k. </math> For example, <math> S_3 </math> is the sequence <math> 1,4,7,10,\ldots. </math> For how many values of <math> == Problem 3 ==
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 2
  ...ose common difference is <math> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <ma ...h>. Thus the requested number of values is <math>12</math>, and the answer is <math>\boxed{012}</math>.
  2 KB (303 words) - 01:31, 5 December 2022
• 2005 AIME I Problems/Problem 3
  ...ivisor]]s (positive integral [[divisor]]s excluding itself), each of which is less than 50? ...so <math>n</math> must be in the form <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>.
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 4
  ...5</math>, so this number works and no larger number can. Thus, the answer is <math>\boxed{294}</math>. ...factors of <math>69</math> are <math>(1,69), (3,23)</math>; <math>x</math> is maximized for the first case. Thus, <math>x = \frac{69 + 1}{2} = 35</math>,
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...te parts to this problem: one is the color (gold vs silver), and the other is the orientation. ...t occur at all, for <math>9</math> total configurations. Thus, the answer is <math>70 \cdot 9 = \boxed{630}</math>.
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 6
  Let <math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>. Thus, we add 1 to each side in ord
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  ...DE</math> is concurrent with line <math>BC</math>. Then, <math>ABED</math> is an isosceles trapezoid so <math>AD=BE=10</math>, and <math>BC=8</math> and ...</math>. The [[Pythagorean Theorem]] yields that <math>GC^2 = 12^2 - \sqrt{3}^2 = 141</math>, so <math>EF = GC = \sqrt{141}</math>. Therefore, <math>AB
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...2^{222x+1} + 1 </math> has three [[real]] [[root]]s. Given that their sum is <math>m/n</math> where <math> m </math> and <math> n </math> are [[relative ...</math> and <math>x_1 + x_2 + x_3 = \frac{2}{111}</math>. Thus the answer is <math>111 + 2 = \boxed{113}</math>.
  1 KB (161 words) - 19:50, 2 January 2022
• 2005 AIME I Problems/Problem 9
  ...[probability]] of the entire [[surface area]] of the larger cube is orange is <math> \frac{p^a}{q^br^c}, </math> where <math> p,q, </math> and <math> r < ...orientations, so from these cubes we gain a factor of <math>\left(\frac{2}{3}\right)^6</math>.
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 10
  ...[[midpoint]] <math>M</math> of [[line segment]] <math>\overline{BC}</math> is <math>\left(\frac{35}{2}, \frac{39}{2}\right)</math>. The equation of the m ...tion for the triangle will give a smaller value of <math>p+q</math>, which is provable by following these steps over again) (alternatively, we could use
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  ...e]] whose sides have length 8. Given the maximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> ...n it touches both other sides of the square. This can happen only when it is arranged so that the center of the semicircle lies on one diagonal of the s
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...squares less than <math>n</math>. So <math>S(1), S(2)</math> and <math>S(3)</math> are odd, while <math>S(4), S(5), \ldots, S(8)</math> are even, and ...t the numbers between <math>1^2</math> and <math>2^2</math>, between <math>3^2</math> and <math>4^2</math>, and so on, all the way up to the numbers bet
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 13
  ...th>U</math> represent a move upwards, and <math>D</math> to be a move that is diagonal. [[Casework]] upon the number of diagonal moves: *'''Case ''' <math>d = 1</math>: It is easy to see only <math>2</math> cases.
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME I Problems/Problem 14
  ...e the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by <math>1000</math>. Consider a point <math>E</math> such that <math>AE</math> is [[perpendicular]] to <math>BD</math>, <math>AE</math> intersects <math>BD</
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  ...re <math> m </math> and <math> n </math> are integers and <math> n </math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, ...thout loss of generality, let <math>AC < AB</math>, so that <math>E</math> is between <math>D</math> and <math>C</math>. Let the length of the median be
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  ...or which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </math> and interna ...centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math>|r_1 - r_2|</math>. So we have
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  ...th> \overline{BC} </math> with <math> CD=6. </math> Point <math> E </math> is on <math> \overline{BC} </math> such that <math> \angle BAE\cong \angle CAD ...{BE} - 1 \Longrightarrow BE = \frac{13^2 \cdot 15}{463}</math>. The answer is <math>q = \boxed{463}</math>.
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases} ...st <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of posit
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...> and <math> c </math> are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> real x = 20 - ((750)^.5)/3, CE = 8*(6^.5) - 4*(5^.5), CD = 8*(6^.5), h = 4*CE/CD;
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  ...oots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{3 ...nomial]] <math>P</math> is very difficult to work with directly, but there is one obvious transformation to make: sum the [[geometric series]]:
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 12
  ...on <math> [z] </math> denotes the [[floor function|greatest integer]] that is less than or equal to <math> z. </math> <math>\left\lfloor\log_2\left(\frac{1}{x}\right)\right\rfloor</math> is even when
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 11
  ...s a 3-inch radius. The entire [[surface]] of the cone, including its base, is painted. A [[plane]] [[parallel]] to the base of the cone divides the cone ...face area]] <math>A = \pi r^2 + \pi r \ell</math>, where <math>\ell</math> is the [[slant height]] of the cone. Using the [[Pythagorean Theorem]], we ge
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 10
  ...[[probability]] that the circle will not touch diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...nter of the circle must be in the <math>34 \times 13</math> rectangle that is one unit away from the sides of rectangle <math>ABCD</math>. We want to fin
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME I Problems/Problem 9
  ...<math> U_1 </math> is similar to <math> U_2 </math> and <math> V_1 </math> is similar to <math> V_2. </math> The minimum value of the area of <math> U_1 ...h>ABC</math>. Thus <math>U_1</math>, and hence <math>U_2</math>, are <math>3-4-5\,\triangle</math>s.
  4 KB (618 words) - 20:01, 4 July 2013
• 2004 AIME I Problems/Problem 1
  ...to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
  2 KB (374 words) - 14:53, 27 December 2019
• 2004 AIME I Problems/Problem 2
  ...atest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find <math>m.</math> ...must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 4
  ...<math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <math> 100k</math>. ...e at each corner of the square. The area enclosed by all of the midpoints is <math>4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86</math> to the n
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 5
  ...</math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>? From here, we see the largest possible value of <math>a+b</math> is <math>349</math>.
  3 KB (436 words) - 18:31, 9 January 2024
• 2004 AIME I Problems/Problem 6
  ...s [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 ...into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  ...ath>Q(x)</math> is some polynomial [[divisibility | divisible]] by <math>x^3</math>. ...x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME I Problems/Problem 8
  There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are sim ...of this <math>n</math>-gon in a counterclockwise direction: <math>0, 1, 2, 3, \ldots, n-1.</math>
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  ...to right. What is the sum of the possible remainders when <math> n </math> is divided by 37? ...t element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m. </math>
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...th>256</math> by <math>1</math> strip of quadruple thickness. This process is repeated <math>8</math> more times. After the last fold, the strip has beco Number the squares <math>0, 1, 2, 3, ... 2^{k} - 1</math>. In this case <math>k = 10</math>, but we will consi
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...ht <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expressi ...g by <math>7</math>, <math>a + 11b + 111c = 1000</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 13
  ...of triangle <math> ABC </math> and the area of triangle <math> EBD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...B \parallel CE, BC \parallel AD, </math> it follows that <math>ABCF</math> is a [[parallelogram]], and so <math>\triangle ABC \cong \triangle CFA</math>.
  3 KB (486 words) - 22:15, 7 April 2023
• 2004 AIME II Problems/Problem 12
  ..., </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any real r = (-60 + 48 * 3^.5)/23;
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...ath> S, </math> the [[probability]] that it is divisible by <math>9</math> is <math> p/q, </math> where <math> p </math> and <math> q </math> are relativ ...{40}{2}</math> because we’re choosing 2 1s to go in 40 digit slots. This is equal to 780; we have found <math>q</math>, our denominator.
  8 KB (1,283 words) - 19:19, 8 May 2024

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