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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 order to complete the fourth power

4 KB (686 words) - 01:55, 5 December 2022
2005 AIME I Problems/Problem 7
label("$10$",(2.5,4.5),W); label("$10$",(18.37,4.5),E);

4 KB (567 words) - 20:20, 3 March 2020
2005 AIME I Problems/Problem 8
...h> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...hen the previous statement says that <math>2^{111\cdot(x_1 + x_2 + x_3)} = 4</math> so taking a [[logarithm]] of that gives <math>111(x_1 + x_2 + x_3) =

1 KB (161 words) - 19:50, 2 January 2022
2005 AIME I Problems/Problem 9
...cube, we need that they show an orange face. This happens in <math>\frac{4}{6} = \frac{2}{3}</math> of all orientations, so from these cubes we gain a ...the corner cubes together contribute a probability of <math>\left(\frac{1}{4}\right)^8 = \frac{1}{2^{16}}</math>

4 KB (600 words) - 21:44, 20 November 2023
2005 AIME I Problems/Problem 10
== Solution 4 == {{AIME box|year=2005|n=I|num-b=9|num-a=11}}

5 KB (852 words) - 21:23, 4 October 2023
2005 AIME I Problems/Problem 11
...mum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> === Solution 4 ===

4 KB (707 words) - 11:11, 16 September 2021
2005 AIME I Problems/Problem 12
...e the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </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 <math>S(9), \ldots, S(15)</math>

4 KB (647 words) - 02:29, 4 May 2021
2005 AIME I Problems/Problem 13
..., and <math>3</math> <math>D</math>'s, so the string is divided into <math>4</math> partitions (<math>-D-D-D-</math>). ...<math>R</math>'s and <math>U</math>'s stay together, then there are <math>4 \cdot 3 = 12</math> places to put them.

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME I Problems/Problem 14
Consider the [[point]]s <math> A(0,12), B(10,9), C(8,0),</math> and <math> D(-4,7). </math> There is a unique [[square]] <math> S </math> such that each of ...th>AE = BD</math>, we have <math>9 - 7 = x_E - 0</math> and <math>10 - ( - 4) = 12 - y_E</math>

3 KB (561 words) - 14:11, 18 February 2018
2005 AIME I Problems/Problem 15
...| divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math> D(MP("A",A,s)--MP("B",B,s)--MP("C",C,N,s)--cycle); D(cir);

5 KB (906 words) - 23:15, 6 January 2024
2005 AIME II Problems/Problem 15
r + 4 &= \sqrt{(x-5)^2 + (y-12)^2} \\ D(CR(A,16));D(CR(B,4));D(shift((0,12)) * yscale(3^.5 / 2) * CR(C,10), linetype("2 2") + d + red)

12 KB (2,000 words) - 13:17, 28 December 2020
2005 AIME II Problems/Problem 14
dotfactor = 4; label("$A$",A,N);

13 KB (2,129 words) - 18:56, 1 January 2024
2004 AIME I Problems/Problem 15
...itive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> an ...icting our assumption that <math>20</math> was the smallest value of <math>n</math>. Using [[complementary counting]], we see that there are only <math>

9 KB (1,491 words) - 01:23, 26 December 2022
2004 AIME I Problems/Problem 14
...> feet. The unicorn has pulled the rope taut, the end of the rope is <math>4</math> feet from the nearest point on the tower, and the length of the rope 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
...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> ...he form <math>\frac m{19}</math> or <math>\frac n {17}</math> for <math>m, n > 0</math>.

2 KB (298 words) - 20:02, 4 July 2013
2004 AIME I Problems/Problem 12
...and <math> n </math> are relatively prime positive integers, find <math> m+n. </math> The notation <math> [z] </math> denotes the [[floor function|great <cmath>x \in \left(\frac{1}{2},1\right) \cup \left(\frac{1}{8},\frac{1}{4}\right) \cup \left(\frac{1}{32},\frac{1}{16}\right) \cup \cdots</cmath>

2 KB (303 words) - 22:28, 11 September 2020
2004 AIME I Problems/Problem 11
...h> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...lid has volume equal to <math>V = \frac13 \pi r^2 h = \frac13 \pi 3^2\cdot 4 = 12 \pi</math> and has [[surface area]] <math>A = \pi r^2 + \pi r \ell</ma

5 KB (839 words) - 22:12, 16 December 2015
2004 AIME I Problems/Problem 9
...<math> n</math> are [[relatively prime]] positive integers. Find <math> m+n. </math> ...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
...from left to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...+ 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
Solving for <math>f+l</math>, we find the sum of the two terms is <math>4</math>. <cmath>2(x-(-x+4)+1) = 1+(x+99)-(-x-99+1)</cmath>

8 KB (1,437 words) - 21:53, 19 May 2023

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