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Perfect square
An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect ...of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>

954 bytes (155 words) - 01:14, 29 November 2023
Uncountable
...or [[countably infinite]]. The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==

2 KB (403 words) - 20:53, 13 October 2019
Modular arithmetic/Intermediate
...quiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...<math>n</math>, the relation <math>a \equiv b</math> (mod <math>n</math>) is an [[equivalence relation]] on the set of integers. This relation gives ri

14 KB (2,317 words) - 19:01, 29 October 2021
Right triangle
A '''right triangle''' is any [[triangle]] with an angle of 90 degrees (that is, a [[right angle]]). A = (0, 3);

3 KB (499 words) - 23:41, 11 June 2022
2006 AIME I Problems
..., \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o

7 KB (1,173 words) - 03:31, 4 January 2023
2006 AIME I Problems/Problem 15
...n that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible Suppose <math>b_{i} = \frac {x_{i}}3</math>.

6 KB (910 words) - 19:31, 24 October 2023
2006 AIME I Problems/Problem 14
...notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) currentprojection = perspective(1,-10,3.3);

6 KB (980 words) - 21:45, 31 March 2020
2006 AIME I Problems/Problem 13
...atest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...h>k</math> is odd, then <math>n+1</math> is even, hence <math>k+n-1</math> is odd, and <math>S_n</math> cannot be a perfect square. Hence <math>k</math>

10 KB (1,702 words) - 00:45, 16 November 2023
2006 AIME I Problems/Problem 11
...h> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: ...s than can be constructed. What is the [[remainder]] when <math> T </math> is divided by 1000?

3 KB (436 words) - 05:40, 4 November 2022
2006 AIME I Problems/Problem 10
...d <math> c </math> are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> int[] array={3,3,2};

4 KB (731 words) - 17:59, 4 January 2022
2006 AIME I Problems/Problem 9
The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[rat ...<math>a, r</math> [[positive integer]]s. <math>a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}</math> so <math>a^{2}r^{11}=2^{1003}</math>.

4 KB (651 words) - 18:27, 22 May 2021
2006 AIME I Problems/Problem 8
...he area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</ma ...(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);

5 KB (730 words) - 15:05, 15 January 2024
2006 AIME I Problems/Problem 7
...[[region]] <math> C </math> to the area of shaded region <math> B </math> is 11/5. Find the ratio of shaded region <math> D </math> to the area of shade pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);

4 KB (709 words) - 01:50, 10 January 2022
2006 AIME I Problems/Problem 5
...rt{10}+144\sqrt{15}+2006}</math> can be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[p <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</cmath>

3 KB (439 words) - 18:24, 10 March 2015
2006 AIME I Problems/Problem 4
...h> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. ...ng into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.

2 KB (278 words) - 08:33, 4 November 2022
2006 AIME I Problems/Problem 3
...]] such that when its leftmost [[digit]] is deleted, the resulting integer is <math>\frac{1}{29}</math> of the original integer. ...7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives <cmath>7 \cdot 10^n = 28N_0</cmath> or <cm

4 KB (622 words) - 03:53, 10 December 2022
2006 AIME I Problems/Problem 2
...<math> \mathcal{A} </math> be a 90-[[element]] [[subset]] of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o ...995</math> are possible values of S, so the number of possible values of S is <math>4995-4095+1=901</math>.

1 KB (189 words) - 20:05, 4 July 2013
2022 AIME I Problems/Problem 1
...x)</math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. so the slope of <math>R(x)</math> is <math>\frac{106-108}{20-16}=-\frac12.</math>

4 KB (670 words) - 13:03, 13 November 2023
2021 JMC 10
What is the value of <cmath>\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?</cmath> <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\te

12 KB (1,784 words) - 16:49, 1 April 2021
2006 AMC 12B Problems
What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>? .../math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>?

13 KB (2,058 words) - 12:36, 4 July 2023

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