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...bda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...he left-hand side of the desired inequality but may contribute to the right-hand side.

4 KB (774 words) - 12:12, 29 October 2016
Mock AMC
Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to {| class="wikitable" style="text-align:center;width:100%"

51 KB (6,175 words) - 20:58, 6 December 2023
2006 AIME I Problems
Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </ma A collection of 8 cubes consists of one cube with edge-length <math> k </math> for each integer <math> k, 1 \le k \le 8. </math> A

7 KB (1,173 words) - 03:31, 4 January 2023
2006 AIME I Problems/Problem 15
Given 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 possi {{AIME box|year=2006|n=I|num-b=14|after=Last Question}}

6 KB (910 words) - 19:31, 24 October 2023
2006 AIME I Problems/Problem 13
...For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 ...math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math>

10 KB (1,702 words) - 00:45, 16 November 2023
LaTeX:Symbols
....)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list of supported commands]. [http://mirrors.ctan.org/info/symbols/comprehensive/symbols-a4.pdf The Comprehensive LaTeX Symbol List].

16 KB (2,324 words) - 16:50, 19 February 2024
2006 AIME I Problems/Problem 10
...will not work in general). Denote the intersection of the line and the [[x-axis]] as <math>(x, 0)</math>. ...ezoid]] and a [[rectangle]], and the areas are <math>\frac{1}{2}((1-x) + (2-x))(3)</math> and <math>2 \cdot 1 = 2</math>, totaling <math>\frac{13}{2} -

4 KB (731 words) - 17:59, 4 January 2022
2006 AIME I Problems/Problem 8
...ath>44.5^2</math> is also less than <math>2006</math>, so we have numbers 1-44, times two because 0.5 can be added to each of time, plus 1, because 0.5 ...have <math>[GIHJ]=0,</math> implying the minimum possible positive-integer-valued area is <math>1.</math>

5 KB (730 words) - 15:05, 15 January 2024
2021 JMC 10
<math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</m ...h>74</math> and <math>83</math> are pretentious. How many pretentious three-digit numbers are odd?

12 KB (1,784 words) - 16:49, 1 April 2021
2006 AMC 12B Problems
...1.99</math>, and <math>\textdollar0.99</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the <ma How many even three-digit integers have the property that their digits, read left to right, are

13 KB (2,058 words) - 12:36, 4 July 2023
2001 AMC 12 Problems
two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit o A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents

13 KB (1,957 words) - 12:53, 24 January 2024
2004 AMC 12B Problems
In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and

13 KB (2,049 words) - 13:03, 19 February 2020
2006 AMC 12B Problems/Problem 14
{{AMC12 box|year=2006|ab=B|num-b=13|num-a=15}}

1 KB (227 words) - 17:21, 8 December 2013
2006 AMC 12B Problems/Problem 25
A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> is a 1-1 correspondence between the odd and even numbers less than and relatively p

5 KB (924 words) - 12:02, 15 June 2022
2005 AMC 12B Problems/Problem 24
...ive x- axis, the answer is <math>\dfrac{\tan(BOJ) + \tan(BOJ+60) + \tan(BOJ-60)}{2}</math>. {{AMC12 box|year=2005|ab=B|num-b=23|num-a=25}}

4 KB (761 words) - 09:10, 1 August 2023
Root (polynomial)
...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. Let <cmath>P(x) = c_nx^n + c_{n-1}x^{n-1} + \dots + c_1x + c_0 = \sum_{j=0}^{n} c_jx^j</cmath> with all <math>c_j

8 KB (1,427 words) - 21:37, 13 March 2022
Prime ideal

967 bytes (176 words) - 18:08, 7 April 2012
2005 AIME II Problems/Problem 12
...0)); pair A=(0,9), B=(9,9), C=(9,0), D=(0,0), E=(2.5-0.5*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), G=(4.5,9), O=(4.5,4.5); draw(A--B--C--D--A);draw(E--O--F);dr ...*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), O=(4.5,4.5), G=O+(E-O)*dir(-90), J=O+(F-O)*dir(-90); draw(A--B--C--D--A);draw(E--O--F);draw(G--O--J);draw(F--G,linet

13 KB (2,080 words) - 21:20, 11 December 2022
2005 AIME I Problems/Problem 13
...n range from <math>l = 5</math> to <math>9</math>; notice that <math>d = 10-l</math> gives the number of diagonals. Let <math>R</math> represent a move ...jacent. The <math>D</math>'s split the string into three sections (<math>-D-D-</math>): by the [[Pigeonhole principle]] all of at least one of the two l

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME II Problems/Problem 14
...from Pythagorean theorem, <math>SC = \frac{18}{5}</math> and <math>AS = 14-SC = \frac{52}{5}</math>. We can also use the Pythagorean theorem on triangl ...and <math>RE</math> as <math>x</math>. <math>RB</math> then equals <math>13-y</math>. Then, we have two similar triangles.

13 KB (2,129 words) - 18:56, 1 January 2024
2004 AIME I Problems/Problem 7
where the left-hand sum can be computed from: <center><math>\sum_{i=1}^{15} S_i = S_{15} + \left(\sum_{i=1}^{7} S_{2i-1} + S_{2i}\right) = -15 + 7 = -8</math></center>

5 KB (833 words) - 19:43, 1 October 2023
2004 AIME II Problems/Problem 10
...math> is clearly never divisible by 9, so we can just focus on <math>1+2^{y-x}</math>. ...1+2^{y-x} \equiv 0 \pmod 9 \implies 2^{y-x} \equiv -1 \pmod 9 \implies 2^{y-x} \equiv 8 \pmod 9</cmath>

8 KB (1,283 words) - 19:19, 8 May 2024
2019 AIME I Problems/Problem 2
...\ne 0</math> because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry. The probability that <math>B-J = 1</math> is <math>\frac{19}{20 \times 19} = \frac{1}{20}</math> because

5 KB (830 words) - 22:15, 28 December 2023
1989 AIME Problems
<center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m

7 KB (1,045 words) - 20:47, 14 December 2023
1990 AIME Problems
Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>. <center><math>\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0</math></center>

6 KB (870 words) - 10:14, 19 June 2021
1998 AIME Problems

7 KB (1,084 words) - 02:01, 28 November 2023
2000 AIME II Problems
...ht distinguishable rings, let <math>n</math> be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order The equation <math>2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n

6 KB (947 words) - 21:11, 19 February 2019
2001 AIME II Problems
...the largest number of students who could study both languages. Find <math>M-m</math>. x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}

8 KB (1,282 words) - 21:12, 19 February 2019
2003 AIME II Problems
...th>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? ...ne{EF,}</math> <math>\overline{CD}\parallel \overline{FA},</math> and the y-coordinates of its vertices are distinct elements of the set <math>\{0,2,4,6

7 KB (1,127 words) - 09:02, 11 July 2023
1985 AIME Problems/Problem 10
...changes when <math>\{x\}=\frac{m}{n}</math>, where <math>m\in\{1,2,3,...,n-1\}</math>, <math>n\in\{2,4,6,8\}</math>. Using [[Euler's Totient Function]] ...6x\rfloor+\lfloor 8x\rfloor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</

12 KB (1,859 words) - 18:16, 28 March 2022
1985 AIME Problems/Problem 5
{{AIME box|year=1985|num-b=4|num-a=6}}

2 KB (410 words) - 13:37, 1 May 2022
1986 AIME Problems/Problem 11
The [[polynomial]] <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_ ...<math>1 - x + x^2 + \cdots - x^{17} = \frac {1 - x^{18}}{1 + x} = \frac {1-x^{18}}{y}</math>. Since <math>x = y - 1</math>, this becomes <math>\frac {1

6 KB (872 words) - 16:51, 9 June 2023
1987 AIME Problems/Problem 7
{{AIME box|year=1987|num-b=6|num-a=8}}

3 KB (547 words) - 22:54, 4 April 2016
1989 AIME Problems/Problem 5
...rac{1}{3}</math>. The answer we are looking for is <math>{5\choose3}(h)^3(1-h)^2 = 10\left(\frac{1}{3}\right)^3\left(\frac{2}{3}\right)^2 = \frac{40}{24 {{AIME box|year=1989|num-b=4|num-a=6}}

2 KB (258 words) - 00:07, 25 June 2023
1990 AIME Problems/Problem 9
...ing this pattern, we find that there are <math>\sum_{i=6}^{11} {i\choose{11-i}} = {6\choose5} + {7\choose4} + {8\choose3} + {9\choose2} + {{10}\choose1} ...th>n-1</math> flips must fall under one of the configurations of <math>S_{n-1}</math>.

3 KB (425 words) - 12:36, 12 May 2024
1991 AIME Problems/Problem 3
...>. Taking logarithms in both sides of this last equation and using the well-known fact <math>\log(a_{}^{}b)=\log a + \log b</math> (valid if <math>a_{}^ ...t[\prod_{j=1}^{k}\frac{(N-j+1)x}{j}\right]=\sum_{j=1}^{k}\log\left[\frac{(N-j+1)x}{j}\right]\, .

5 KB (865 words) - 12:13, 21 May 2020
2006 AMC 10B Problems/Problem 22
{{AMC10 box|year=2006|ab=B|num-b=21|num-a=23}}

2 KB (394 words) - 00:51, 25 November 2023
1993 AIME Problems/Problem 13
...h>, <math>CX=AC\cdot\left(\frac{CD}{AB-CD}\right)=200\cdot\left(\frac{t}{3t-t}\right)=100</math>. ...f this line with <math>AB</math> as <math>S</math>. Then <math>SB=AB-AS=3t-t=2t</math>.

8 KB (1,231 words) - 20:06, 26 November 2023
1994 AIME Problems/Problem 4
...th> such that <math>\lfloor\log_2{a}\rfloor=j</math>, and there are <math>n-2^k+1</math> such integers such that <math>\lfloor\log_2{a}\rfloor=k</math>. ...3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor= \sum_{j=0}^{k-1}(j\cdot2^j) + k(n-2^k+1) = 1994</math>.

2 KB (264 words) - 13:33, 11 August 2018
1995 AIME Problems/Problem 12
...>AP = 1.</math> It follows that <math>\triangle OPA</math> is a <math>45-45-90</math> [[right triangle]], so <math>OP = AP = 1,</math> <math>OB = OA = \ Without loss of generality, place the pyramid in a 3-dimensional coordinate system such that <math>A = (1,0,0),</math> <math>B =

8 KB (1,172 words) - 21:57, 22 September 2022
1996 AIME Problems/Problem 12
<cmath>|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.</cmath> ...obtained from these paired sequences are also obtained in another <math>2^5-1</math> ways by permuting the adjacent terms <math>\{a_1,a_2\},\{a_3,a_4\},

5 KB (879 words) - 11:23, 5 September 2021
1996 AIME Problems/Problem 7
...l be <br /> three other equivalent boards.</font></td><td><font style="font-size:85%">For those symmetric about the center, <br /> there is only one oth ...re rotationally symmetric about the center square; there are <math>\frac{49-1}{2}=24</math> such pairs. There are then <math>{49 \choose 2}-24</math> pa

4 KB (551 words) - 11:44, 26 June 2020
1996 AIME Problems/Problem 3
...itive [[integer]] <math>n</math> for which the expansion of <math>(xy-3x+7y-21)^n</math>, after like terms have been collected, has at least 1996 terms. ...3)^n</math>. Both [[binomial expansion]]s will contain <math>n+1</math> non-like terms; their product will contain <math>(n+1)^2</math> terms, as each t

3 KB (515 words) - 04:29, 27 November 2023
1998 AIME Problems/Problem 15
...s a path that retraces no segment. Each time that such a path reaches a non-terminal vertex, it must leave it. ...it can be arranged that <math>n-2</math> segments will emanate from <math>n-2</math> of the vertices and that an odd number of segments will emanate fro

9 KB (1,671 words) - 22:10, 15 March 2024
1999 AIME Problems/Problem 7
...math> take on the values <math>0, 1, \ldots, 9</math>. At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are ...}{d_{i}}= 2^{9-x_{i}}3^{9-y_{i}}5^{9-z_{i}}</math>. In general, the divisor-count of <math>\frac{N}{d}</math> must be a multiple of 4 to ensure that a s

3 KB (475 words) - 13:33, 4 July 2016
2001 AIME I Problems/Problem 15
.../math>, which can be done in <math>4! = 24</math> ways. Then choose a three-edge path along tetrahedron <math>DBEG</math> which, because it must start a ...faces, and one face adjacent to the three B-faces, which we will call the C-face.

11 KB (1,837 words) - 18:53, 22 January 2024
2001 AIME I Problems/Problem 12
...ween a plane and a point <math>I</math> can be calculated as <math>\frac{(I-G) \cdot P}{|P|}</math>, where G is any point on the plane, and P is a vecto ...rpendicular to plane <math>ABC</math> can be found as <math>V=(A-C)\times(B-C)=\langle 8, 12, 24 \rangle</math>

6 KB (1,050 words) - 18:44, 27 September 2023
2003 AIME I Problems/Problem 3
...1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math> she writes on her list the gre Each [[element]] of the [[set]] will appear in <math>7</math> two-element [[subset]]s, once with each other number.

2 KB (317 words) - 00:09, 9 January 2024
2003 AIME II Problems/Problem 15
...that <math> \left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right) \cdot (x-1) = x^{24} - 1 </math>. The five-element sum is just <math>\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \s

4 KB (675 words) - 17:23, 30 July 2022
2003 AIME II Problems/Problem 13
...ing the starting vertex in the next move. Thus <math>P_n=\frac{1}{2}(1-P_{n-1})</math>. ...-1</math> steps plus the number of ways to get to <math>C</math> in <math>n-1</math> steps.

15 KB (2,406 words) - 23:56, 23 November 2023

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