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Physics books
.../www.amazon.com/exec/obidos/ASIN/0756613647/artofproblems-20 Chaos: Making a New Science] by James Gleick ...ics&qid=1583572681&sr=8-1 Physics for Scientists and Engineers] by Randall D. Knight

10 KB (1,410 words) - 13:07, 20 February 2024
Cauchy-Schwarz Inequality
..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro

13 KB (2,048 words) - 15:28, 22 February 2024
Complex number
...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.

5 KB (860 words) - 15:36, 10 December 2023
2004 AMC 12A Problems
{{AMC12 Problems|year=2004|ab=A}} ...A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D) } 2.9 \qquad \text{(E) } 29</math>

13 KB (1,953 words) - 00:31, 26 January 2023
2003 AMC 12A Problems
{{AMC12 Problems|year=2003|ab=A}} ...m{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>

13 KB (1,955 words) - 21:06, 19 August 2023
2002 AMC 12A Problems
{{AMC12 Problems|year=2002|ab=A}} ...\frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </math>

12 KB (1,792 words) - 13:06, 19 February 2020
2001 AMC 12 Problems
...A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D)} 2S + 6 \qquad \text{(E)}\ 2S + 12</math> ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a

13 KB (1,957 words) - 12:53, 24 January 2024
2002 AMC 12B Problems
{{AMC12 Problems|year=2002|ab=B}} ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc

10 KB (1,547 words) - 04:20, 9 October 2022
2004 AMC 12B Problems
...athrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \qquad (\mathrm {E}) 15</math> ...>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and 3, although not necessarily in that order. What is

13 KB (2,049 words) - 13:03, 19 February 2020
2006 AMC 12A Problems/Problem 24
...uad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad\mathrm{(E) \ } 2,015,028</math> <cmath>\sum_{a+b+c=2006}{\frac{2006!}{a!b!c!}x^ay^bz^c}</cmath>

8 KB (1,332 words) - 17:37, 17 September 2023
Set
...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <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
2002 AIME I Problems
{{AIME Problems|year=2002|n=I}} ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-

8 KB (1,374 words) - 21:09, 27 July 2023
2003 AIME I Problems
...centric circles with radii <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius 1 is colored red, and each regi ...et <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math>

6 KB (965 words) - 16:36, 8 September 2019
2002 AIME I Problems/Problem 15
...CD</math> is a square with <math>AB = 12;</math> face <math>ABFG</math> is a trapezoid with <math>\overline{AB}</math> parallel to <math>\overline{GF},< ...= (6,-6,0), E = (2,0,12), H=(-6+2*sqrt(19),0,12), H1=(-6-2*sqrt(19),0,12), F, G, E1 = (6,0,12);

7 KB (1,181 words) - 20:32, 8 January 2024
2002 AIME I Problems/Problem 13
...{CE}</math> to intersect the [[circumcircle]] of <math>ABC</math> at <math>F</math>. The area of triangle <math>AFB</math> is <math>m\sqrt{n}</math>, wh ...,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C));

6 KB (974 words) - 13:01, 29 September 2023
2002 AIME I Problems/Problem 10
...AD}</math> bisects angle <math>CAB</math>. Points <math>E</math> and <math>F</math> are on <math>\overline{AB}</math> and <math>\overline{AC}</math>, re ...> is <math>10/3</math> that of triangle <math>AEG</math>, since they share a common side and angle, so the area of triangle <math>AGF</math> is <math>10

4 KB (643 words) - 22:44, 8 August 2023
IMO Problems and Solutions
* [[1960 IMO Problems/Problem 3 | Problem 3]] proposed by Gheorghe D. Simionescu, Romania * [[1961 IMO Problems/Problem 6 | Problem 6]] proposed by Gheorghe D. Simionescu, Romania

35 KB (4,009 words) - 20:25, 21 February 2024
P versus NP
.... <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math> ver Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are

6 KB (1,104 words) - 15:11, 25 October 2017
2003 AMC 10A Problems
{{AMC10 Problems|year=2003|ab=A}} ...m{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>

13 KB (1,900 words) - 22:27, 6 January 2021
2002 IMO Shortlist Problems/N3
...ry divisor <math>a </math> of <math>m </math>, <math>ka </math> is clearly a divisor of <math>km </math>, but not <math>m </math>. d(2^{uv} + 1) \ge d \left[ (2^{u}+1)(2^{v}+1)/3 \right] \ge 4^{n}

10 KB (1,739 words) - 06:38, 12 November 2019
2002 USAMO Problems/Problem 5
...h>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by ...ive]] (<math> a \leftrightarrow b \leftrightarrow c </math> implies <math> a \leftrightarrow c </math> ).

7 KB (1,194 words) - 15:39, 28 March 2015
Mock AIME 2 Pre 2005/Problems
<math>x</math> is a real number with the property that <math>x + \frac{1}{x} = 3</math>. Let < ...l, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed.

6 KB (990 words) - 15:23, 11 November 2009
2002 USA TST Problems
Problems from the 2002 USA [[TST]]. Let <math> \displaystyle ABC </math> be a triangle. Prove that

3 KB (544 words) - 06:58, 3 August 2017
2003 USAMO Problems
A convex polygon <math> \mathcal{P} </math> in the plane is dissected into sm A = a_0,a_1,a_2,\dots, a_n

3 KB (487 words) - 09:21, 14 May 2021
2002 PMWC Problems
[[2002 PMWC Problems/Problem I1|Solution]] [[2002 PMWC Problems/Problem I2|Solution]]

5 KB (725 words) - 16:07, 23 April 2014
2002 AMC 12B Problems/Problem 24
...] [[quadrilateral]] <math>ABCD</math> with area <math>2002</math> contains a point <math>P</math> in its interior such that <math>PA = 24, PB = 32, PC = ...8465} \qquad \mathrm{(C)}\ 2</math> <math>(48+\sqrt{2002}) \qquad \mathrm{(D)}\ 2\sqrt{8633} \qquad \mathrm{(E)}\ 4(36 + \sqrt{113})</math>

2 KB (313 words) - 10:23, 4 July 2013
2007 AMC 10A Problems
{{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?

13 KB (2,058 words) - 17:54, 29 March 2024
2002 AMC 12B Problems/Problem 25
Let <math>f(x) = x^2 + 6x + 1</math>, and let <math>R</math> denote the [[set]] of [[po <cmath>f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0</cmath>

2 KB (365 words) - 14:48, 7 March 2022
2003 AMC 12B Problems/Problem 24
...ive integers <math>a,b,</math> and <math>c</math> are chosen so that <math>a<b<c</math>, and the system of [[equation]]s <center><math>2x + y = 2003 \quad</math> and <math>\quad y = |x-a| + |x-b| + |x-c|</math></center>

7 KB (1,183 words) - 11:47, 15 February 2016
2002 AMC 10A Problems/Problem 25
pair A,B,C,D; A=(0,0);

8 KB (1,308 words) - 07:05, 19 December 2022
Aczel's Inequality
Consider the function <math>f(x)=(a_1 x - b_1)^2-\sum_{i=2}^n(a_i x - b_i)^2=</math> <math>(a_1^2-a_2^2-\ ...x)</math> must have at least one root, <math>\Leftrightarrow </math> <math>D=(a_1b_1-a_2b_2-\cdots -a_nb_n)^2- (a_1^2-a_2^2-\cdots -a_n^2)(b_1^2-b_2^2-\

2 KB (428 words) - 16:36, 29 December 2021
2002 USAMO Problems
Problems of the [[2002 USAMO | 2002]] [[USAMO]]. ...elements, and let <math>N </math> be an integer with <math> 0 \le N \le 2^{2002} </math>. Prove that it is possible to color every subset of <math>S </mat

3 KB (486 words) - 09:21, 14 May 2021
2002 AMC 10A Problems
{{AMC10 Problems|year=2002|ab=A}} The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers?

11 KB (1,733 words) - 11:04, 12 October 2021
2002 AMC 10A Problems/Problem 20
Points <math>A,B,C,D,E</math> and <math>F</math> lie, in that order, on <math>\overline{AF}</math>, dividing it into ...text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2</math>

3 KB (439 words) - 22:15, 9 June 2023
2004 AMC 10B Problems
...math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club? ...} 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726 </math>

13 KB (1,988 words) - 20:19, 15 May 2024
2001 AMC 10 Problems
...\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 10 \qquad\textbf{(E) } 11</math> A number <math>x</math> is <math>2</math> more than the product of its recipr

14 KB (1,983 words) - 16:25, 2 June 2022
2002 AMC 10B Problems/Problem 17
A regular octagon <math>ABCDEFGH</math> has sides of length two. Find the ar ...d \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2</math>

2 KB (273 words) - 13:27, 21 May 2021
2002 AMC 12A Problems/Problem 14
...h>n</math>, let <math>f(n)=\log_{2002} n^2</math>. Let <math>N=f(11)+f(13)+f(14)</math>. Which of the following relations is true? \text{(A) }N<1

694 bytes (106 words) - 20:43, 25 August 2022
2002 AMC 12A Problems/Problem 19
...math>f</math> is shown below. How many solutions does the equation <math>f(f(x))=6</math> have? \text{(A) }2

2 KB (350 words) - 11:09, 18 July 2023
2002 AMC 12A Problems/Problem 23
...and the [[perpendicular bisector]] of <math>BC</math> meet in point <math>D</math>, and <math>BD</math> bisects <math>\angle ABC</math>. If <math>AD=9< ...h>\text{(A)}\ 14 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 14\sqrt5 \qquad \text{(E)}\ 28\sqrt5</math>

6 KB (899 words) - 01:41, 5 July 2023
2011 AMC 12A Problems/Problem 23
...a</math> and <math>b</math> are complex numbers. Suppose that <math>\left| a \right| = 1</math> and <math>g(g(z))=z</math> for all <math>z</math> for wh \textbf{(A)}\ 0 \qquad

3 KB (519 words) - 15:49, 5 November 2023
2002 AMC 10B Problems/Problem 24
...any seconds does it take a rider to travel from the bottom of the wheel to a point <math>10</math> vertical feet above the bottom? ...(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15 </math>

3 KB (559 words) - 02:44, 8 February 2024
2003 AMC 8 Problems
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of ...thrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22 \qquad\mathrm{(E)}\ 26</math>

16 KB (2,236 words) - 12:02, 19 February 2024
2002 AMC 8 Problems/Problem 20
The area of triangle <math>XYZ</math> is 8 square inches. Points <math>A</math> and <math>B</math> are midpoints of congruent segments <math>\overli /* AMC8 2002 #20 Problem */

9 KB (1,488 words) - 23:09, 28 December 2023
2002 Romanian NMO Problems
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h [[2002 Romanian NMO Problems/Grade 7/Problem 1|Solution]]

10 KB (1,695 words) - 10:03, 10 May 2012
2018 AIME I Problems/Problem 10
...mber of paths with <math>15</math> steps that begin and end at point <math>A</math>. Find the remainder when <math>n</math> is divided by <math>1000.</m for(int d = 90; d < 360 + 90; d += 72){

11 KB (1,934 words) - 12:18, 29 March 2024
2001 JBMO Problems
Solve the equation <math>a^3 + b^3 + c^3 = 2001</math> in positive integers. Let <math>ABC</math> be a triangle with <math>\angle C = 90^\circ</math> and <math>CA \ne CB</math>.

2 KB (258 words) - 18:09, 8 December 2018
2003 JBMO Problems
...ath>n</math> digits, each of which is 8. Prove that <math>A+2B+4</math> is a perfect square. Suppose there are <math>n</math> points in a plane no three of which are collinear with the property that if we label th

2 KB (330 words) - 13:59, 25 August 2018
2002 Pan African MO Problems
...all <math>n \in N_0</math> and the minimum of the set <math>\{ f(0), f(1), f(2) \cdots \}</math> is <math>1</math>. [[2002 Pan African MO Problems/Problem 1|Solution]]

2 KB (329 words) - 13:53, 4 December 2019
Mock AIME 2 Pre 2005 Problems
<math>x</math> is a real number with the property that <math>x+\tfrac1x = 3</math>. Let <math>S ...ll, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. T

6 KB (1,052 words) - 13:52, 9 June 2020

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