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• Order of operations
  The '''order of operations''' is a [[mathematical convention]] for [[arithmetic]] computation. The below li ...m PEMDAS. An AoPS mnemonic you can use to remember the order of operations is "Please Evaluate, My Dear AoPS Students".
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• Octahedron
  An '''octahedron''' is a type of [[polyhedron]]. ...eometry]], an octahedron is any polyhedron with eight [[face]]s. The term is most frequently to refer to a polyhedron with eight [[triangular]] faces, w
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• Circumradius
  ...f the [[circle]] that [[circumscribe]]s the triangle. Since every triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle] ...rea of the triangle. Then, the measure of the circumradius of the triangle is simply <math>R=\frac{abc}{4A}</math>. This can be rewritten as <math>A=\fra
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• Sum and difference of powers
  If <math>p</math> is a positive integer and <math>x</math> and <math>y</math> are real numbers, <math>x^3-y^3=(x-y)(x^2+xy+y^2)</math>
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• LaTeX:Layout
  ...style are items that are set in the preamble. Often, much of the preamble is placed in a separate file and included using the \usepackage statement. Th ...defaults. One option that might need modification is the font size, which is 10pt by default but can be increased to 11pt or 12pt. A reference on other
  30 KB (5,171 words) - 10:16, 4 April 2021
• LaTeX:Packages
  ...ds or a basic preamble that you can include in any of your LaTeX documents is simple. Just follow these steps: ...X 2.9\tex\latex'''. (If you have an older MiKTeX installation, this folder is probably '''C:\texmf\tex\latex'''.)</li>
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• LaTeX:Commands
  | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2} |<math>\sqrt{3}</math>||\sqrt{3}
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• Imaginary number
  An '''imaginary number''' is a [[complex number]] whose [[real part]] is equal to 0. In the [[complex plane]], these numbers lie on the [[imaginary ...= \sqrt{-1}</math> is the [[imaginary unit]] and <math>\textrm{Im}</math> is the [[imaginary part]] [[function]].
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• Science Olympiad
  '''Science Olympiad''' is a national team-based science competition primarily for middle high school ...school is allowed to have up to five ninth graders on its team. Division C is generally for high school students in grades 10-12, though ninth graders ar
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• Chicken McNugget Theorem
  ...e form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. ...t in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.
  17 KB (2,748 words) - 19:22, 24 February 2024
• Dodecagon
  ...dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\circ}</math>. Each of its exterior angles has measure <math>3 D=dir(360/12*3);
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• Triangular number
  The <math>n^{th}</math> triangular number is the sum of all natural numbers from one to n. That is, the <math>n^{th}</math> triangle number is
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• 2005 AMC 12A Problems/Problem 24
  ...math>Q(x)</math> does there exist a polynomial <math>R(x)</math> of degree 3 such that <math>P(Q(x))=P(x) \cdot R(x)</math>? <math>P(Q(x))=(Q(x)-1)(Q(x)-2)(Q(x)-3)=P(x)\cdot R(x)=(x-1)(x-2)(x-3)\cdot R(x)</math>.
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• 2005 AMC 12A Problems/Problem 23
  ...}\}</math>. What is the [[probability]] that <math>\mathrm{log}_a b</math> is an [[integer]]? ...<math>\sum_{x=1}^{12} \lfloor\frac {25}x-1\rfloor = 24 + 11 + 7 + 5 + 4 + 3 + 2 + 2 + 1 + 1 + 1 + 1 = 62</math>.
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• 2007 AMC 12A Problems/Problem 2
  ...with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? ...th>4000 cm^3</math>; therefore, when the volume increases by <math>8000 cm^3</math>, the water level rises <math>2 cm \Rightarrow\fbox{D}</math>
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• 2007 AMC 12A Problems/Problem 3
  The larger of two consecutive odd integers is three times the smaller. What is their sum? *Thus, the answer is <math>1+(1+2)=4 \mathrm{(A)}</math>
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• 2007 AMC 12A Problems/Problem 6
  ...measures 40 degrees, and angle <math>ADC</math> measures 140 degrees. What is the degree measure of angle <math>BAD</math>? Since triangle <math>ABC</math> is isosceles we know that angle <math>\angle BAC = \angle BCA</math>.
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• 2007 AMC 12A Problems/Problem 18
  ...</math> has real coefficients, and <math>f(2i) = f(2 + i) = 0.</math> What is <math>a + b + c + d?</math> <math>x^4 - 4x^3 + 9x^2 - 16x + 20 = 0</math></div>
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• 2007 AMC 12A Problems/Problem 9
  ...s fast as he walks, and both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from the stad ...-1 = \frac{4}{3}</math>, so the answer is <math>\boxed{\mathrm{(B)}\ \frac{3}{4}}</math>
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• Proportion
  Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers <math>x</mat where <math>k</math> is some [[real number]].
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• Parity
  '''Parity''' refers to whether a number is [[even]] or [[odd]]. While this may seem highly basic, checking the parity of numbers is often an useful tactic for solving problems, especially with [[proof by con
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• 2007 AMC 12A Problems/Problem 13
  ...ouse starts getting farther from the cheese rather than closer to it. What is <math>a+b</math>? ...th>. The perpendicular has slope <math>\frac{1}{5}</math>, so its equation is <math>y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}{5}</math>. The <math>x<
  2 KB (387 words) - 18:20, 27 November 2023
• 2007 AMC 12A Problems/Problem 14
  What is <math>a+b+c+d+e</math>? ...th>e</math> are <math>9, 7, 5, 3,</math> and <math>1,</math> and their sum is <math>\fbox{25 (C)}</math>
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• 2007 AMC 12A Problems/Problem 22
  ...sum of the digits of <math>n.</math> For how many values of <math>n</math> is <math>n + S(n) + S(S(n)) = 2007?</math> <math>\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5</math>
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• 2007 AMC 12A Problems/Problem 23
  ...y = 2\log_{a}x,</math> and <math>y = 3\log_{a}x,</math> respectively. What is <math>a?</math> ...A)}\ \sqrt [6]{3}\qquad \mathrm{(B)}\ \sqrt {3}\qquad \mathrm{(C)}\ \sqrt [3]{6}\qquad \mathrm{(D)}\ \sqrt {6}\qquad \mathrm{(E)}\ 6</math>
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• Proportion/Intermediate
  ...ath>. If <math>x = 8</math> when <math>y = 1/2</math> and <math>z = \sqrt {3}/2</math>, find <math>y</math> when <math>x = 1</math> and <math>z = 6</mat ...the problem. Using the second gives us <math>1*y=y=4</math>, so the answer is <math>y=4</math>.
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• 2007 AMC 12A Problems/Problem 24
  ...<math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>? <math>F(2)=3</math>
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• 2007 AMC 12A Problems/Problem 16
  ...ee-digit numbers are composed of three distinct digits such that one digit is the [[average]] of the other two? We can find the number of increasing [[arithmetic sequence]]s of length 3 possible from 0 to 9, and then find all the possible permutations of these
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• 2007 AMC 12A Problems/Problem 17
  ...n b = \sqrt{\frac{5}{3}}</math> and <math>\cos a + \cos b = 1</math>. What is <math>\cos (a - b)</math>? <math>\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad \mathrm{(B)}\ \frac 13\qquad \mathrm{(C)}\ \frac 12\qquad \math
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• 2007 AMC 12A Problems/Problem 19
  ...),</math> <math>D = (680,380),</math> and <math>E = (689,389).</math> What is the sum of all possible x coordinates of <math>A</math>? ...= \frac 12bh</math>, we have that the height of <math>\triangle ABC</math> is <math>h = \frac{2k}{b} = \frac{2007 \cdot 2}{223} = 18</math>. Thus <math>A
  4 KB (565 words) - 17:01, 2 April 2023
• 2007 AMC 12A Problems/Problem 25
  ...out of any three consecutive integers. How many [[subset]]s of <math>\{1,2,3,\ldots,12\},</math> including the [[empty set]], are spacy? ...ath>\{ 1, 2, ... n \}</math>. We have <math>S_{0} = 1, S_{1} = 2, S_{2} = 3</math>.
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• 2005 AMC 12A Problems/Problem 18
  ...ber ''prime-looking'' if it is [[composite]] but not divisible by <math>2, 3,</math> or <math>5.</math> The three smallest prime-looking numbers are <ma ...ath>1000 - (168-3) - 1 - |S_2 \cup S_3 \cup S_5|</math> (note that <math>2,3,5</math> are primes).
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• 2005 AMC 12A Problems/Problem 19
  A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odo ...r both the other two [[intersection]]s. The intersection of all three sets is just <math>2</math>. So we get:
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• 2005 AMC 12A Problems/Problem 21
  ...\ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4</math> ...onent of <math>a</math> becomes huge, and since <math>a \ge 2</math> there is no way we can satisfy the second condition. Hence we have two ordered tripl
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• 2005 AMC 12A Problems/Problem 2
  ...C 12A Problems|2005 AMC 12A #2]] and [[2005 AMC 10A Problems|2005 AMC 10A #3]]}} ...+ 7 = 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <math>b</math>?
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• Tau
  ...of [[radians]] in a circle. For a convincing proof that <math>\tau</math> is a better circle constant than <math>\pi</math>, see [http://www.tauday.com == Why <math>\tau</math> Is Better Than <math>\pi</math>==
  2 KB (386 words) - 21:18, 22 November 2021
• 2005 AMC 12A Problems/Problem 10
  ...ly one-fourth of the total number of faces of the unit cubes are red. What is <math>n</math>? (\mathrm {A}) \ 3 \qquad (\mathrm {B}) \ 4 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 6 \
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• Regular tetrahedron/Introductory problem
  <math>\mathrm {A} =\dfrac{4\sqrt{3}}{4}=\sqrt{3}</math> <math>\mathrm {lh}=\sqrt{2^2-1^2}=\sqrt{3}</math>
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• 2005 AMC 12A Problems/Problem 8
  What is <math>A</math>? (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 5
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• 2005 AMC 12A Problems/Problem 9
  ...^2 + ax + 8x + 9 = 0</math> has only one solution for <math>x</math>. What is the sum of these values of <math>a</math>? ...o <math>\pm 12 = a + 8 \Longrightarrow a = 4, -20</math>. The sum of these is <math>-20 + 4 = -16 \Rightarrow \mathrm{(A)}</math>.
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• 2005 AMC 12A Problems/Problem 11
  How many three-digit numbers satisfy the property that the middle digit is the [[mean|average]] of the first and the last digits? ...<math>C</math> and is unique for each <math>(A,C)</math>. Thus our answer is <math>9 \cdot 5 \cdot 1 = 45 \implies E</math>.
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• 2005 AMC 12A Problems/Problem 12
  (\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 9 The slope of the line is<math>
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• 2005 AMC 12A Problems/Problem 13
  numbers 3, 5, 6, 7 and 9, although not necessarily in that order. The sums of the arithmetic sequence, although not necessarily in that order. What is the middle
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• 2005 AMC 12A Problems/Problem 14
  ...om with each dot equally likely to be chosen. The die is then rolled. What is the [[probability]] that the top face has an odd number of dots? There are <math>1 + 2 + 3 + 4 + 5 + 6 = 21</math> dots total. [[Casework]]:
  2 KB (245 words) - 20:07, 4 March 2024
• 2005 AMC 12A Problems/Problem 15
  ...rline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the [[ratio]] of the area of <math>\triangle DCE</math> to the area of <mat dotfactor=3;
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• 2005 AMC 12A Problems/Problem 17
  ...r along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex <math>W</math>? path d= (0,0)--(3,13)--(13,13)--(10,0);
  2 KB (215 words) - 13:56, 19 January 2021
• 2004 AMC 12A Problems/Problem 10
  The sum of <math>49</math> consecutive integers is <math>7^5</math>. What is their [[median]]? <math>\text {(A)}\ 7 \qquad \text {(B)}\ 7^2\qquad \text {(C)}\ 7^3\qquad \text {(D)}\ 7^4\qquad \text {(E)}\ 7^5</math>
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• 2006 AIME II Problems/Problem 3
  ...st integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math> ...irst <math>100</math> positive odd integers can be written as <math>1\cdot 3\cdot 5\cdot 7\cdots 195\cdot 197\cdot 199=\frac{1\cdot 2\cdots200}{2\cdot4\
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• 2006 AIME II Problems/Problem 6
  ...b, </math> and <math> c </math> are positive integers and <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> dotfactor=3;
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• 2006 AIME II Problems/Problem 14
  ...the smallest positive integer <math>n</math> for which <math> S_n </math> is an integer. ...th>. Since we're looking for the smallest such <math>n</math>, the answer is <math>\boxed{063}</math>.
  2 KB (263 words) - 23:32, 28 February 2021
• 2006 AIME II Problems/Problem 8
  ...paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/ ...choices for the three outer triangles, if two are one color and the third is a different color.
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• 2006 AIME II Problems/Problem 12
  .../math> and <math> r</math> are [[relatively prime]], and <math> q </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D;
  6 KB (1,033 words) - 02:36, 19 March 2022
• 2006 AIME II Problems/Problem 5
  .../math>. Given that the [[probability]] of obtaining face <math> F </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relat ...er die. 7 can be obtained by rolling a 2 and 5, 5 and 2, 3 and 4, or 4 and 3. Each has a probability of <math>\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{3
  5 KB (712 words) - 12:10, 5 November 2023
• 2006 AIME II Problems/Problem 9
  ...h>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime, find <math>p+q+r.</math> ...n do the same thing to get <math>O_3s_1 = 4</math> and <math>s_1s = 4\sqrt{3}</math>.
  3 KB (553 words) - 10:45, 26 August 2015
• 2006 AIME II Problems/Problem 13
  ...frac{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)</math>. Thus, <math>j</math> is a factor of <math>N</math>. ...must also be odd. For every odd value of <math>j</math>, <math>2n+j</math> is also odd, making this case valid for all odd <math>j</math>. Looking at the
  4 KB (675 words) - 10:40, 14 July 2022
• 2006 AIME II Problems/Problem 15
  ...m </math> and <math> n </math> are positive integers and <math> n </math> is not divisible by the square of any prime, find <math> m+n.</math> ...of length <math>x, y</math> and <math>z</math>, and suppose this triangle is acute (so all [[altitude]]s are in the interior of the triangle).
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• 2006 AIME II Problems/Problem 11
  ...361, </math> find the [[remainder]] when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000. Define the sum as <math>s</math>. Since <math>a_n\ = a_{n + 3} - a_{n + 2} - a_{n + 1} </math>, the sum will be:
  3 KB (417 words) - 10:07, 12 October 2023
• 2006 AIME II Problems/Problem 4
  ...<math> (a_1,a_2,a_3,\ldots,a_{12}) </math> be a permutation of <math> (1,2,3,\ldots,12) </math> for which An example of such a permutation is <math> (6,5,4,3,2,1,7,8,9,10,11,12). </math> Find the number of such permutations.
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• 2006 AIME II Problems/Problem 7
  ...time both <math>a</math> and <math>b</math> can have a 0 in the tens digit is when they are divisible by 100, which falls into the above category, so we ...</math> numbers in every hundred numbers that have a tens digit of 0 (this is true from 100 to 900), totaling <math>9 \cdot 9 = 81</math> such numbers; c
  7 KB (1,114 words) - 03:41, 12 September 2021
• 2006 AIME II Problems/Problem 10
  ...team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...th>A</math> and <math>B</math> end with the same score in these five games is <math>1-2p</math>.
  6 KB (983 words) - 13:42, 8 December 2021
• 2006 AIME II Problems
  ...and <math>\angle F</math> are congruent. The area of the hexagonal region is <math>2116(\sqrt{2}+1).</math> Find <math>AB</math>. ...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>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 2005 PMWC Problems
  The following is a list of [[PMWC]] problems from the year 2005 What is the greatest possible number one can get by discarding <math>100</math> dig
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• 1988 IMO Problems/Problem 6
  is the square of an integer. ...>b'=\min(a,b), a'=\max(a,b)</math>. Thus, <math>a'^2-kb'a'+b'^2-k=0</math> is a quadratic in <math>a'</math>. Should there be another root, <math>c'</mat
  4 KB (720 words) - 12:26, 7 April 2024
• 1998 PMWC Problems
  Calculate: <math>\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}</math> dot((z.x+j -1 + i/2 ,z.y + i*sqrt(3)/2));
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  ...> different factors (including the number <math>1</math> and itself). What is the smallest possible value of <math>y</math>? ...^{6972593}-1</math>. What is the remainder when <math>2^{6972593}-1</math> is divided by <math>5</math>?
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2001 PMWC Problems
  for(int j = 0; j < 3; ++j) draw(origin--(-1.3*3,0)--(-1.3*3,3)--(0,3)--cycle);
  4 KB (641 words) - 21:24, 21 April 2014
• 2005 PMWC Problems/Problem T6
  1+2 &=& 3 \\ ...number in the <math>80</math>th row (e.g. the last number of the third row is <math>15</math>).
  882 bytes (140 words) - 19:07, 10 March 2015
• 1971 Canadian MO Problems
  <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circ == Problem 3 ==
  3 KB (519 words) - 08:58, 13 September 2012
• 1997 PMWC Problems/Problem I11
  ...given figure. Find the perimeter, in cm, of <math>ABCD</math> if its area is <math>6750\text{ cm}^2</math>. pair A=(0,0),B=(0,-2.5),C=(3,-2.5),D=(3,0);
  944 bytes (154 words) - 12:44, 13 August 2014
• 1997 PMWC Problems/Problem I12
  ...math>1</math> and <math>6</math>, <math>2</math> and <math>5</math>, <math>3</math> and <math>4</math> appear on opposite faces. When <math>2</math> dic What is the sum of these <math>4</math> products ?
  1 KB (241 words) - 14:35, 20 April 2014
• 1997 PMWC Problems/Problem I15
  for(int i = 0; i < 3; ++i){ draw((4/3,i+1)--(4/3,i)--(8/3,i+1)--(8/3,i));
  929 bytes (135 words) - 14:37, 20 April 2014
• 2005 PMWC Problems/Problem T10
  ...t of the first prime). Note that <math>9</math> may only map one time, but is mapped to from both <math>1</math> and <math>7</math>, so <math>9</math> mu ...math>7</math> from being the second digit. These together imply that <math>3,7,9</math> cannot be the first digit, so the second digit must be <math>1</
  2 KB (234 words) - 15:30, 3 July 2012
• 1997 PMWC Problems/Problem T2
  && 1 \left(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1} &+& 3\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}
  1 KB (139 words) - 19:07, 10 March 2015
• 2005 PMWC Problems/Problem I2
  ...nd <math>c</math> is a five-digit positive integer (natural number). What is the number <math>c</math>? :''The following solution is non-rigorous''.
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• 2005 PMWC Problems/Problem I9
  ...<math>9</math> and the last number is a multiple of <math>11</math>. What is the first of these four numbers? n \equiv -3 &\equiv& 8 \pmod{11}\\\end{eqnarray*}</cmath>
  1 KB (202 words) - 19:07, 10 March 2015
• 1961 IMO Problems
  Let ''a'',''b'', and ''c'' be the lengths of a triangle whose area is ''S''. Prove that <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
  3 KB (425 words) - 21:18, 20 August 2020
• 1961 IMO Problems/Problem 6
  ...ath> such that the points <math>L,M,N</math> do not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independ ...y point on plane <math>\epsilon</math>. Thus, the locus of <math>G</math> is the plane parallel to <math>\epsilon</math> and halfway between the centroi
  2 KB (301 words) - 23:29, 18 July 2016
• 1999 USAMO Problems/Problem 5
  ...pell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy. ...S in <math>l + 3</math> and if <math>l < m</math> he fills an S in <math>l-3</math>.
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• 2006 Cyprus MO/Lyceum/Problem 12
  &=f(3,1)\\ &=1 & \text{Thus the answer is}\mathrm{(E)}
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• 2006 Cyprus MO/Lyceum/Problem 19
  ...to the circle <math>(B,B\Delta )</math>, the [[area]] of the shaded region is ...{(B)}\ 4\left(\sqrt2-\frac{\pi}{3}\right)\qquad\mathrm{(C)}\ \frac{8\sqrt2-3\pi}{16}\qquad\mathrm{(D)}\ \frac{\pi}{8}\qquad\mathrm{(E)}\ \text{None of t
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• 2004 AMC 10A Problems/Problem 5
  ...shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? There are <math>\binom{9}{3}</math> ways to choose three points out of the 9 there. There are 8 combina
  1,017 bytes (143 words) - 14:14, 21 April 2021
• 2004 AMC 10A Problems/Problem 21
  ...f the acute angle formed by the two lines? (Note: <math>\pi</math> radians is <math>180</math> degrees.) ...he area of the unshaded region be <math>U</math>, and the acute angle that is formed by the two lines be <math>\theta</math>. We can set up two equations
  3 KB (476 words) - 03:50, 23 January 2023
• 2004 AMC 12A Problems/Problem 18
  ...</math> intersects side <math>\overline{AD}</math> at <math>E</math>. What is the length of <math>\overline{CE}</math>? ...frac{1}{2}</math>. Solving <math>EF=\frac{1}{2}</math>. Adding, the answer is <math>\frac{5}{2}</math>.
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• 2004 AMC 10A Problems/Problem 20
  ...<math>ABCD</math> so that <math>\triangle BEF</math> is equilateral. What is the ratio of the area of <math>\triangle DEF</math> to that of <math>\trian unitsize(3 cm);
  4 KB (710 words) - 02:47, 18 April 2024
• 1989 USAMO Problems/Problem 5
  <cmath> (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. </cmath> ...e, with proof, which of the two numbers, <math>u</math> or <math>v</math>, is larger.
  2 KB (300 words) - 19:16, 18 July 2016
• 1990 USAMO Problems
  A sequence of functions <math>\, \{f_n(x) \} \,</math> is defined recursively as follows: (Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive square root.) For each positive intege
  3 KB (386 words) - 20:47, 3 July 2013
• 2007 AMC 12B Problems
  Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls ==Problem 3==
  12 KB (1,814 words) - 12:58, 19 February 2020
• 2002 AMC 12A Problems/Problem 2
  ...the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the ...orrect result is <math>\frac{138-3}{9}=\frac{135}{9}=15</math>. Our answer is <math>\boxed{\textbf{(A) }15}</math>.
  1 KB (163 words) - 12:46, 8 November 2021
• 2002 AMC 12A Problems/Problem 4
  Find the degree measure of an angle whose complement is 25% of its supplement. ...om the supplementary angle, we have <math>90^{\circ}</math> as <math>\frac{3}{4}</math> of the supplementary angle.
  1,019 bytes (140 words) - 19:14, 20 August 2019
• 1990 USAMO Problems/Problem 2
  A sequence of [[function]]s <math>\, \{f_n(x) \} \,</math> is defined [[recursion|recursively]] as follows: (Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive [[square root]].) For each positive in
  2 KB (322 words) - 19:14, 18 July 2016
• 1999 AHSME Problems
  <math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math> Which of the following statements is false?
  13 KB (1,945 words) - 18:28, 19 June 2023
• Cramer's Rule
  '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]]. ...>. Here, <math>A</math> is the coefficient matrix, <math>\mathbf{b}</math> is a column vector.
  2 KB (352 words) - 18:22, 11 October 2023
• 2001 IMO Problems/Problem 2
  <cmath>\geq ^{Cheff} \frac{1}{3}(a+b+c)\left( \frac{1}{\sqrt{S+6bc}}+\frac{1}{\sqrt{S+6ca}}+\frac{1}{\sqrt{ <cmath>\geq^{AH} \frac{1}{3}(a+b+c) \left( \frac{9}{\sqrt{S+6bc}+\sqrt{S+6ca}+\sqrt{S+6ab}} \right)</cm
  7 KB (1,174 words) - 12:14, 3 August 2023
• Omega
  ...[[ring theory]], omega is a constant the represents <math>e^{\frac{2i\pi}{3}}</math>. ...never one item in the ring is taken to any power, another item in the ring is the result.
  613 bytes (96 words) - 15:54, 5 June 2020
• 1960 IMO Problems/Problem 1
  ...rty that <math>N</math> is divisible by 11, and <math>\dfrac{N}{11}</math> is equal to the sum of the squares of the digits of <math>N</math>. ...digits in the odd positions minus the sum of digits in the even positions is divisible by <math>11</math>. Thus we get: <math>b = a + c</math> or <math>
  4 KB (751 words) - 05:01, 17 August 2022
• 1960 IMO Problems/Problem 2
  If <math>x \neq 0</math>, then the LHS is defined and rewrites as follows: ...}</math> is imaginary. So the original inequality holds iff <math>x</math> is in <math>[-1/2, 0) \cup (0, 45/8).</math>
  2 KB (239 words) - 05:51, 7 April 2024
• 1960 IMO Problems/Problem 5
  ...>, where <math>X</math> is any point of <math>AC</math> and <math>Y</math> is any point of <math>B'D'</math>; ...t unique points <math>X</math> and <math>Y</math> such that <math>P</math> is the midpoint of <math>XY</math>.
  2 KB (402 words) - 23:28, 18 July 2016
• 1960 IMO Problems/Problem 6
  ...ution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base o <cmath>V_1 = \frac{1}{3}\pi R^2 h</cmath>
  7 KB (1,214 words) - 18:49, 29 January 2018
• 1999 AHSME Problems/Problem 1
  <math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math> ...-1) + \cdots + 99</math>, and since there are 49 pairs of terms the answer is <math>-49 + 99 = 50 \Rightarrow \mathrm{\textbf{(E)}}</math>.
  1 KB (170 words) - 21:35, 9 February 2024
• Asymptote: Useful functions
  ''t'' is the radius of the first line, and s[] stores the radius of subsequent lines ...f the angle mark. However, anglemark() makes a bit more abstract sense and is more flexible.
  4 KB (646 words) - 21:18, 26 March 2024
• Excircle
  An '''excircle''' is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [ pair X=(0,0), Y=(10,0), Z=(3,6);
  5 KB (843 words) - 03:02, 1 July 2020
• 1997 USAMO Problems/Problem 2
  <math>\triangle ABC</math> is a triangle. Take points <math>D, E, F</math> on the perpendicular bisectors But this is clearly true, since D lies on the perpendicular bisector of BC, BD = DC.
  2 KB (347 words) - 08:13, 26 August 2021
• 2001 USAMO Problems/Problem 2
  ...D_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>. It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</math>. (Proof
  11 KB (2,091 words) - 08:35, 16 November 2017
• Factor Theorem
  In algebra, the '''Factor theorem''' is a theorem regarding the relationships between the factors of a polynomial a ...ath>f(a)</math> (<math>a</math> is constant, <math>f</math> is polynomial) is <math>0</math> using polynomial division rather than plugging in large valu
  3 KB (508 words) - 12:31, 2 May 2024
• L'Hôpital's Rule
  '''L'Hopital's Rule''' is a theorem dealing with [[limit]]s that is very important to [[calculus]]. The definition of a derivative is <math>f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}</math> which can
  2 KB (475 words) - 15:04, 24 March 2022
• Summation
  A '''summation''' is the [[sum]] of a number of terms (addends). Summations are often written us ...index of summation, <math>a</math> is the lower bound, and <math>b</math> is the upper bound.
  3 KB (482 words) - 16:39, 8 October 2023
• Symmetric sum
  ...h>, where <math>\sigma</math> ranges over all permutations of <math>(1, 2, 3, \dots, n)</math>. ...generally, a '''symmetric sum''' of <math>n</math> variables is a sum that is unchanged by any [[permutation]] of its variables.
  1 KB (255 words) - 12:52, 8 October 2023
• Concurrence/Problems
  <math>5*1-1=4</math> ...the side lengths are all distinct, and that the side across from vertex C is 13 inches in length. How long are the other two sides?
  3 KB (458 words) - 15:44, 1 December 2015
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Choice]]. ...per bound]], i.e., an upper bound <math>a</math> so that if <math>b</math> is an upper bound of <math>T</math>, then <math>a \le b</math>.
  9 KB (1,669 words) - 19:02, 1 August 2018
• Median
  ''This article is about the median used in statistics. For other medians, check [[Median (dis A '''median''' is a measure of central tendency used frequently in statistics.
  2 KB (370 words) - 15:30, 12 November 2023
• Root (operation)
  ...mbers and a partial [[inverse]] to [[exponentiation]]. (The proper inverse is the [[logarithm]]) A known method to compute all the roots of <math>\sqrt[n]{x}</math> is by the DeMoivre's formula.
  3 KB (532 words) - 16:52, 20 May 2020
• 1962 IMO Problems
  ...s erased and placed in front of the remaining digits, the resulting number is four times as large as the original number <math>n</math>. <math>\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac{1}{2}</math>
  3 KB (431 words) - 21:17, 20 August 2020
• 2004 AMC 12A Problems/Problem 11
  ...ge value of all the pennies, nickels, dimes, and quarters in Paula's purse is <math>20</math> cents. If she had one more quarter, the average value would ...t {(A)}\ 0 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 2 \qquad \text {(D)}\ 3\qquad \text {(E)}\ 4</math>
  2 KB (310 words) - 17:39, 31 December 2022
• 2004 AMC 12A Problems/Problem 2
  ...ct answer is worth <math>0</math> points, and each problem left unanswered is worth <math>2.5</math> points. If Charlyn leaves <math>8</math> of the <mat {{AMC12 box|year=2004|ab=A|num-b=1|num-a=3}}
  837 bytes (122 words) - 20:14, 3 July 2013
• 2004 AMC 12A Problems/Problem 12
  ...and <math>\overline{BB'}</math> intersect at <math>C = (2,8)</math>. What is the length of <math>\overline{A'B'}</math>? ...sqrt2 \qquad \text {(C)} 3 \qquad \text {(D)} 2 + \sqrt 2\qquad \text {(E)}3\sqrt 2</math>
  1 KB (181 words) - 20:17, 3 July 2013
• 2004 AMC 12A Problems/Problem 13
  *'''Case 1''': The line is horizontal or vertical, clearly <math>3 \cdot 2 = 6</math>. *'''Case 3''': The only remaining lines pass through two points, a vertex and a non-ve
  2 KB (381 words) - 00:58, 19 October 2020
• 2004 AMC 12A Problems/Problem 14
  ...d term, the three resulting numbers form a [[geometric progression]]. What is the smallest possible value for the third term in the geometric progression ...allest possible value occurs when <math>d = -14</math>, and the third term is <math>2(-14) + 29 = 1\Rightarrow\boxed{\mathrm{(A)}\ 1}</math>.
  4 KB (689 words) - 03:35, 16 January 2023
• 2004 AMC 12A Problems/Problem 17
  What is the value of <math>f(2^{100})</math>? f\left(2^3\right) &= 2^2\cdot f\left(2^2\right) &&= 2^{2+1}, \\
  2 KB (233 words) - 08:14, 6 September 2021
• 2004 AMC 12A Problems/Problem 20
  ...and <math>c</math>, respectively, are rounded to the nearest integer. What is the [[probability]] that <math>A + B = C</math>? ...=c=\frac 12</math>, and we want <math>c < \frac 12</math>. Thus the chance is <math>\frac{\frac{1}{4}}2 = \frac 18</math>.
  3 KB (552 words) - 23:26, 28 December 2020
• 2004 AMC 12A Problems/Problem 22
  ...a horizontal plane. A sphere of radius <math>2</math> rests on them. What is the distance from the plane to the top of the larger sphere?<!-- don't remo ...\frac {\sqrt {123}}{4}\qquad \text {(D)}\ \frac {52}{9}\qquad \text {(E)}\ 3 + 2\sqrt2</math>
  2 KB (382 words) - 19:59, 29 January 2023
• 2004 AMC 12A Problems/Problem 25
  ...th> are positive integers and <math>n</math> is as small as possible. What is <math>m</math>? ...<math>= \frac{x^2 + 3x + 3}{x^3 - 1}</math> <math>= \frac{(x+1)^3 - 1}{x(x^3 - 1)}</math>.
  3 KB (480 words) - 14:50, 17 August 2020
• 2007 AMC 12B Problems/Problem 9
  ...at <math>f(3x-1)=x^2+x+1</math> for all real numbers <math>x</math>. What is <math>f(5)</math>? <math>f(3(2)-1) = 2^2+2+1=7 \implies (A)</math>
  460 bytes (67 words) - 01:01, 19 October 2020
• 2004 AMC 12A Problems/Problem 8
  ...C}</math> and <math>\overline{BE}</math> intersect at <math>D</math>. What is the difference between the areas of <math>\triangle{ADE}</math> and <math>\ ...>[\triangle EBA]</math> is 16 and the area of <math>[\triangle ABC]</math> is 12. Set the area of <math>[\triangle ADB]</math> to be x. We want to find <
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• Binet's Formula
  '''Binet's formula''' is an explicit formula used to find the <math>n</math>th term of the Fibonacci It is so named because it was derived by mathematician Jacques Philippe Marie Bin
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• 1992 USAMO Problems/Problem 2
  pair O=(0,0), a=expi(0), b=expi(1/6), c=expi(2/6), d=expi(3/6), y=expi(32/30), z= expi(34/30); ...rc</math>. It then follows that the area of triangle <math>M_aOM_b</math> is <math>\tfrac{1}{2} \sin(b-a)^\circ \cdot OM_a \cdot OM_b = \tfrac{1}{2} \si
  4 KB (628 words) - 07:41, 19 July 2016
• 2005 Alabama ARML TST Problems/Problem 8
  ...and <math>y \pmod{3}</math>, since <math>21</math> is a multiple of <math>3</math>. * Case 1: <math>x\equiv 0\pmod{3}</math>
  1 KB (250 words) - 00:38, 28 October 2015
• 1995 AHSME Problems/Problem 4
  ...<math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math> ...) \ 1 } \qquad \mathrm{(D) \ \frac {6}{5} } \qquad \mathrm{(E) \ \frac {4}{3} } </math>
  2 KB (232 words) - 22:30, 16 February 2018
• De Longchamps point
  </asy>|right|The de Longchamps <br />point (<math>L</math>) is the the <br />[[orthocenter]] (<math>H</math>) reflected <br /> through the The '''de Longchamps point''' of a [[triangle]] is the [[reflection (mathematics)|reflection]] of the triangle's [[orthocenter
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• 1999 USAMO Problems/Problem 4
  Let <math>a_{1}, a_{2}, \dots, a_{n}</math> (<math>n > 3</math>) be real numbers such that Since <math>-a_i</math> is a positive real for all <math>k+1 \le i \le n</math>, it follows that
  3 KB (499 words) - 09:47, 20 July 2016
• 1999 IMO Problems/Problem 2
  The answer is <math>C=1/8</math>, and equality holds exactly when two of the <math>x_i</m \sum_{i=1}^{k-1} x_i^3 (x_k + x_{k+1}) + S(x_k^3 + x_{k+1}^3) \\
  4 KB (838 words) - 01:04, 17 November 2023
• Proofs of AM-GM
  ...me proofs of the weighted [[AM-GM Inequality]]. The inequality's statement is as follows: for all nonnegative reals <math>a_1, \dotsc, a_n</math> and non ...ht-hand side of the inequality is zero and the left hand of the inequality is greater or equal to zero, with equality if and only if <math>a_j = 0 = a_i<
  12 KB (2,171 words) - 07:55, 11 May 2023
• 1976 USAMO Problems/Problem 5
  <cmath> P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x), </cmath> prove that <math>x-1</math> is a factor of <math>P(x)</math>.
  3 KB (572 words) - 17:14, 16 August 2015
• Gravity
  ...tism]]), but it is the only one which is always attractive, which means it is much more noticeable than the others. ...rmulas so that they covered slight discrepancies with actual results. This is the theory generally accepted today.
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• 2003 Polish Mathematical Olympiad Problems/Problem 3
  ...condition: for every [[natural number]] <math>n</math>, <math>2^n-1</math> is divisible by <math>W(n)</math>. ...on "natural number" is ambiguous, but this is irrelevant, as every integer is a divisor of <math>2^0 -1 = 0</math>.''
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• 2000 AMC 12 Problems/Problem 9
  The first number is divisible by <math>1</math>. The sum of the first two numbers is even.
  4 KB (601 words) - 14:27, 19 February 2024
• 2000 AMC 12 Problems/Problem 10
  ...<math>x</math>-axis to produce <math>R</math>, and finally, <math>R</math> is translated by 5 units in the positive-<math>y</math> direction to produce < ...C) } ( - 1, - 2,8) \qquad \text {(D) } ( - 1,3,3) \qquad \text {(E) } (1,3,3)
  1 KB (173 words) - 08:56, 21 August 2023
• 2008 AMC 12B Problems/Problem 23
  ...logarithms of the divisors of <math>10^n</math> is <math>792</math>. What is <math>n</math>? ...>2^{n-a} \times 5^{n-b}</math>. Note this is not true if <math>10^n</math> is a perfect square. When these are multiplied, they equal <math>2^{a+n-a} \ti
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• 2000 AMC 12 Problems/Problem 11
  ...nd <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>? ...- 2 \qquad \textbf{(B)} \ \frac { -1 }{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2</math>
  2 KB (405 words) - 21:48, 21 April 2024
• Graph (graph theory)
  In [[graph theory]], a '''graph''' is a (usually [[finite]]) [[empty set | nonempty]] [[set]] of [[vertex|vertice ...>. Note that this definition describes ''simple, loopless'' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself,
  8 KB (1,428 words) - 10:26, 27 August 2020
• 2000 AMC 12 Problems/Problem 13
  <math>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \t ...ollows that <math>6-p</math> and <math>p-4</math> are also positive, which is only possible when <math>p = 5\ \mathrm{(C)}</math>.
  6 KB (996 words) - 18:14, 29 June 2023
• 2000 AMC 12 Problems/Problem 15
  Let <math>f</math> be a function for which <math>f\left(\dfrac{x}{3}\right) = x^2 + x + 1</math>. Find the sum of all values of <math>z</math> ...\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\ 5/9 \qquad \text {(E)}\ 5/3</cmath>
  3 KB (441 words) - 21:11, 29 April 2023
• 2000 AMC 12 Problems/Problem 16
  ...board. If the board is renumbered so that the left column, top to bottom, is <math>1,2,\ldots,13,</math>, the second column <math>14,15,\ldots,26</math> Index the rows with <math>i = 1, 2, 3, ..., 13</math>
  2 KB (310 words) - 11:28, 3 August 2021
• 2000 AMC 12 Problems/Problem 17
  ...</math> and contains the point <math>A</math>. The segment <math>AB</math> is tangent to the circle at <math>A</math> and <math>\angle AOB = \theta</math dotfactor=3;
  6 KB (979 words) - 12:50, 17 July 2022
• 2000 AMC 12 Problems/Problem 19
  ...ngle bisector|bisector]] of angle <math>BAC</math>. Which of the following is closest to the area of the triangle <math>ADE</math>? ...{(A)}\ 2 \qquad \text {(B)}\ 2.5 \qquad \text {(C)}\ 3 \qquad \text {(D)}\ 3.5 \qquad \text {(E)}\ 4</math>
  3 KB (547 words) - 17:37, 17 February 2024
• 2000 AMC 12 Problems/Problem 20
  ...d y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}</cmath> Then what is the value of <math>xyz</math> ?
  2 KB (244 words) - 07:34, 4 November 2022
• 2000 AMC 12 Problems/Problem 21
  ...]] of the area of the other small right triangle to the area of the square is draw((0,0)--(6,0)--(0,3)--cycle);
  5 KB (804 words) - 01:22, 13 May 2024
• 2000 AMC 12 Problems/Problem 22
  ...l]] <math>P(x) = x^4 + ax^3 + bx^2 + cx + d</math>. Which of the following is the smallest? Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. We approximate each of the above expressions:
  1 KB (212 words) - 14:37, 5 June 2022
• 2000 AMC 12 Problems/Problem 23
  ...he same property— the sum of the base-ten logarithms is an integer. What is the [[probability]] that Professor Gamble holds the winning ticket? <math>\textbf {(A)}\ 1/5 \qquad \textbf {(B)}\ 1/4 \qquad \textbf {(C)}\ 1/3 \qquad \textbf {(D)}\ 1/2 \qquad \textbf {(E)}\ 1</math>
  2 KB (348 words) - 22:26, 24 October 2021
• 2000 AMC 12 Problems/Problem 24
  ...verarc{BC}</math> is <math>12</math>, then the circumference of the circle is label("C", (32, 32*sqrt(3)), N);
  2 KB (263 words) - 19:59, 18 April 2024
• 2000 AMC 12 Problems/Problem 25
  Since the octahedron is indistinguishable by rotations, without loss of generality fix a face to be ...e three possible rotations about the fixed face, so the answer is <math>7!/3 = 1680</math>.
  7 KB (998 words) - 21:04, 23 December 2023
• 2007 AMC 10A Problems/Problem 23
  <math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text <cmath>m^2 - n^2 = (m+n)(m-n) = 96 = 2^{5} \cdot 3</cmath>
  2 KB (352 words) - 14:20, 3 July 2023
• 2003 JBMO Problems/Problem 1
  ...<math>n</math> digits, each of which is 8. Prove that <math>A+2B+4</math> is a perfect square. &= \left(\frac{2(10^n + 2)}{3}\right)^2.
  1,002 bytes (145 words) - 00:14, 28 August 2018
• 2007 AMC 10A Problems/Problem 20
  ...<math>a</math> satisfies the equation <math>4 = a + a^{ - 1}</math>. What is the value of <math>a^{4} + a^{ - 4}</math>? == Solution 3 (Detailed Explanation of Solution 2) ==
  4 KB (663 words) - 07:32, 4 November 2022
• 2007 AMC 10A Problems
  ...<math>a@b = ab - b^{2}</math> and <math>a\#b = a + b - ab^{2}</math>. What is <math>\frac {6@2}{6\#2}</math>? == Problem 3 ==
  13 KB (2,058 words) - 17:54, 29 March 2024
• 2007 AMC 10A Problems/Problem 21
  ...th> square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? <math>\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text
  4 KB (609 words) - 14:41, 3 December 2023
• 2007 AMC 10A Problems/Problem 12
  ...act the <math>2</math> cases where a guide has no tourist. Thus the answer is <math>2^6 - 2 = \boxed{62}\ \mathrm{(D)}</math>. ...ts go to tour guide A. Thus, we can see that the total number of groupings is:
  3 KB (411 words) - 18:07, 14 March 2023
• 2007 AMC 10A Problems/Problem 11
  ...h a manner that the sum of the four numbers on each face is the same. What is this common sum? ...</math> numbers represent all of the vertices of the cube. Thus the answer is <math>\frac{1 + 2 + \cdots + 8}{2} = 18\ \mathrm{(C)}</math>.
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• 2007 AMC 10A Problems/Problem 10
  ...<math>48</math> years old, and the average age of the mother and children is <math>16</math>. How many children are in the family? <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math>
  2 KB (411 words) - 20:54, 19 August 2023
• 2007 AMC 10A Problems/Problem 9
  ...<math>3^{a} = 81^{b + 2}</math> and <math>125^{b} = 5^{a - 3}</math>. What is <math>ab</math>? <cmath>81^{b+2} = 3^{4(b+2)} = 3^a \Longrightarrow a = 4b+8</cmath>
  2 KB (268 words) - 12:40, 3 June 2021
• 2007 AMC 10A Problems/Problem 4
  The larger of two consecutive odd integers is three times the smaller. What is their sum? ...+2</math>. Then <math>a+2 = 3a</math>, so <math>a = 1</math> and their sum is <math>4\ \mathrm{(A)}</math>.
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• 2007 AMC 10A Problems/Problem 2
  ...<math>a@b = ab - b^{2}</math> and <math>a\#b = a + b - ab^{2}</math>. What is <math>\frac {6@2}{6\#2}</math>? ...equal to <math>\frac{8}{-16} = -\frac{1}{2}</math>. Therefore the solution is <math>A</math>.
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• 1995 AHSME Problems
  ...mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math> If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math>
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• 1995 AHSME Problems/Problem 24
  What is <math>A + B + C</math>? ...</math> and <math>C</math> are relatively prime, <math>C=1</math>, <math>B=3</math>, <math>A=2</math>. <math>A+B+C=6 \Rightarrow \mathrm{(A)}</math>
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• 1995 AHSME Problems/Problem 28
  ...dway between them is of length <math>\sqrt {a}</math> where <math>a</math> is ...f the other leg. Now the length of the leg of <math>\triangle OAA_1</math> is either <math>6 + x</math> or <math>6 - x</math> depending whether or not <m
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• 1995 AHSME Problems/Problem 20
  ...th>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is ...1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math>
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• 1995 AHSME Problems/Problem 1
  ...mathrm{(C) \ \text{increase by 2} } \qquad \mathrm{(D) \ \text{increase by 3} } \qquad \mathrm{(E) \ \text{increase by 4} } </math> ...cores is <math>\frac{88+83+87}{3}=86</math>. The average of all four exams is <math>\frac{87+83+88+90}{4}=87</math>. It increased by one point. <math>\ma
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• 1995 AHSME Problems/Problem 12
  ..., f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true? ...if <math>f(a)=f(b)</math> for <math>a\neq b</math>, then <math>f(x)</math> is a constant function. Since <math>f(5)=5</math>, <math>f(0)=5\Rightarrow \ma
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• 1995 AHSME Problems/Problem 21
  ...e]]s of the other two vertices are integers. The number of such rectangles is <math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math>
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• 1995 AHSME Problems/Problem 29
  ...ny three-element sets of distinct positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>? ...to figure out the number of ways to distribute these prime factors amongst 3 different integers, without over counting triples which are simply permutat
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• 1997 PMWC Problems
  Peter is ill. He has to take medicine <math>A</math> every <math>8</math> hours, med ...)--(1.45,0.54)--(1.38,0.46)--(1.31,0.46)--(1.35,0.38)--(1.31,0.3)--(1.38,0.3)--cycle);
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• 1995 AHSME Problems/Problem 5
  ...the number that most closely approximates the number of ants in the field is The rectangular field is <math>300 \text{ feet} \cdot \frac{12 \text{ inches}}{1 \text{ foot}} = 36
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• 1995 AHSME Problems/Problem 14
  ...x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math> ...} \qquad \mathrm{(B) \ -2 } \qquad \mathrm{(C) \ 1 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 8 } </math>
  1 KB (202 words) - 18:05, 6 April 2016
• 1995 AHSME Problems/Problem 10
  ...]] bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is label("$y=x$",(3,3),(1,-1));label("$y=-x$",(-3,3),(-1,-0.5));
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• Euclid's Elements
  ...eometry]] and was compiled by [[Euclid]] in the time of ancient Greece. It is divided into thirteen volumes, each consisting of definitions, "common noti ...sibly translated into Latin during the reign of the Roman Empire, but this is doubtful. The Arabs acquired copies of ''The Elements'' circa 750 AD, and i
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• 1951 AHSME Problems
  The percent that <math>M</math> is greater than <math>N</math> is: ...ed by <math>x</math> yards of fencing. The area in terms of <math>x</math> is:
  23 KB (3,641 words) - 22:23, 3 November 2023
• 1951 AHSME Problems/Problem 2
  ...ed by <math>x</math> yards of fencing. The area in terms of <math>x</math> is: ...2w^2 = 2\left(\frac{x^2}{36}\right) = \frac{x^2}{18}</math>, so the answer is <math>\mathrm{D}</math>.
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• 1990 USAMO Problems/Problem 3
  19. Prove that for any odd integer <math>n \geq 1</math>, there is a way to number so that each integer is used once, and adjacent beads correspond to
  3 KB (460 words) - 19:14, 18 July 2016
• 1990 USAMO Problems/Problem 4
  ...h> is a <math>k</math>-good sequence which starts on <math>a</math>, so it is a permutation of <math>k</math> consecutive integers, say <math>m, \dotsc, ...<math>k</math> digits. Thus the total number of desired positive integers is
  3 KB (529 words) - 19:15, 18 July 2016
• Algebraic manipulation
  ...plication is a common method of solving [[proportions]]. Essentially, one is multiplying both sides by the denominators of both sides. \frac{x}{3} &= \frac{10}{5} \\
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• 1959 AHSME Problems
  ...<math>50</math>%. The percent of increase of the surface area of the cube is: Through a point <math>P</math> inside the <math>\triangle ABC</math> a line is drawn parallel to the base <math>AB</math>, dividing the triangle into two
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  ...o of <math>3x - 4</math> to <math>y + 15</math> is constant, and <math>y = 3</math> when <math>x = 2</math>, then, when <math>y = 12</math>, <math>x</ma ...sed 10% and the altitude to this base is decreased 10%, the change in area is
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1966 AHSME Problems/Problem 10
  ...>1</math> and their product is <math>1</math>, then the sum of their cubes is: ...i\sqrt {3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ - \frac {3i\sqrt {3}}{4} \qquad \text{(E)} \ - 2</math>
  611 bytes (97 words) - 12:40, 5 July 2013
• 1966 AHSME Problems/Problem 1
  ...] of <math>3x - 4</math> to <math>y + 15</math> is constant, and <math>y = 3</math> when <math>x = 2</math>, then, when <math>y = 12</math>, <math>x</ma ...frac{1}{9} = \frac{3x - 4}{(12) + 15}</math>. Solving gives <math>3x - 4 = 3 \Longrightarrow x = \frac 73 \Rightarrow \mathrm{(B)}</math>.
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• 1991 USAMO Problems/Problem 3
  is eventually constant. [The tower of exponents is defined by <math>a_1 = 2, \; a_{i+1} = 2^{a_i}</math>. Also <math>a_i \pmod
  3 KB (559 words) - 19:18, 5 June 2023
• 1991 USAMO Problems/Problem 1
  ..., angle <math>A</math> is twice angle <math>B</math>, angle <math>C</math> is [[obtuse triangle|obtuse]], and the three side lengths <math>a, b, c</math> //y = 4x/3 and x+2y = 2 (sides AC and BC, respectively) intersect here:
  5 KB (755 words) - 08:58, 6 May 2023
• 1991 USAMO Problems/Problem 2
  ...\frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), </cmath> ...denotes a sum involving all nonempty subsets <math>S</math> of <math>\{1,2,3, \ldots,n\}</math>.
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• 1991 USAMO Problems
  ...h>\,A\,</math> is twice angle <math>\,B,\,</math> angle <math>\,C\,</math> is obtuse, and the three side lengths <math>\,a,b,c\,</math> are integers. De ...\frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), </cmath>
  3 KB (512 words) - 19:17, 18 July 2016
• 2002 AMC 12B Problems/Problem 20
  == Solution 3 (Solution 1 but shorter) == ...her variable to get <math>15x^2=960</math> and <math>x^2=64</math>. Now XY is equal to the square root of four times these quantities, so <math>(105+64)
  3 KB (447 words) - 15:02, 17 August 2023
• 2002 AMC 12B Problems/Problem 25
  The area of <math>R</math> is closest to <cmath>x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16</cmath>
  2 KB (365 words) - 14:48, 7 March 2022
• 2002 AMC 12B Problems/Problem 23
  ...om <math>A</math> to <math>\overline{BC}</math> have the same length. What is <math>BC</math>? \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2
  4 KB (598 words) - 09:45, 23 January 2024
• 2002 AMC 12B Problems/Problem 22
  <cmath>\begin{align*}b- c &= \left(\frac{1}{\log 2002}\right)(\log 2 + \log 3 + \log 4 + \log 5 - \log 10 - \log 11 - \log 12 - \log 13 - \log 14)\\ &= \left(\frac{1}{\log 2002}\right)\left(\log \frac{2 \cdot 3 \cdot 4 \cdot 5}{10 \cdot 11 \cdot 12 \cdot 13 \cdot 14}\right)\\
  2 KB (312 words) - 18:26, 13 January 2022
• 2002 AMC 12B Problems/Problem 16
  ...is the [[probability]] that the product of the two rolls is a multiple of 3? ...frac{2}{8} = \frac{1}{4}</math> be the probability that Juan rolls a <math>3</math> or a <math>6</math>, and <math>P(b) = \frac{2}{6} = \frac 13</math>
  2 KB (317 words) - 10:26, 5 November 2023
• 2002 AMC 12B Problems/Problem 14
  ...ge--><onlyinclude>Four distinct [[circle]]s are drawn in a [[plane]]. What is the maximum number of points where at least two of the circles intersect?<! ...t 2 = 12</math>. We can construct such a situation as below, so the answer is <math>\boxed{\mathrm{(D)}\ 12}</math>.
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• 2002 AMC 12B Problems/Problem 18
  ...<math>P</math> is closer to the [[origin]] than it is to the point <math>(3,1)</math>? ...le is <math>2 - \frac 12 = \frac 32</math>. The probability is <math>\frac{3/2}{2} = \frac 34 \Rightarrow \mathrm{(C)}</math>.
  3 KB (376 words) - 19:16, 20 August 2019
• 2002 AMC 12B Problems/Problem 11
  ...> and <math>A+B</math> are all prime numbers. The sum of these four primes is \qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3
  3 KB (470 words) - 11:57, 10 August 2022
• 2002 AMC 12B Problems/Problem 12
  For how many integers <math>n</math> is <math>\dfrac n{20-n}</math> the [[perfect square|square]] of an integer? \qquad\mathrm{(C)}\ 3
  4 KB (579 words) - 05:54, 17 October 2023
• 2002 AMC 12B Problems/Problem 1
  ...C 12B Problems|2002 AMC 12B #1]] and [[2002 AMC 10B Problems|2002 AMC 10B #3]]}} ...ne 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 disti
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• 2002 AMC 12B Problems/Problem 2
  What is the value of <math>(3x - 2)(4x + 1) - (3x - 2)4x + 1</math> when <math>x=4 <cmath>(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \boxed{\mathrm{(D)}\ 11}</cmath>.
  802 bytes (105 words) - 10:46, 9 August 2022
• 2002 AMC 12B Problems/Problem 3
  {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #3]] and [[2002 AMC 10B Problems|2002 AMC 10B #6]]}} For how many positive integers <math>n</math> is <math>n^2 - 3n + 2</math> a [[prime]] number?
  1 KB (247 words) - 21:30, 24 August 2022
• 2002 AMC 12B Problems/Problem 5
  ...(as a pentagon can be split into <math>5- 2 = 3</math> triangles) is <math>3 \cdot 180 = 540^{\circ}</math>. If we let <math>v = x - 2d, w = x - d, y = ...the middle term of an arithmetic sequence with an odd number of terms, it is simply the average of the sequence.
  2 KB (263 words) - 05:58, 8 August 2023
• 2002 AMC 12B Problems/Problem 4
  ...frac 17 + \frac 1n</math> is an integer. Which of the following statements is '''not ''' true: \qquad\mathrm{(B)}\ 3\ \text{divides\ }n
  2 KB (321 words) - 10:54, 9 August 2022
• 2002 AMC 12B Problems/Problem 6
  ...utions <math>a</math> and <math>b</math>. Then the pair <math>(a,b)</math> is ...<math>-a - b = a</math> and that <math>ab = b</math>. Since <math>b</math> is nonzero, <math>a = 1</math>, and <math>-1 - b = 1 \Longrightarrow b = -2</m
  3 KB (457 words) - 14:53, 17 August 2023
• Mock AIME 1 Pre 2005 Problems/Problem 1
  ...gers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>. ...e tens and units digits. Thus the sum of the hundreds places is <math>(1+2+3+\cdots+9)(72) \times 100 = 45 \cdot 72 \cdot 100 = 324000</math>.
  1 KB (194 words) - 13:44, 5 September 2012
• 2007 AMC 10A Problems/Problem 15
  ...nd externally tangent to a circle of radius <math>2</math>, as shown. What is the area of the square? real h=3*sqrt(2)/2;
  2 KB (326 words) - 10:29, 4 June 2021
• Three Greek problems of antiquity
  ...h>\cos 20^{\circ}</math> is <math>8x^3-6x-1</math>, which has degree <math>3</math>. ...of <math>\sqrt[3]{2}</math> is <math>x^3-2</math>, which has degree <math>3</math>.
  2 KB (359 words) - 17:06, 11 June 2022
• Construction
  A '''compass''' is a tool that can draw circles and arcs of circles. A '''straightedge''' is an unmarked ruler that can draw line segments.
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• 2007 AMC 8 Problems
  ...spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number o ...} \frac{1}{2} \qquad \textbf{(C)} \frac{5}{4} \qquad \textbf{(D)} \frac{5}{3} \qquad \textbf{(E)} \frac{5}{2}</math>
  12 KB (1,800 words) - 20:01, 8 May 2023
• 2008 iTest Problems/Problem 97
  ...values are there of <math>k</math>, where <math>k</math> is at least <math>3</math> and at most <math>2008</math>? ...that student can either stay still, for which the number of rearrangements is simply the number of rearrangements for the other <math>n</math> students (
  3 KB (497 words) - 13:29, 20 October 2019
• 2001 AMC 12 Problems/Problem 4
  The mean of three numbers is <math>10</math> more than the least of the numbers and <math>15</math> ...than the greatest. The median of the three numbers is <math>5</math>. What is their
  2 KB (311 words) - 21:53, 10 February 2024
• 2004 AMC 10A Problems/Problem 16
  Since there are five types of squares: <math>1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4,</math> and <math>5 \times 5.</math> We must find how many of e *There is <math>1</math> of all <math>1\times 1</math> squares, containing the black
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• 2002 AMC 12B Problems/Problem 8
  ...ates: <math>(1,8,15,22,29)</math>, <math>(2,9,16,23,30)</math>, and <math>(3,10,17,24,31)</math>. The only day of the week that is guaranteed to appear five times is therefore <math>\boxed{\textrm{(D)}\ \text{Thursday}}</math>.
  2 KB (329 words) - 15:48, 14 October 2023
• 2002 AMC 12B Problems/Problem 9
  ...d <math>a,b,d</math> form a geometric sequence, then <math>\frac ad</math> is We can let <math>a=1</math>, <math>b=2</math>, <math>c=3</math>, and <math>d=4</math>. <math>\frac{a}{d}=\boxed{\frac{1}{4}} \Longr
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• 2003 AMC 12B Problems/Problem 22
  ...math> and <math>\overline{BD}</math>, respectively. Which of the following is closest to the minimum possible value of <math>PQ</math>? ...sector]]s. Thus <math>\angle POQ = 90^{\circ}</math>, so <math>OPNQ</math> is a [[rectangle]]. Since the diagonals of a rectangle are of equal length, <m
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• 2003 AMC 12B Problems/Problem 21
  ...h>, and moves <math>5</math> cm in a straight line to <math>C</math>. What is the [[probability]] that <math>AC < 7</math>? \qquad\mathrm{(D)}\ \frac{1}{3}
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• 2002 AMC 12B Problems/Problem 10
  ...sults by 3, we obtain the set <math>\{-3, -2, -1, 0, 1, 2, 3\}</math>. It is easy to see that we can get any integer between <math>-6</math> and <math>6 ...e numbers will be a multiple of <math>3</math>. All the multiples of <math>3</math> from <math>1+4+7=12</math> to <math>13+16+19=48</math> are possible,
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• 2007 AMC 10A Problems/Problem 18
  ...ine{AG}</math> and <math>\overline{CH}</math> meet at <math>M</math>. What is the area of quadrilateral <math>ABCM</math>? pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
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• 2003 AMC 12B Problems/Problem 20
  Part of the graph of <math>f(x) = ax^3 + bx^2 + cx + d</math> is shown. What is <math>b</math>? <cmath>a(x-1)(x+1)(x-n) = ax^3-anx^2-ax+an = ax^3 + bx^2 + cx + d = 0</cmath>
  2 KB (382 words) - 18:01, 23 November 2020
• 2003 AMC 12B Problems/Problem 17
  If <math>\log (xy^3) = 1</math> and <math>\log (x^2y) = 1</math>, what is <math>\log (xy)</math>? <cmath>3\log(xy) + 2\log y + 2\log x = 3 \Longrightarrow 5\log(xy) = 3</cmath>
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• 2003 AMC 12B Problems/Problem 1
  Which of the following is the same as <cmath>\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?</cmath>
  970 bytes (134 words) - 00:09, 14 September 2015
• 2004 AMC 10A Problems/Problem 19
  ...ted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet? draw(ellipse(origin, 3, 1));
  2 KB (220 words) - 14:19, 21 April 2021
• 2007 AMC 10A Problems/Problem 24
  ...vely, and <math>EF</math> is a common [[tangent (geometry)|tangent]]. What is the area of the shaded region <math>ECODF</math>? <math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qqu
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• 2007 AMC 10A Problems/Problem 19
  ...etric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? unitsize(3 cm);
  4 KB (560 words) - 14:57, 3 June 2021
• 2003 AMC 12B Problems/Problem 19
  ...second term is <math>2</math>, in lowest terms, is <math>a/b</math>. What is <math>a+b</math>? ...the answer is <math>\frac{3 \cdot 3!}{4 \cdot 4!} = \frac {18}{96} = \frac{3}{16}</math>, and <math>a+b=19 \Rightarrow \mathrm{(E)}</math>.
  2 KB (320 words) - 22:59, 5 May 2024
• 2003 AMC 12B Problems/Problem 24
  has exactly one solution. What is the minimum value of <math>c</math>? When <math>x<a</math>, the slope is <math>-3</math>.
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• 2003 AMC 12B Problems/Problem 13
  ...eam occupies <math>75\%</math> of the volume of the frozen ice cream. What is the ratio of the cone’s height to its [[radius]]? \qquad\mathrm{(B)}\ 3:1
  960 bytes (154 words) - 15:20, 17 October 2016
• 1998 AHSME Problems/Problem 22
  What is the value of the [[expression]] Thus the sum is
  1 KB (138 words) - 10:41, 22 April 2016
• 1998 AHSME Problems/Problem 26
  In [[quadrilateral]] <math>ABCD</math>, it is given that <math>\angle A = 120^{\circ}</math>, angles <math>B</math> and < ...ath>\triangle ABE \sim \triangle CDE</math>, so <math>\triangle CDE</math> is also a <math>30-60-90</math> triangle.
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• 1974 USAMO Problems/Problem 1
  .../math> denote a polynomial having all integral coefficients. Show that it is impossible that <math>P(a)=b</math>, <math>P(b)=c</math>, and <math>P(c)=a< If <math>P</math> is a polynomial with integral coefficients, then <cmath>a - b | P(a) - P(b).</
  7 KB (1,291 words) - 20:30, 27 April 2020
• 1998 AHSME Problems/Problem 27
  ...as each center cube are removed. The [[surface area]] of the final figure is: ...th>. Thus the total surface area for each <math>3 \times 3 \times 3</math> is <math>72</math>.
  3 KB (403 words) - 23:00, 10 August 2020
• 1995 USAMO Problems
  ...ng terms. Prove that, for all <math>\, n, \,</math> <math>\, a_n \,</math> is the number obtained by writing <math>\, n \,</math> in base <math>\, p-1 \, A calculator is broken so that the only keys that still work are the <math>\, \sin, \; \cos
  3 KB (540 words) - 13:31, 4 July 2013
• 2004 AMC 12B Problems/Problem 25
  ...h>2^{2004}</math> is a <math>604</math>-[[digit]] number whose first digit is <math>1</math>, how many [[element]]s of the [[set]] <math>S = \{2^0,2^1,2^ ...,9\}</math> [[iff|if and only if]] the first digit of <math>2^{k-1}</math> is <math>4</math>, so there are <math>\boxed{195} \Rightarrow \mathrm{(B)}</ma
  3 KB (383 words) - 20:12, 15 October 2016
• 2004 AMC 12B Problems/Problem 23
  ...three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of <math>n</math> are possible? <center><math>(x - r)(x - s)(x - (r + s))</math> <math>= x^3 - (r + s + r + s) x^2 + (rs + r(r + s) + s(r + s))x - rs(r + s) = 0</math><
  3 KB (533 words) - 14:52, 29 October 2023
• 2004 AMC 12B Problems/Problem 24
  ...ath> <math>\cot \angle DBC</math> form an [[arithmetic progression]]. What is the area of <math>\triangle ABC</math>? ..., D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0);
  2 KB (302 words) - 19:59, 3 July 2013
• 2004 AMC 12B Problems/Problem 19
  ...angent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? \qquad\mathrm{(E)}\ 6\sqrt{3}</math>
  3 KB (520 words) - 19:12, 20 November 2023
• 1990 USAMO Problems/Problem 5
  An acute-angled triangle <math>ABC</math> is given in the plane. The circle with diameter <math>\, AB \,</math> intersec ...ath>, <math>H</math>, <math>A'</math> are concurrent, where <math>H</math> is the orthocentre.
  3 KB (604 words) - 20:52, 24 October 2018
• 1995 USAMO Problems/Problem 2
  A calculator is broken so that the only keys that still work are the <math>\, \sin, \; \cos ...th>n=1</math> and <math>m=0</math>, and <math>\sqrt{m/n} = 0</math>, which is initially shown on the screen. For the inductive step, we consider separat
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• 2004 AMC 12B Problems/Problem 21
  ...th>\frac yx</math> over all points <math>(x,y)</math> on the ellipse. What is the value of <math>a+b</math>? <math>\mathrm{(A)}\ 3
  2 KB (256 words) - 16:23, 13 March 2023
• 2004 AMC 12B Problems/Problem 13
  ...>f^{-1}(x) = bx+a</math> with <math>a</math> and <math>b</math> real, what is the value of <math>a+b</math>? ...ath>f(f^{-1}(x))=-(-x-1)-1=x+1-1=x.</cmath><cmath>f^{-1}(f(x))=-(-x-1)-1=x+1-1=x.</cmath>Therefore <math>a+b=\boxed{-2}</math>.
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• 2007 AMC 10B Problems/Problem 11
  ...o sides of length <math>3</math> and a base of length <math>2</math>. What is the area of this circle? ...} \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>
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• 1994 USAMO Problems
  ...let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> for <math>\, m = 1,2,3, \ldots \; \;</math>. Prove that, for each positive integer <math>\, n, \, ...t sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
  2 KB (391 words) - 07:58, 19 July 2016
• 2002 AMC 10A Problems/Problem 25
  ...ath>CD = 39</math>, and <math>DA = 5</math>. The area of <math>ABCD</math> is ...e hard to use [[trigonometry]] to bash this and find the height, but there is a much easier way. Extend <math>\overline{AD}</math> and <math>\overline{BC
  8 KB (1,308 words) - 07:05, 19 December 2022
• 2002 AMC 10A Problems/Problem 22
  A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered w The pattern is quite simple to see after listing a couple of terms.
  3 KB (430 words) - 23:13, 13 September 2023
• Isogonal conjugate
  Sometimes it is convenient to take one pair of isogonals as the base one, for example, <mat ...h> into a point with coordinates <math>(\frac{1}{x}, \frac {y}{x}),</math> is projective.
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 10A Problems
  ...qquad\mathrm{(B)}\ \text{3:00}\ {\small\text{PM}}\qquad\mathrm{(C)}\ \text{3:30}\ {\small\text{PM}}\qquad\mathrm{(D)}\ \text{4:30}\ {\small\text{PM}}\qq ...gth to its width is <math>2:1</math>. What percent of the rectangle's area is in the square?
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 2
  Clearly, <math>P_1(x)</math> has 2 real solutions, where 1 is positive and 1 is negative. The absolute values of these two solutions are also both less tha ...have that <math>-2<s<2</math>, so <math>0<r^2<4</math>, so <math>r</math> is real and <math>|r|<2</math>. Therefore all of the roots of <math>P_{k+1}</m
  3 KB (596 words) - 16:19, 28 July 2015
• 1976 IMO Problems/Problem 3
  ...f the box, then exactly <math>40</math> percent from the volume of the box is occupied. Determine the possible dimensions of the box. x &= \left\lfloor\frac{a}{\sqrt[3]{2}}\right\rfloor \\
  4 KB (605 words) - 16:27, 29 January 2021
• 1976 IMO Problems/Problem 4
  ...who is the product of some positive integers, and the sum of these numbers is <math>1976.</math> ...>3*3=2*2*2+1</math>, 3's are more efficient than 2's. We try to prove that 3's are more efficient than anything:
  4 KB (668 words) - 04:29, 2 January 2023
• 1976 IMO Problems/Problem 6
  A sequence <math>(u_{n})</math> is defined by <cmath>\lfloor u_{n} \rfloor = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}</cmath>
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• Partition (combinatorics)
  ...ee partitions of 3: <math>3 = 2+1 =1+1+1</math>. Each of the [[summand]]s is a ''part'' of the partition. ...formula! No simpler formula is known, and the existence of such a formula is doubtful.
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2001 AMC 12 Problems/Problem 5
  What is the product of all positive odd integers less than <math>10000</math>? ...dot 4 \cdot 6 \cdots 10000}= \dfrac{10000!}{2^{5000} \cdot 1 \cdot 2 \cdot 3 \cdots 5000}= \dfrac{10000!}{2^{5000}\cdot5000!}</math>
  2 KB (226 words) - 00:30, 30 December 2023
• 2001 AMC 12 Problems/Problem 6
  order; that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</m ...1, 2, 3, 4, 5, 6, 7, 8, 9</math>. Clearly, the most restrictive condition is the consecutive odd digits, so we create casework based on that.
  2 KB (340 words) - 03:02, 28 June 2023
• 2008 AMC 12A Problems
  ...qquad\mathrm{(B)}\ \text{3:00}\ {\small\text{PM}}\qquad\mathrm{(C)}\ \text{3:30}\ {\small\text{PM}}\qquad\mathrm{(D)}\ \text{4:30}\ {\small\text{PM}}\qq What is the [[reciprocal]] of <math>\frac{1}{2}+\frac{2}{3}</math>?
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 1
  ...qquad\mathrm{(B)}\ \text{3:00}\ {\small\text{PM}}\qquad\mathrm{(C)}\ \text{3:30}\ {\small\text{PM}}\qquad\mathrm{(D)}\ \text{4:30}\ {\small\text{PM}}\qq ...8:30}=\text{2:40}</math> hours. Thus, the entire job is completed in <math>3\cdot(\text{2:40})=\text{8:00}</math> hours.
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• 2008 AMC 12A Problems/Problem 2
  What is the [[reciprocal]] of <math>\frac{1}{2}+\frac{2}{3}</math>? ...mathrm{(B)}\ \frac{7}{6}\qquad\mathrm{(C)}\ \frac{5}{3}\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ \frac{7}{2}</math>
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• 2008 AMC 12A Problems/Problem 19
  what is the [[coefficient]] of <math>x^{28}</math>? ...<math>x^{28}</math>. Thus the coefficient of the <math>x^{28}</math> term is <math>224 \Rightarrow \boxed{C}</math>.
  5 KB (895 words) - 22:54, 9 January 2021
• 2008 AMC 12A Problems/Problem 17
  ...sitive integers <math>a_1 \le 2008</math> is it true that <math>a_1</math> is less than each of <math>a_2</math>, <math>a_3</math>, and <math>a_4</math>? ...4n+1</math>, <math>4n+2</math>, or <math>4n+3</math>, where <math>n</math> is a nonnegative integer.
  2 KB (386 words) - 13:52, 21 December 2020
• 2002 AMC 12B Problems/Problem 17
  ...<math>\frac{x}{2}</math>, and Carlos's lawn has an area of <math>\frac{x}{3}</math>. ...ed of <math>\frac{y}{3}</math>, and Beth's cuts at a speed <math>\frac{2y}{3}</math>.
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• 2008 AMC 12A Problems/Problem 8
  What is the [[volume]] of a [[cube]] whose [[surface area]] is twice that of a cube with volume 1? A cube with volume <math>1</math> has a side of length <math>\sqrt[3]{1}=1</math> and thus a surface area of <math>6 \cdot 1^2=6</math>.
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• 2008 AMC 12A Problems/Problem 16
  ...h>12^\text{th}</math> term of the sequence is <math>\log{b^n}</math>. What is <math>n</math>? ...</math>, and <math>8A + 15B</math>, and the <math>12^\text{th}</math> term is <math>nB</math>.
  3 KB (577 words) - 16:33, 9 October 2022
• 2008 AMC 12A Problems/Problem 4
  Which of the following is equal to the [[product]] Thus, the product is <math>\frac{2008}{4}=502</math>, and the answer is <math>\textbf{(B)}</math>.
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• 2008 AMC 12A Problems/Problem 21
  .../math> is <u>heavy-tailed</u> if <math>a_1 + a_2 < a_4 + a_5</math>. What is the number of heavy-tailed permutations? ..._2,a_3,a_4,a_5)</math> such that <math>a_1 + a_2 < a_4 + a_5</math>, there is exactly one permutation such that <math>a_1 + a_2 > a_4 + a_5</math>. Thus
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• 2008 AMC 12A Problems/Problem 13
  ...nt to both <math>\overline{OA}</math> and <math>\overline{OB}</math>. What is the ratio of the area of the smaller circle to that of the larger circle? ...A=(3,0), B=(3/2,3/2*3^.5), C=(3^.5,1), D=(3^.5,0), F=(1.5*3^.5,1.5), G=(2*3^.5,2);
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• 2008 AMC 10A Problems/Problem 20
  ...math>, and the area of <math>\triangle AKD</math> is <math>24.</math> What is the area of trapezoid <math>ABCD</math>? pair D=(0,0),C=(12,0), K=(7,16/3); /* note that K.x is arbitrary, as generator for A,B */
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• 2008 AMC 12A Problems/Problem 18
  ...e positive <math>z</math>-axis. Let <math>O</math> be the [[origin]]. What is the volume of [[tetrahedron]] <math>OABC</math>? draw((0,10)--(0,0)--(8,0));draw((-3,-4)--(0,0));draw((0,10)--(-3,-4)--(8,0)--cycle);
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• 2008 AMC 12A Problems/Problem 12
  ...f{(C)}\ [0,2],[-1,0]\qquad\mathrm{(D)}\ [1,3],[-1,0]\qquad\mathrm{(E)}\ [1,3],[0,1]</math> ...(x)</math> is defined if <math>f(x + 1)</math> is defined. Thus the domain is all <math>x| x + 1 \in [0,2] \rightarrow x \in [ - 1,1]</math>.
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• 2008 AMC 12A Problems/Problem 7
  ...ant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they ar We set up the following equation, where <math>x</math> is the answer:
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• 2008 AMC 12A Problems/Problem 20
  ...math> have radii <math>r_a</math> and <math>r_b</math>, respectively. What is <math>r_a/r_b</math>? ...uad\mathrm{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right)\\\mathrm{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right)</math>
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• 2008 AMC 12A Problems/Problem 23
  ...i=0</math> are the vertices of a convex polygon in the complex plane. What is the area of the polygon? ...hrm{(C)}\ 2\qquad\mathrm{(D)}\ 2^{\frac{5}{4}}\qquad\mathrm{(E)}\ 2^{\frac{3}{2}}</math>
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• 2008 AMC 12A Problems/Problem 15
  Let <math>k={2008}^{2}+{2}^{2008}</math>. What is the units digit of <math>k^2+2^k</math>? ...th>k^2 \equiv 0 \pmod{10}</math>. Since <math>k = 2008^2+2^{2008}</math> is a multiple of four and the units digit of powers of two repeat in cycles of
  4 KB (547 words) - 04:19, 30 September 2023
• 2008 AMC 12A Problems/Problem 22
  ...that the inner corners each touch an inner corner of an adjacent mat. What is <math>x</math>? path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
  10 KB (1,518 words) - 17:00, 16 May 2023
• 2008 AMC 12A Problems/Problem 11
  ...t the <math>13</math> visible numbers have the greatest possible sum. What is that sum? ...ubes each have a pair of opposite faces that are covered up. When the cube is folded, <math>(1,32)</math>; <math>(2,16)</math>; and <math>(4,8)</math> ar
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• 2007 AMC 12B Problems/Problem 24
  ...ath>\text{gcd}(a,b)=1</math> and <math>\frac{a}{b} + \frac{14b}{9a}</math> is an integer? ...| 9a^2 + 14b^2 \quad\Longrightarrow\quad 9 | b^2 \quad\Longrightarrow\quad 3 | b</math>
  9 KB (1,522 words) - 22:46, 12 May 2022
• 2007 AMC 12B Problems/Problem 15
  ...involving odd powers of <math>r</math> have a sum of <math>3</math>. What is <math>a+r</math>? ...}\ \frac{12}{7}\qquad\textbf{(C)}\ \frac{3}{2}\qquad\textbf{(D)}\ \frac{7}{3}\qquad\textbf{(E)}\ \frac{5}{2} </math>
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• 2008 AMC 12A Problems/Problem 14
  What is the area of the region defined by the [[inequality]] <math>|3x-18|+|2y+7|\l <math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ \fra
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• 2008 AMC 12A Problems/Problem 9
  ...n on an older television screen with a <math>27</math>-inch diagonal. What is the height, in inches, of each darkened strip? ...B)}\ 2.25\qquad\mathrm{(C)}\ 2.5\qquad\mathrm{(D)}\ 2.7\qquad\mathrm{(E)}\ 3</math>
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• 2007 AMC 12B Problems/Problem 18
  ...h>acb</math> lies two thirds of the way between the same two squares. What is <math>a+b+c</math>? The difference between <math>acb</math> and <math>abc</math> is given by
  5 KB (758 words) - 16:35, 15 February 2021
• 2007 AMC 12B Problems/Problem 14
  ...ven that <math>PQ=1</math>, <math>PR=2</math>, and <math>PS=3</math>, what is <math>AB</math>? ...\mathrm{(B)}\ 3\sqrt{3} \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 4\sqrt{3} \qquad \mathrm{(E)}\ 9</math>
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• 2007 AMC 10B Problems/Problem 18
  ...rrounded by <math>4</math> circles of radius <math>r</math> as shown. What is <math>r</math>? ...d\textbf{(B) } 1+\sqrt{2} \qquad\textbf{(C) } \sqrt{6} \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 2+\sqrt{2}</math>
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• 1992 USAMO Problems/Problem 3
  ...math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible value of <math>a_{10}</math>? ...math> is a <math>2</math>-<math>3</math> set and <math>\{1,2,4,10\}</math> is a <math>4</math>-<math>8</math> set)
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• 2008 AMC 12A Problems/Problem 24
  ...math>. Point <math>D</math> is the [[midpoint]] of <math>BC</math>. What is the largest possible value of <math>\tan{\angle BAD}</math>? ...}\ \frac{\sqrt{3}}{2\sqrt{2}}\qquad\mathrm{(D)}\ \frac{\sqrt{3}}{4\sqrt{2}-3}\qquad\mathrm{(E)}\ 1</math>
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• 2008 AMC 12A Problems/Problem 25
  ...+ 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)</math> for <math>n = 1,2,3,\ldots</math>. Suppose that <math>(a_{100},b_{100}) = (2,4)</math>. What is <math>a_1 + b_1</math>?
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• 2008 AMC 12B Problems/Problem 13
  ...th> is between <math>\frac{1}{3}</math> and <math>\frac{2}{3}</math>. What is the area of <math>R</math>? \textbf{(D)}\ \frac{3-\sqrt3}{9} \qquad
  2 KB (328 words) - 10:54, 4 July 2013
• 2007 AMC 12B Problems/Problem 13
  ...watch the light. What is the probability that the color changes while she is watching? ...frac{1}{10} \qquad \mathrm{(D)}\ \frac{1}{7} \qquad \mathrm{(E)}\ \frac{1}{3}</math>
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• 2007 AMC 12B Problems/Problem 19
  ...> by taping <math>\overline{AB}</math> to <math>\overline{DC}</math>. What is <math>\sin(\angle ABC)</math>? ...pi}{6} \qquad \mathrm{(D)}\ \frac{\pi}{4} \qquad \mathrm{(E)}\ \frac{\sqrt{3}}{2}</math>
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• 2007 AMC 12B Problems/Problem 21
  ...f these base-<math>3</math> representations are palindromes? (A palindrome is a number that reads the same forward and backward.) <math>2007_{10} = 2202100_{3}</math>
  2 KB (267 words) - 16:35, 15 February 2021
• 2008 AMC 10A Problems/Problem 22
  ...tracts 1. What is the probability that the fourth term in Jacob's sequence is an [[integer]]? ...{(C)}\ \frac{1}{2}\qquad\mathrm{(D)}\ \frac{5}{8}\qquad\mathrm{(E)}\ \frac{3}{4}</math>
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• 2007 AMC 12B Problems/Problem 23
  ...ositive integer leg lengths have areas that are numerically equal to <math>3</math> times their perimeters? We have <math>\frac{1}{2}ab = 3(a+b+c)</math>.
  4 KB (725 words) - 19:59, 4 January 2024
• 2007 AMC 12B Problems/Problem 16
  Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistingu <math>4:0:0</math>, <math>3:1:0</math>, <math>2:2:0</math>, or <math>2:1:1</math>
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• 2004 AMC 12A Problems/Problem 19
  ...adius <math>1</math> and passes through the center of <math>D</math>. What is the radius of circle <math>B</math>? pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
  5 KB (785 words) - 00:29, 31 July 2023
• 2004 AMC 12A Problems/Problem 24
  ...us <math>1</math> in the plane that cover <math>\overline{AB}</math>. What is the area of <math>S</math>? ...textbf {(C) } 3\pi - \frac {\sqrt3}{2} \qquad \textbf {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \textbf {(E) }4\pi - 2\sqrt3</math>
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• 2008 AMC 10B Problems
  ...askets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. How many different numbers could represent the total points <math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6</math>
  12 KB (1,838 words) - 16:52, 7 October 2022
• 2008 AMC 12B Problems
  ...askets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. How many different numbers could represent the total points <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math>
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AMC 12B Problems/Problem 2
  A <math>4\times 4</math> block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows ar 1&2&3&4\\\hline
  2 KB (219 words) - 23:53, 31 July 2023
• 2008 AMC 12B Problems/Problem 4
  ...\angle AOC = 30^\circ</math>, and <math>\angle DOB = 45^\circ</math>. What is the ratio of the area of the smaller sector <math>COD</math> to the area of pair C = 3*dir (30);
  1 KB (183 words) - 22:35, 10 June 2017
• 2005 Alabama ARML TST Problems/Problem 13
  ...ural number with exactly 6 positive divisors, the sum of whose reciprocals is 2. Find that natural number. ...p_1</math> and <math>p_2</math> be primes. Therefore, one of the following is true:
  2 KB (279 words) - 19:14, 10 March 2015
• 2005 Alabama ARML TST Problems/Problem 14
  <math>x^2-2y^2=1</math> means that <math>x</math> is odd. We can let <math>x=2x_1-1</math> for some <math>x_1>0</math>: y is even, <math>y=2y_1</math> for some <math>y_1>0</math>.
  3 KB (478 words) - 23:41, 5 January 2014
• 2006 Alabama ARML TST
  ...urnament]], one of two tests given to choose the [[Alabama ARML]] team. It is similar to an easy [[AIME]], Alabaman scores average about 5 right, while t * [[2006 Alabama ARML TST Problems/Problem 3]]
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• 2006 Alabama ARML TST Problems
  ==Problem 3== River draws four cards from a standard 52 card deck of playing cards. Exactly 3 of them are 2’s. Find the probability River drew exactly one spade and on
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• 2008 AMC 12B Problems/Problem 5
  ...rs to buy flowers for a classmate who is in the hospital. Roses cost <math>3</math> dollars each, and carnations cost <math>2</math> dollars each. No ot
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• 2008 AMC 12B Problems/Problem 11
  ...math>\frac{1}{8}</math> of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet? ...of the cone formed by cutting off the mountain at <math>4,000</math> feet is half that of the original mountain). Therefore, volume varies as the invers
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• 2008 AMC 12B Problems/Problem 1
  ...askets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. How many different numbers could represent the total points <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math>
  1 KB (176 words) - 16:02, 19 August 2017
• 2008 AMC 12B Problems/Problem 21
  ...om the line segment joining <math>(0,1)</math> to <math>(2,1)</math>. What is the probability that circles <math>A</math> and <math>B</math> intersect? ...)} \; \frac {2 + \sqrt {3}}{4} \qquad \textbf{(E)} \; \frac {4 \sqrt {3} - 3}{4}</math>
  6 KB (1,008 words) - 11:46, 24 December 2020
• 2008 AMC 12B Problems/Problem 9
  ...oint <math>C</math> is the midpoint of the minor arc <math>AB</math>. What is the length of the line segment <math>AC</math>? ...ot 5\cdot \frac{4}{5}} = \sqrt{50-50\frac{4}{5}} = \sqrt{10}</math>, which is answer choice <math>\boxed{\text{A}}</math>.
  2 KB (319 words) - 13:48, 15 February 2021
• 2006 Alabama ARML TST Problems/Problem 4
  ...sing order. The number of ways to take three digits out is <math>\binom{9}{3}=\boxed{84}</math> ...ay that <math>b</math> is the second smallest integer. Then <math>b</math> is the ten-thousands digit. etc.
  1 KB (200 words) - 19:37, 12 April 2012
• 2008 AMC 12B Problems/Problem 25
  ...ath>\angle B</math> and <math>\angle C</math> meet at <math>Q</math>. What is the area of hexagon <math>ABQCDP</math>? ...C)}\ 32\sqrt{3}\qquad \textbf{(D)}\ 35\sqrt{3}\qquad \textbf{(E)}\ 36\sqrt{3}</math>
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• 2008 AMC 12B Problems/Problem 24
  ...ve integer <math>n,\ A_{n-1}B_nA_n</math> is an equilateral triangle. What is the least <math>n</math> for which the length <math>A_0A_n\geq100</math>? ...lateral triangle, can be compacted as follows: <cmath>\left(a_n\frac{\sqrt{3}}{2}\right)^2=\frac{a_n}{2}+a_{n-1}+a_{n-2}+\cdots+a_1</cmath>
  9 KB (1,482 words) - 13:52, 4 April 2024
• 2008 AMC 12B Problems/Problem 22
  ...er SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? ...4}{7} \qquad \textbf{(C)} \; \frac {81}{140} \qquad \textbf{(D)} \; \frac {3}{5} \qquad \textbf{(E)} \; \frac {17}{28}</math>
  4 KB (653 words) - 11:06, 15 October 2022
• 1985 AJHSME Problems
  <math>\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=</math> == Problem 3 ==
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• Gravitational constant
  ...m^3 kg^{-1} s^{-2}</math>. It is commonly denoted as <math>G</math>. There is some uncertainty associated with the value of the constant, due to limitati ...large mountain", thus estimating <math>G = (6.7 \pm 0.6) \cdot 10^{-11} m^3 kg^{-1} s^{-2}</math>.
  2 KB (274 words) - 12:56, 9 June 2021
• Mock AIME 1 Pre 2005 Problems/Problem 4
  ...004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</math>. ...= 400</math>, by [[Fermat's Little Theorem|Fermat-Euler's Theorem]], this is equivalent to finding <math>\frac{7^{400 \cdot 5 + 5} - 1}{6} \equiv \frac{
  685 bytes (81 words) - 10:51, 11 June 2013
• Mock AIME 1 Pre 2005 Problems/Problem 7
  # None of the first four letters is an <math>A</math>. # None of the next five letters is a <math>B</math>.
  1 KB (221 words) - 17:27, 23 February 2013
• Mock AIME 1 Pre 2005 Problems/Problem 8
  ...</math> denote the apex of <math>P</math>. If the volume of <math>P</math> is eight times that of <math>P'</math>, then the value of <math>XT</math> can ...with volumes, the ratio of the volume of <math>P'</math> to <math>P</math> is the cube of the ratio of the height of <math>P'</math> to <math>P</math>.
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• 1993 USAMO Problems/Problem 1
  is larger. It is [[Trivial Inequality|trivial]] that
  2 KB (393 words) - 07:54, 19 July 2016
• 2008 AIME I Problems
  ...by <math>20</math> more boy students, all of whom like to dance, the party is now <math>58\%</math> girls. How many students now at the party like to dan ...the area common to triangle <math>GEM</math> and square <math>AIME</math> is <math>80</math> square units. Find the length of the altitude to <math>EM<
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• 2008 AIME II Problems
  ...nd subtractions alternate in pairs. Find the remainder when <math>N</math> is divided by <math>1000</math>. ...te break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minut
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 1
  ...by <math>20</math> more boy students, all of whom like to dance, the party is now <math>58\%</math> girls. How many students now at the party like to dan ...gives <math>k = 116</math>. Thus, the number of people that like to dance is <math>2k + 20 = \boxed{252}</math>.
  2 KB (378 words) - 22:23, 19 December 2022
• 2008 AIME I Problems/Problem 2
  ...the area common to triangle <math>GEM</math> and square <math>AIME</math> is <math>80</math> square units. Find the length of the altitude to <math>EM< ..., then the maximum area of the intersection of the triangle and the square is <math>5\cdot10=50</math>.
  2 KB (250 words) - 19:37, 29 December 2020
• 2008 AIME I Problems/Problem 3
  ...ath> kilometers after jogging for <math>2</math> hours, swimming for <math>3</math> hours, and biking for <math>4</math> hours. Their biking, jogging, ..., we need <math>s\equiv1\pmod{4}</math>. Thus, <math>(j,s) = (13,1),(8,5),(3,9)</math>.
  3 KB (564 words) - 22:15, 28 November 2023
• 2008 AIME I Problems/Problem 4
  ..., one of the factors is even. A [[parity]] check shows that if one of them is even, then both must be even. Since <math>244 = 2^2 \cdot 61</math>, the fa Indeed, by solving, we find <math>(x,y) = (18,62)</math> is the unique solution.
  4 KB (732 words) - 22:17, 28 November 2023
• 2008 AIME I Problems/Problem 6
  ...]] array of numbers has a first row consisting of the odd integers <math>1,3,5,\ldots,99</math> in increasing order. Each row below the first has one f ...ontains one entry that is a multiple of <math>67</math>, and so the answer is <math>\frac{33+1}{2} = \boxed{017}</math>.
  3 KB (509 words) - 17:21, 22 March 2018
• 2008 AIME I Problems/Problem 8
  <cmath>\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac Applying this to the first two terms, we get <math>\arctan{\dfrac{1}{3}} + \arctan{\dfrac{1}{4}} = \arctan{\dfrac{7}{11}}</math>.
  3 KB (490 words) - 22:36, 28 November 2023
• 2008 AIME I Problems/Problem 9
  ...Let <math>\frac {m}{n}</math> be the probability that the stack of crates is exactly <math>41\mathrm{ft}</math> tall, where <math>m</math> and <math>n</ Only the heights matter, and each crate is either 3, 4, or 6 feet tall with equal probability. We have the following:
  6 KB (909 words) - 15:39, 8 August 2022
• 2008 AIME I Problems/Problem 10
  ...<math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>. ...A=(0,0), D=(4,0), B= A+2 expi(1/3*pi), C= D+2expi(2/3*pi), E=(-4/3,0), F=(3,0);
  4 KB (629 words) - 22:38, 28 November 2023
• 2008 AIME I Problems/Problem 11
  ...AA</math>, <math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? 3&1&2&
  7 KB (1,173 words) - 22:39, 28 November 2023
• 2008 AIME I Problems/Problem 12
  ...s the photoelectric eye in one hour. Find the quotient when <math>M</math> is divided by <math>10</math>. ...unit, the CAR comes first, THEN the empty space. So at time zero, the car is right at the eye.
  4 KB (669 words) - 18:35, 8 October 2023
• 2008 AIME I Problems/Problem 13
  ...a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.</cmath> There is a point <math>\left(\frac {a}{c},\frac {b}{c}\right)</math> for which <math
  8 KB (1,218 words) - 00:07, 11 April 2024
• 2008 AIME I Problems/Problem 15
  ...th>n</math> are positive integers, <math>m<1000</math>, and <math>m</math> is not divisible by the <math>n</math>th power of any prime. Find <math>m+n</m ...er, and <math>M</math> and <math>N</math> be the two points whose distance is <math>\sqrt{17}</math> from <math>P</math>. Also, let <math>R</math> be the
  6 KB (1,041 words) - 00:54, 1 February 2024
• Math Contest Camp 2016
  New in 2016 is the option of a Computer Science Track July 12-22 only for students enterin ...by Dr. Kevin Hopkins, the chairman of the math department at SBU, but SBU is not a sponsor of the camp.
  2 KB (343 words) - 22:29, 23 January 2016
• 2008 AIME I Problems/Problem 14
  ...ne <math>CT</math> is tangent to <math>\omega</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. ...math>k = \frac{2x-27}{x^2} \Longrightarrow kx^2 - 2x + 27 = 0</math>; this is a quadratic, and its [[discriminant]] must be nonnegative: <math>(-2)^2 - 4
  8 KB (1,333 words) - 00:18, 1 February 2024
• Mock AIME 5 Pre 2005 Problems
  Function <math>g(y)</math> is given such way that for all <math>y</math>, ...{abcde}</math> and last two digits on the <math>\text {fghij}</math>? That is, <math>b + c + d + i + j</math>.
  6 KB (909 words) - 07:27, 12 October 2022
• Mock AIME 5 Pre 2005 Problems/Problem 12
  ...60401 + 256 = 104060657, \end{aligned}</math> and so the sum of the digits is <math>1+4+6+6+5+7 = \boxed{29}.</math>
  517 bytes (55 words) - 20:01, 23 March 2017
• Mock AIME 1 2007-2008 Problems/Problem 4
  If <math>x</math> is an odd number, then find the largest integer that always divides the expres <cmath>4(5x + 1)(5x + 3)(5x+5)</cmath>
  1 KB (179 words) - 00:04, 10 August 2019
• Mock AIME 1 2007-2008 Problems/Problem 9
  : <math>\frac n2</math> is a perfect square. : <math>\frac n3</math> is a perfect cube.
  2 KB (323 words) - 00:32, 31 August 2015
• Mock AIME 1 2007-2008 Problems
  What is the coefficient of <math>x^3y^{13}</math> in <math>\left(\frac 12x + y\righ ...^n</math> has 3 consecutive terms with coefficients in the ratio <math>1:2:3</math> that can be written in the form <cmath>{n\choose k} : {n\choose k+1}
  6 KB (992 words) - 14:15, 13 February 2018
• 2008 AIME II Problems/Problem 1
  ...nd subtractions alternate in pairs. Find the remainder when <math>N</math> is divided by <math>1000</math>. ...- 93^2 + 92^2 + 91^2 + \ldots - 10^2 - 9^2 + 8^2 + 7^2 - 6^2 - 5^2 + 4^2 + 3^2 - 2^2 - 1^2 \mbox{, and reordering, we get}\\
  4 KB (575 words) - 16:41, 14 April 2024
• 2008 AIME II Problems/Problem 2
  ...te break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minut ...>r</math> be the time Rudolph takes disregarding breaks and <math>\frac{4}{3}r</math> be the time Jennifer takes disregarding breaks. We have the equati
  3 KB (432 words) - 11:53, 19 June 2019
• 2008 AIME II Problems/Problem 3
  ...se. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese af ...\boxed{729}</math>. Equality is achieved when <math>a=b=c=9</math>, which is possible if we make one slice perpendicular to the <math>10</math> cm edge,
  2 KB (282 words) - 12:56, 3 March 2015
• 2008 AIME II Problems/Problem 4
  ...th>3</math>, we find that <math>\overline{2008}_{10} = \overline{2202101}_{3}</math>. In other words, ...2008 = 2 \cdot 3^{6} + 2 \cdot 3^{5} + 2 \cdot 3^3 + 1 \cdot 3^2 + 1 \cdot 3^0</math></center>
  1 KB (167 words) - 21:47, 21 September 2020
• 2008 AIME II Problems/Problem 5
  ...th>, note that the midpoint of <math>\overline{AD}</math>, <math>N</math>, is the center of the [[circumcircle]] of <math>\triangle AED</math>. We can do ...ath>, <math>M</math>, to the midpoint of <math>\overline{AD}</math>, which is <math>N</math>, then <math>E,M,N</math> are [[collinear]].
  8 KB (1,338 words) - 23:15, 28 November 2023
• 2008 AIME II Problems/Problem 6
  The sequence <math>\{a_n\}</math> is defined by The sequence <math>\{b_n\}</math> is defined by
  1 KB (223 words) - 23:34, 4 July 2013
• 2008 AIME II Problems/Problem 7
  8x^3 + 1001x + 2008 = 0. Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>.
  7 KB (1,251 words) - 19:18, 2 January 2024
• 2008 AIME II Problems/Problem 8
  is an integer. ...\cdot 251</math>. Therefore, it is clear that <math>n = \boxed{251}</math> is the smallest such integer.
  4 KB (621 words) - 18:26, 16 January 2023
• 2008 AIME II Problems/Problem 9
  ...direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the greatest integer less than or equal to <math>| ...ta</math> be the inclination of OP to the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have
  5 KB (725 words) - 22:37, 28 January 2024
• 2008 AIME II Problems/Problem 10
  ...w shows a <math>4\times4</math> rectangular array of points, each of which is <math>1</math> unit away from its nearest neighbors. ...the property that the distance between consecutive points of the sequence is strictly increasing. Let <math>m</math> be the maximum possible number of p
  4 KB (569 words) - 09:44, 25 November 2019
• 2008 AIME II Problems/Problem 11
  ...math>n</math>, and <math>k</math> are positive integers and <math>k</math> is the product of distinct primes. Find <math>m + nk</math>. pair A=(0,96),B=(-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M=foot(Q,X,P);
  6 KB (1,065 words) - 20:12, 9 August 2022
• 2008 AIME II Problems/Problem 12
  ...flags on either pole are adjacent. Find the remainder when <math>N</math> is divided by <math>1000</math>. ...ns where either pole has no flags, we have to count these separately. This is the same as choosing our extra blue as one of the two ends, and ordering th
  10 KB (1,550 words) - 12:58, 15 July 2023
• 2008 AIME II Problems/Problem 15
  :(ii) <math>2n + 79</math> is a perfect square. ...h>, or equivalently, <math>(2n + 1)(2n - 1) = 4n^2 - 1 = 12m^2 + 12m + 3 = 3(2m + 1)^2</math>.
  4 KB (628 words) - 16:23, 2 January 2024
• Ultrafilter
  An '''ultrafilter''' is a [[set theory | set theoretic]] structure. An ultrafilter on a set <math>X</math> is a non-empty [[filter]] <math>\mathcal{F}</math> on <math>X</math> with the
  9 KB (1,685 words) - 20:28, 13 October 2019
• 1974 USAMO Problems/Problem 5
  pair P=(7,0), Q=(14,0), R=P+7expi(pi/3), M=(10,1.2); We rotate figure <math>PRQM</math> by a clockwise angle of <math>\pi/3</math> about <math>Q</math> to obtain figure <math>RR'QM'</math>:
  7 KB (1,221 words) - 18:57, 3 July 2013
• 2005 USAMO Problems/Problem 3
  ...ath>B_1</math> in such a way that convex quadrilateral <math>APCB_1</math> is cyclic, <math>QB_1 \parallel BA </math>, and <math>B_1 </math> and <math>Q ...f the circumcircle of <math>B_1' C_1P</math> and line <math>BC</math>. It is enough to show that <math>B_1'=B_1</math> and <math>Q' =Q</math>. All our
  6 KB (1,117 words) - 01:17, 11 October 2021
• 2005 USAMO Problems/Problem 2
  \begin{align*}x^6 + x^3 + x^3y + y &= 147^{157} \\ x^3 + x^3y + y^2 + y + z^9 &= 157^{147}\end{align*}
  7 KB (1,053 words) - 10:38, 12 August 2015
• Asian Pacific Mathematics Olympiad
  The '''Asian Pacific Mathematics Olympiad''' ('''APMO''') is a mathematics olympiad for high school students in countries on and near th ...|type=Proof|difficulty=6 - 8|breakdown=<u>Problem 1</u>: 6<br><u>Problem 2/3</u>: 7<br><u>Problem 4</u>: 7.5<br><u>Problem 5</u>: 8}}
  1 KB (196 words) - 10:24, 12 October 2020
• 1989 APMO Problems
  ...th>(1+x_1)(1+x_2)(1+x_3)\cdots (1+x_n)\leq 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots +\dfrac{S^n}{n!}</cmath>. == Problem 3 ==
  2 KB (340 words) - 10:27, 11 April 2016
• 2005 USAMO Problems/Problem 1
  ...g'') Determine all composite positive integers <math>n</math> for which it is possible to arrange all divisors of <math>n</math> that are greater than 1 ...</math>, start with the circular arrangement of <math>n,p_{1}p_{2},p_{2}p_{3}\ldots,p_{k-1}p_{k}</math> as shown.
  4 KB (650 words) - 13:40, 4 July 2013
• 1990 APMO Problems
  ...math>, how many non-similar triangles are there in which <math>AEGF</math> is a cyclic quadrilateral? == Problem 3 ==
  2 KB (364 words) - 23:07, 12 September 2016
• 1997 PMWC Problems/Problem T7
  ...and fix a cuboid, <b>keeping the surfaces of the cuboid red</b>. Now what is the maximum possible volume of the cuboid? ...nt 1:''' The number of cubes with at least one red face is <math>5^3-(5-2)^3=125-27=98</math>. Therefore, the volume of the cube cannot more than <math>
  2 KB (283 words) - 21:47, 13 July 2023
• Curvature
  ...or a [[circle]] of [[radius]] <math>r</math>, the curvature at every point is <math>\frac{1}{r}</math>. Intuitively, this grows smaller as <math>r</math ...<math>y = f(x)</math> of the function at the point <math>(x, f(x))</math> is given by the formula
  1 KB (196 words) - 21:53, 20 November 2015
• 1995 AHSME Problems/Problem 19
  ...ea of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is ...1}{6} } \qquad \mathrm{(B) \ \frac {1}{4} } \qquad \mathrm{(C) \ \frac {1}{3} } \qquad \mathrm{(D) \ \frac {2}{5} } \qquad \mathrm{(E) \ \frac {1}{2} }
  1 KB (200 words) - 20:58, 10 February 2019
• Cone
  .... Thus, the cone is the special case of the [[pyramid]] in which the base is circular. import three; currentprojection = perspective(0,-3,1); defaultpen(linewidth(0.7)); triple vertex = (0,0,1.5);
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2006 Alabama ARML TST Problems/Problem 13
  <math>\sqrt[4]{x+27}=1\Rightarrow x=-26, \sqrt[4]{55-x}=\sqrt[4]{81}=3</math> <math>\sqrt[4]{x+27}=3\Rightarrow x=54, \sqrt[4]{55-x}=1</math>
  1 KB (167 words) - 09:56, 15 December 2018
• 2007 Alabama ARML TST
  ...urnament]], one of two tests given to choose the [[Alabama ARML]] team. It is similar to an easy [[AIME]], Alabaman scores average about 5 right, while t * [[2007 Alabama ARML TST Problems/Problem 3]]
  1 KB (167 words) - 00:59, 19 June 2018
• 2008 AIME II Problems/Problem 14
  ...a^2</math>, so <math>b^2 \ge \frac {3}{4}a^2 \implies \rho^2 \le \frac {4}{3}</math>. ...e equation, so <math>\rho^2 = \frac {4}{3}</math>, and the answer is <math>3 + 4 = \boxed{007}</math>.
  8 KB (1,387 words) - 11:56, 29 January 2024
• 2008 AIME II Problems/Problem 13
  ...plex plane]] has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let <math>R</math> be the region outside th ...{cis}\, \theta = \cos \theta + i \sin \theta</math>). Since <math>R</math> is symmetric every <math>60^{\circ}</math> about the origin, it suffices to co
  6 KB (894 words) - 18:56, 25 December 2022
• Mass points
  ...greatly simplify the proofs of many theorems concerning [[polygon]]s, and is helpful in solving complex geometry problems involving lengths. In essence ...h>AB</math> with point <math>C</math> on it, and the mass put on a point P is denoted as <math>m_P</math>,
  5 KB (815 words) - 22:31, 1 April 2024
• Shoelace Theorem
  The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a simple [[polygon]] given the ...sted in clockwise order. Then the area (<math>A</math>) of <math>P</math> is
  8 KB (1,358 words) - 15:32, 22 February 2024
• Mock AIME 1 2007-2008 Problems/Problem 11
  <math>\triangle DEF</math> is inscribed inside <math>\triangle ABC</math> such that <math>D,E,F</math> li draw(A--E,StickIntervalMarker(1,3,size=6));draw(B--D,StickIntervalMarker(1,3,size=6));
  2 KB (265 words) - 20:02, 29 January 2020
• 0.999...
  <math>0.999\ldots</math> (or <math>0.\overline{9}</math>) is an equivalent representation of the [[integer]] <math>1</math>. ...roblem, one needs mathematics beyond the elementary school level math that is needed to understand the question.
  3 KB (577 words) - 20:04, 4 February 2023
• 2008 AMC 10B Problems/Problem 3
  ...ath> is a [[positive]] [[real number]]. Which is equivalent to <math>\sqrt[3]{x\sqrt{x}}</math>? <math>\mathrm{(A)}\ x^{1/6}\qquad\mathrm{(B)}\ x^{1/4}\qquad\mathrm{(C)}\ x^{3/8}\qquad\mathrm{(D)}\ x^{1/2}\qquad\mathrm{(E)}\ x</math>
  707 bytes (97 words) - 19:54, 24 April 2024
• 2008 AMC 10B Problems/Problem 7
  An equilateral triangle of side length <math>10</math> is completely filled in by non-overlapping equilateral triangles of side lengt ...s <math>\frac{10^2\sqrt3}{4}</math>, while the area of each small triangle is <math>\frac{1^2\sqrt3}{4}</math>. Dividing these two quantities results in
  2 KB (323 words) - 20:23, 14 July 2021
• 2008 AMC 10B Problems/Problem 8
  ...dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to b ...Of course, the number of roses must be such that the number of carnations is non-negative. We get the inequality <math>25-3r \geq 0</math> which solves
  2 KB (401 words) - 20:32, 26 July 2022
• 2008 AMC 10B Problems/Problem 9
  ...atic equation <math>ax^2 - 2ax + b = 0</math> has two real solutions. What is the average of these two solutions? ...formulas, the sum of the roots is <math>2</math>, therefore their average is <math>1\Rightarrow \boxed{A}</math>.
  1 KB (218 words) - 16:59, 13 April 2024
• 2008 AMC 10B Problems/Problem 10
  ...<math>C</math> is the [[midpoint]] of the minor arc <math>AB</math>. What is the length of the line segment <math>AC</math>? ...h>\overline{AB}</math> and <math>\overline{OC}</math> (then <math>D</math> is the midpoint of <math>\overline{AB}</math>). <math>OA=OB=5</math>, since th
  1 KB (247 words) - 12:36, 7 June 2021
• 2008 AMC 10B Problems/Problem 11
  Suppose that <math>(u_n)</math> is a [[sequence]] of real numbers satifying <math>u_{n+2}=2u_{n+1}+u_n</math>, and that <math>u_3=9</math> and <math>u_6=128</math>. What is <math>u_5</math>?
  913 bytes (135 words) - 19:42, 24 October 2022
• 2008 AMC 10B Problems/Problem 13
  ...an of the first <math>n</math> terms of a sequence is <math>n</math>. What is the <math>2008^{\text{th}}</math> term of the sequence? ...08^2</math>. Hence, the <math>2008^{\text{th}}</math> term of the sequence is <math>2008^2-2007^2=(2008+2007)(2008-2007)=4015\Rightarrow \boxed{\textbf{(
  2 KB (387 words) - 05:10, 1 October 2023
• 2008 AMC 10B Problems/Problem 14
  ...</math> and <math>\angle AOB=30^\circ</math>. Suppose that <math>OA</math> is rotated <math>90^\circ</math> counterclockwise about <math>O</math>. What a \mathrm{(A)}\ \left( - \frac {10}{3}\sqrt {3},5\right)
  3 KB (444 words) - 04:45, 21 August 2023
• 2008 AMC 10B Problems/Problem 15
  ...ht side of the equation is odd. <math>2b</math> is even. <math>2b+1</math> is odd. So <math>a=1,3,5,7,9,11,13</math>, but if <math>a=1</math>, then <math>b=0</math>. Thus <m
  1,016 bytes (166 words) - 11:39, 8 October 2023
• 2008 AMC 10B Problems/Problem 16
  ...t the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.) ...\qquad \mathrm{(D)}\ {{{\frac{5} {8}}}} \qquad \mathrm{(E)}\ {{{\frac{2} {3}}}}</math>
  2 KB (357 words) - 22:58, 4 October 2023
• 2008 AMC 10B Problems/Problem 17
  ...rk. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter appr ...ach of these is <math>(0.7)(0.3)^2=0.063</math>. Thus, the answer is <math>3\cdot0.063=\boxed{\mathrm{(B)}\ {{{0.189}}}}</math>
  1 KB (202 words) - 12:42, 30 January 2024
• 2008 AMC 10B Problems/Problem 19
  ...ide. The tank is filled with water to a depth of <math>2</math> feet. What is the volume of water, in cubic feet? \mathrm{(B)}\ 24\pi - 24 \sqrt {3}
  2 KB (373 words) - 13:21, 7 June 2021
• 2008 AMC 10B Problems/Problem 20
  ..., <math>4</math>, <math>5</math>, <math>6</math>, and <math>8</math>. What is the probability that the sum of the top two numbers will be <math>5</math>, ...\ 5/18\qquad\mathrm{(B)}\ 7/18\qquad\mathrm{(C)}\ 11/18\qquad\mathrm{(D)}\ 3/4\qquad\mathrm{(E)}\ 8/9</math>
  4 KB (572 words) - 17:44, 14 June 2023
• 2008 AMC 10B Problems/Problem 21
  ...ouples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arran ...ssible arrangements for the five women. The answer is <math>10\cdot 4\cdot 3\cdot 2\cdot 1\cdot 2 = \boxed{(\text{C}) 480}</math>.
  2 KB (262 words) - 22:37, 6 November 2021
• 2008 AMC 10B Problems/Problem 22
  ...wo white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? ...\ 1/12\qquad\mathrm{(B)}\ 1/10\qquad\mathrm{(C)}\ 1/6\qquad\mathrm{(D)}\ 1/3\qquad\mathrm{(E)}\ 1/2</math>
  4 KB (663 words) - 13:49, 7 June 2021
• 2008 AMC 10B Problems/Problem 23
  <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> ...the painted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are <math>a-2</math> by <math>b-2</
  2 KB (247 words) - 23:47, 17 October 2023
• 2008 AMC 10B Problems/Problem 24
  ...angle ABC = 70^\circ</math> and <math>m\angle BCD = 170^\circ</math>. What is the degree measure of <math>\angle BAD</math>? ...eir intersection point <math>Y</math>. Note that triangle <math>BCD</math> is an isosceles triangle so angles <math>CDB</math> and <math>CBD</math> are e
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• 2003 IMO Problems/Problem 2
  <cmath> \frac{a^2}{2ab^2-b^3+1} </cmath> is a positive integer.
  2 KB (397 words) - 00:23, 16 December 2017
• 2008 AMC 10A Problems/Problem 2
  ...gth to its width is <math>2:1</math>. What percent of the rectangle's area is inside the square? ...t <math>1</math>. So the percent of the rectangle's area inside the square is <math>\frac{1}{8} \times 100 = 12.5 \longrightarrow \fbox{A}</math>
  954 bytes (160 words) - 17:20, 4 June 2021
• 2008 AMC 10A Problems/Problem 3
  ...angle=1+2=3</math> and <math>\langle 12 \rangle =1+2+3+4+6=16</math>. What is <math>\langle\langle\langle 6\rangle\rangle\rangle</math>? Since <math>6</math> is a perfect number, any such operation where <math>n=6</math> will yield <mat
  969 bytes (143 words) - 15:12, 1 July 2021
• 2008 AMC 10A Problems/Problem 10
  ...math>S_2</math> to construct an even smaller square <math>S_3</math>. What is the area of <math>S_3</math>? ...\ \frac{1}{2}\qquad\mathrm{(B)}\ 1\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ 4</math>
  1 KB (203 words) - 15:15, 1 July 2021
• 2008 AMC 10A Problems/Problem 6
  ...hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the <math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 5\qquad\mathrm{(D)}\ 6\qquad\mathrm
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• 2008 AMC 10A Problems/Problem 15
  ...th>I</math>, and set Han's speed as <math>H</math>. Therefore, Jan's speed is <math>H+5.</math> We get the following equation for how much Han is ahead of Ian:
  6 KB (1,033 words) - 15:19, 1 July 2021
• 2008 AMC 10A Problems/Problem 17
  ...ontaining all points that are outside the triangle but not more than <math>3</math> units from a point of the triangle? ...uad\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\\\mathrm{(E)}\ 9\left(\sqrt{3}+1\right)^2\pi</math>
  2 KB (348 words) - 19:59, 4 June 2021
• 2008 AMC 10A Problems/Problem 18
  ...riangle]] has [[perimeter]] <math>32</math> and area <math>20</math>. What is the length of its [[hypotenuse]]? ...he hypotenuse is <math>\sqrt{a^2+b^2}</math>, and the area of the triangle is <math>\frac 12 ab</math>. So we have the two equations
  8 KB (1,339 words) - 14:15, 1 August 2022
• 2008 AMC 10A Problems/Problem 19
  ...ise about the point <math>S</math> moved to after the first rotation. What is the length of the path traveled by point <math>P</math>? ...ad\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi</math>
  2 KB (321 words) - 17:52, 7 November 2021
• 2008 AMC 10A Problems/Problem 21
  ...site edges not containing <math>A</math> or <math>C</math>, as shown. What is the area of [[quadrilateral]] <math>ABCD</math>? ...hrm{(C)}\ \sqrt{2}\qquad\mathrm{(D)}\ \frac{5}{8}\qquad\mathrm{(E)}\ \frac{3}{4}</math>
  2 KB (300 words) - 19:42, 4 June 2021
• 2008 AMC 10A Problems/Problem 23
  ...ath>S=\lbrace a,b,c,d,e\rbrace</math> are to be chosen so that their union is <math>S</math> and their intersection contains exactly two elements. In how ...th of those subsets contain the two chosen elements, so their intersection is the two chosen elements). Unfortunately, we have over-counted (Take for ex
  4 KB (603 words) - 14:18, 10 November 2022
• 2001 Iran NMO (Round 3) Problems/Problem 5
  pair A=(1,2.7), B=(0,0), C=(3,0); pair S=IntersectionPoint(N--(N+3(I-N)),O,1);
  3 KB (437 words) - 15:47, 27 April 2008
• 2008 USAMO Problems/Problem 1
  ...all strictly greater than 1, such that <math>k_0 k_1 \dotsm k_n -1</math> is the product of two consecutive integers. for some integer <math>a</math>. Let <math>k_n = a^2 + 3a+3</math>. Evidently, <math>k_n >1</math>. Also,
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 USAMO Problems/Problem 2
  Without Loss of Generality, assume <math>AB >AC</math>. It is sufficient to prove that <math>\angle OFA = 90^{\circ}</math>, as this woul Here, we used that <math>BM=MC</math>, as <math>M</math> is the midpoint of <math>BC</math>. Now, since <math>EC =EA</math> and <math>B
  20 KB (3,565 words) - 11:54, 1 May 2024
• 2008 USAMO Problems/Problem 4
  ...h>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles tr ...angle in <math>Q</math>, from which it follows that the isosceles triangle is <math>\triangle P_aP_{a+k/2}\,P_{a+k}</math>, and so <math>2|k</math>. Repe
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2008 USAMO Problems/Problem 3
  ...>n</math> paths (a partition of <math>S_n</math> into <math>m</math> paths is a set <math>\mathcal{P}</math> of <math>m</math> nonempty paths such that e ...-1,1)--(0,1)--(1,1)--(1,0),linewidth(0.7)); draw((1,0)--(1,-1),linetype("3 3")+linewidth(0.7)); draw((1,-1)--(1,-2)--(2,-2),linewidth(0.7));
  9 KB (1,585 words) - 01:00, 14 August 2014
• 2008 USAMO Problems/Problem 6
  ...mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <math>k</math>). .../math> be an [[indicator function]] with <math>f(v) = 0</math> if a vertex is in the first [[partition]] and <math>f(v) = 1</math> otherwise (this corres
  13 KB (2,414 words) - 14:37, 11 July 2016
• 2008 USAMO Problems
  ...h>, all strictly greater than 1, such that <math>k_0k_1\cdots k_n-1</math> is the product of two consecutive integers. ===Problem 3===
  4 KB (674 words) - 21:48, 12 August 2014
• Normal subgroup
  ...ively. The [[quotient group]] of <math>{\rm G}</math> under this relation is often denoted <math>{\rm G/H}</math> (said, "<math>{\rm G}</math> mod <math An equivalent definition of normal subgroups is this-
  15 KB (2,840 words) - 12:22, 9 April 2019
• Free magma
  A '''free magma''' is [[magma]] structure that is as general as possible&mdash;a magma generated from an initial set with no The free magma generated from a [[set]] <math>X</math> is constructed as follows.
  4 KB (887 words) - 13:19, 6 July 2016
• Mathleague.org
  ...2<br><u>MS Harder</u>: 2-3<br><u>HS Easier</u>: 0.5-3<br><u>HS Harder</u>: 3-5}} ...imple arithmetic problems and/or word problems. Every ten questions, there is an estimation question which requires the estimation answer to be within <m
  5 KB (801 words) - 12:47, 23 September 2023
• Alternating group
  ...e [[symmetric group]]; and for <math>n=3</math> or <math>n\ge 5</math>, it is in fact a [[simple group]]. <math>A_n</math> is also the group of [[determinant]]-preserving permutations of the rows of an
  776 bytes (127 words) - 11:47, 4 March 2022
• Mock ARML
  A '''Mock ARML''' is a test written by an AoPS member, designed to mimic the format of the indiv Starting from 2015, trumpeter is releasing a Mock ARML every year some time from March to May of the year (s
  1 KB (153 words) - 11:44, 16 June 2019
• 2008 Mock ARML 1 Problems/Problem 1
  ...range does not include negative numbers, it follows that the negative root is extraneous, and thus we have <math>x = \boxed{\frac{1+\sqrt{17}}{2}}</math> <math>(x^2-x-4)(x^2+x-3)</math>
  1 KB (196 words) - 21:49, 4 December 2016
• 2008 Mock ARML 1 Problems/Problem 2
  ...ue of the difference between any two consecutive digits of <math>n</math> is at least <math>7</math> . Compute the number of <math>8</math>-digit yo-yos ...<math>7</math> if <math>n</math> is even. By symmetry, the desired answer is <math>2(a_8 + b_8 + c_8) - a_8</math>, to exclude the integers with leftmos
  2 KB (326 words) - 19:28, 10 March 2015
• 2008 Mock ARML 1 Problems/Problem 3
  D(MP("A",v(0))--MP("B",v(1),N)--MP("C",v(2),N)--MP("D",v(3),SW)--MP("E",v(4))--MP("F",v(5))--cycle); D(v(0)--v(3));D(v(1)--v(5));D(v(1)--v(3));D(v(2)--v(4));
  3 KB (425 words) - 22:32, 5 December 2020
• 2008 Mock ARML 1 Problems/Problem 6
  ...>BC</math>. <math>P</math> is on <math>MN</math> such that <math>N</math> is between <math>M</math> and <math>P</math>, and <math>m\angle MAN = m\angle pair A=(0,2), D=(0,0), C=(2,0), B=(2,2), M=(C+D)/2, N=(B+C)/2, P=8/3*(N-M)+M;
  2 KB (283 words) - 22:51, 13 April 2015
• 2008 Mock ARML 1 Problems/Problem 8
  ...math>, which is also the area, so the answer is <math>\boxed{\frac {\sqrt {3}}{2}}</math>. ...tem reduces and we find that the desired sum is <math>\boxed{\frac {\sqrt {3}}{2}}</math>.
  2 KB (267 words) - 20:17, 4 December 2016
• 2008 Mock ARML 2 Problems
  <math>ABCD</math> is a convex quadrilateral such that <math>|AB| = 5</math>, <math>|BC| = 17</ma ...(a proper divisor of <math>n</math> is a positive integer that divides but is not equal to <math>n</math>).
  4 KB (582 words) - 21:57, 8 May 2019
• 2003 AMC 12A Problems/Problem 17
  ...nd center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance from <math>P</math> to <math>\overline{AD}</math>? ...ac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math>
  6 KB (1,026 words) - 22:35, 29 March 2023
• 2003 AMC 12A Problems/Problem 21
  <cmath>P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e</cmath> has five distinct <math>x</math>-intercepts, one of which is at <math>(0,0)</math>. Which of the following [[coefficient]]s cannot be ze
  1 KB (210 words) - 21:12, 29 September 2021
• Burnside's Lemma
  ...a''' is a [[combinatorics |combinatorial]] result in [[group theory]] that is useful for counting the [[orbit]]s of a [[set]] on which a [[group]] [[grou ...e it is also called the '''Cauchy-Frobenius Lemma''', or '''the lemma that is not Burnside's'''. The lemma was (mistakenly) attributed to Burnside becau
  5 KB (757 words) - 18:11, 23 October 2023
• Sylow Theorems
  ...m.''' In every finite group, the number of Sylow <math>p</math>-subgroups is equivalent to 1 (mod <math>p</math>). Evidently, a subset of <math>A \times T</math> is stable under this action if and only if <math>A= G</math>. Thus the fixed
  11 KB (2,071 words) - 12:25, 9 April 2019
• 2003 AMC 12B Problems/Problem 18
  ...alue of <math>x</math> has a prime factorization <math>a^cb^d.</math> What is <math>a + b + c + d?</math> ...\equiv 0 \pmod{5}</math>. Hence, we can substitute the quantity <math>11^{3} \cdot 7^{2}</math> into <math>y</math>. Doing so gets us:
  6 KB (914 words) - 11:07, 7 September 2023
• 2006 Alabama ARML TST Problems/Problem 3
  River draws four cards from a standard 52 card deck of playing cards. Exactly 3 of them are 2’s. Find the probability River drew exactly one spade and on The probability is equal to the number of successful outcomes(<math>S</math>) divided by the n
  1 KB (223 words) - 19:43, 12 April 2012
• 1995 AHSME Problems/Problem 27
  Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacen & & 3 & & 4 & & 4 & & 3 & & \\
  5 KB (682 words) - 09:45, 18 February 2022
• KGS math club
  The meaning of this page is to collect the problems posed there and save hints and solution suggestions ...do the wiki table formatting more elegantly, be my guest; after all, this is wikiiii.
  22 KB (3,358 words) - 15:17, 18 July 2017
• KGS math club/solution 2 1
  Lemma 1: 2007 = 223 * 3^2 (trivial) For n = 3: n(3) = 6 * 20 - 12 * 4 + 8 = 80 = 2^3 * (3^2 + 1)
  2 KB (393 words) - 12:54, 11 August 2023
• 1995 AHSME Problems/Problem 18
  ...y, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is ...\ \sqrt{3} } \qquad \mathrm{(D) \ 2 } \qquad \mathrm{(E) \ \frac{4}{\sqrt{3}} } </math>
  1 KB (167 words) - 13:59, 5 July 2013
• How should I prepare?
  ...ld also try to strengthen in the areas you are not as good at. This guide is intended to help you get started. ...osttests to see if the books are on the right level for you. '''Alcumus''' is a good resource even if you do not own any of the AoPS books.
  13 KB (1,926 words) - 11:22, 30 November 2023
• 1999 AHSME Problems/Problem 25
  There are unique integers <math>a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}</math> such that <cmath>\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7
  4 KB (595 words) - 01:43, 4 October 2023
• 1999 AHSME Problems/Problem 19
  ...Among all such triangles, the smallest possible value of <math>AC</math> is A=(5,12); B=origin; C=(10,0); D=(8.52071005917,3.55029585799);
  1 KB (173 words) - 14:35, 5 July 2013
• 1999 AHSME Problems/Problem 29
  ...probability that <math>P</math> lies inside one of the five small spheres is closest to ...qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4 </math>
  3 KB (522 words) - 11:39, 3 October 2023
• 2008 IMO Problems
  === Problem 3 === [[2008 IMO Problems/Problem 3 | Solution]]
  4 KB (613 words) - 00:30, 18 February 2021
• 1999 AHSME Problems/Problem 27
  ...math>4 \sin B + 3 \cos A = 1</math>. Then <math>\angle C</math> in degrees is Thus <math>\frac 12 = \sin A \cos B + \sin B \cos A</math>. This is the sine addition identity, so <math>\frac 12 = \sin (A + B) = \sin (180 -
  2 KB (296 words) - 10:21, 3 October 2023
• 1998 AHSME Problems
  ...thout rotating or reflecting, in position as below. Which of the rectangle is the top leftmost one? draw((0,5)--(0,7)--(3,7)--(3,5)--cycle);
  15 KB (2,222 words) - 10:40, 11 August 2020
• Legendre's Formula
  ...in the [[prime factorization]] of <math>n!</math> and <math>S_p(n)</math> is the [[sum]] of the [[digit]]s of <math>n</math> when written in [[base]] <m ...ht\rfloor+\left\lfloor\frac{27}{2^2}\right\rfloor+\left\lfloor\frac {27}{2^3}\right\rfloor+\left\lfloor\frac{27}{2^4}\right\rfloor\\
  4 KB (699 words) - 17:55, 5 August 2023
• 1998 AHSME Problems/Problem 28
  ...angle <math>CAD</math> is twice angle <math>DAB</math>. If <math>AC/AD = 2/3</math>, then <math>CD/BD = m/n</math>, where <math>m</math> and <math>n</ma ...ath>. Then, it is given that <math>\cos 2\theta = \frac{AC}{AD} = \frac{2}{3}</math> and
  4 KB (662 words) - 00:51, 3 October 2023
• 1998 AHSME Problems/Problem 8
  A square with sides of length <math>1</math> is divided into two congruent trapezoids and a pentagon, which have equal area Then <math>2[I]+2[II] = [I]+3[II] \Longrightarrow [I]=[II]</math>. Let the shorter side of <math>I</math>
  2 KB (347 words) - 11:28, 2 December 2019
• 1998 AHSME Problems/Problem 4
  ...math> to mean <math>\frac {a+b}c</math>, where <math>c \neq 0</math>. What is the value of <center><math>\left[[60,30,90],[2,1,3],[10,5,15]\right]?</math></center>
  714 bytes (95 words) - 14:28, 5 July 2013
• 1995 AHSME Problems/Problem 13
  The addition below is incorrect. The display can be made correct by changing one digit <math>d</m + & 8 & 2 & 9 & 4 & 3 & 0 \\
  1 KB (223 words) - 13:59, 5 July 2013
• 2001 IMO Shortlist Problems/C1
  ...h> be a sequence of positive integers. Let <math>m</math> be the number of 3-element subsequences <math>(a_i, a_j, a_k)</math> with <math>1 \le i < j < ...ue if and only if they belong to the same block. We can do this because it is unstrictly increasing. For example, if the sequence consists of 1000 ones,
  4 KB (703 words) - 12:45, 27 November 2017
• 2003 USAMO Problems/Problem 4
  ...h>MC = MF</math> if and only if <math>\angle GCD = \angle FDA</math>, that is, <math>\angle FDA + \angle CGF = 180^\circ</math>. Because quadrilateral <math>ABED</math> is cyclic, <math>\angle FDA = \angle ABE</math>. It follows that <math>MC = MF
  5 KB (820 words) - 02:39, 10 January 2023
• 2003 USAMO Problems/Problem 5
  Since all terms are homogeneous, we may assume WLOG that <math>a + b + c = 3</math>. ...- a)^2} = \sum \frac {a^2 + 6a + 9}{3a^2 - 6a + 9} = \sum \left(\frac {1}{3} + \frac {8a + 6}{3a^2 - 6a + 9}\right)</math>.
  6 KB (1,063 words) - 02:36, 9 August 2023
• 2003 USAMO Problems/Problem 6
  ...gular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and repla ...ginal numbers are <math>a,b,c,d,e,f</math>. Since <math>a+b+c+d+e+f</math> is odd, either <math>a+c+e</math> or <math>b+d+f</math> must be odd. WLOG let
  5 KB (739 words) - 13:39, 4 July 2013
• 2000 USAMO Problems/Problem 2
  ...th>5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},</cmath> where <math>r</math> is the inradius and <math>P, Q, R</math> are the points of tangency of the inc
  4 KB (703 words) - 18:40, 3 January 2019
• 2001 IMO Shortlist Problems/A5
  We claim that there is only one such sequence: <math>a_1=2, a_2=5, a_3=56, a_4=56\times 1400</math ...k that <math>(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)</math> for <math>k=1,2,3</math>.
  3 KB (477 words) - 17:42, 8 August 2019
• 2001 IMO Shortlist Problems/A6
  ...} + \frac {b}{\sqrt {b^2 + kca}} + \frac {c}{\sqrt {c^2 + kab}} \geq \frac{3}{\sqrt{1+k}}.</math></center> ...the function <math>f(x)=\frac{1}{\sqrt{x}}</math>. Note that this function is convex and monotonically decreasing which implies that if <math>a > b</math
  3 KB (453 words) - 00:12, 28 March 2021
• 2001 IMO Shortlist Problems/C3
  ...that there are two or fewer people at the party whose departure leaves no 3-clique remaining. ...e exists only one 3-clique, remove anyone in that clique. (If there are no 3-cliques, we are done!) Otherwise, consider the following cases:
  3 KB (479 words) - 15:01, 23 November 2017
• 2001 IMO Shortlist Problems/C4
  ...ee nonnegative integers <math>\{x,y,z\}</math> with <math>x < y < z</math> is called <i>historic</i> if <math>\{z - y,y - x\} = \{1776,2001\}</math>. Sh ..., y, or z position in its historic set (e.g. in <math>\{1,2,4\}</math>, 4 is in column z, 2 in column y, and 1 in column x).
  2 KB (494 words) - 17:34, 5 July 2020
• 2001 IMO Shortlist Problems/G6
  ...PCE</math>, <math>PAF</math> are all equal. Prove that each of these areas is equal to the area of triangle <math>ABC</math> itself. ...(0,1,0)</math>, <math>C</math> is <math>(0,0,1)</math>, and <math>P</math> is <math>(p,q,r)</math>, with <math>p+q+r=1</math>.
  3 KB (565 words) - 23:25, 2 April 2012
• 2001 IMO Shortlist Problems/N1
  Prove that there is no positive integer <math>n</math> such that, for <math>k = 1,2,\ldots,9</m Suppose that there is such a number <math>n</math>. Let <math>a</math> be the number of digits of
  3 KB (440 words) - 08:14, 1 April 2022
• 2001 IMO Shortlist Problems/N2
  ...th>, we discard the second inequality and have that <math>3+2\sqrt2</math> is a lower bound for <math>r</math> ...nominators and keep the same value of <math>r</math>. Thus, <math>\boxed{m=3+2\sqrt2}</math>.
  2 KB (267 words) - 14:38, 1 September 2017
• 2001 IMO Shortlist Problems/N4
  ...at neither <math>a^{p - 1} - 1</math> nor <math>(a + 1)^{p - 1} - 1</math> is divisible by <math>p^2</math>. ...math>(\mathbb{Z}/p^2\mathbb{Z})^{\times}</math> is cyclic, we know that it is isomorphic to <math>(\mathbb{Z}/(p^2-p)\mathbb{Z})^{+}</math>.
  3 KB (454 words) - 22:27, 16 October 2020
• 2001 IMO Shortlist Problems/N6
  Is it possible to find 100 positive integers not exceeding 25,000, such that a The answer is yes.
  2 KB (443 words) - 13:08, 17 August 2011
• 2008 IMO Problems/Problem 4
  ...nctions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that ...he only possible solution is <math>f(1)=1</math> since <math>f(1)=0</math> is impossible.
  4 KB (661 words) - 01:10, 19 November 2023
• Millennium Problems
  ..."setting the agenda for the twentieth century". (Devlin 2003, pp. 2&ndash;3) These problems he believed to be the most significant and important unsolv ...y hard to isolate one problem that captures the program." (Devlin 2003, p. 3)
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1984 IMO Problems/Problem 4
  ...he circle on diameter <math>AB</math>. Prove that the line <math>AB</math> is tangent to the circle on diameter <math>CD</math> if and only if the lines ...hat the length of the perpendicular from <math>M</math> to <math>CD</math> is <math>\frac{1}{2}AB</math>. Let the foot of the perpendicular from <math>C<
  4 KB (750 words) - 23:49, 29 January 2021
• Carnot's Theorem
  ...to the sides (i.e., signed lengths of the pedal lines from <math>O</math>) is: c=(1,3);
  4 KB (723 words) - 01:45, 18 February 2021
• 1988 IMO Problems/Problem 4
  is a union of disjoint intervals, the sum of whose length is <math>1988</math>. ...ve that the solution set to the inequality <math>f(x)\ge\frac{5}{4}</math> is the union of the intervals <math>(n,r_n]</math> (since if <math>f(x)<\frac{
  3 KB (518 words) - 11:36, 30 January 2021
• 1989 IMO Problems/Problem 4
  ...>, the intersection of circle <math>A</math> with the line <math>AB</math> is the same as the intersection of circle <math>B</math> with the line <math>A ...<math>P</math> to <math>CD</math>, we have that the circle <math>P</math> is tangent to <math>CD</math>. From the conditions <math>AP=h+AD</math> and <
  4 KB (771 words) - 11:57, 30 January 2021
• Farey sequence
  A '''Farey sequence''' of order <math>n</math> is the sequence of all completely reduced fractions between 0 and 1 where, whe <math>F_3=\{0/1, 1/3, 1/2, 2/3, 1/1\}</math>
  1 KB (199 words) - 04:16, 30 January 2021
• 1989 AHSME Problems/Problem 29
  What is the value of the sum <math>S=\sum_{k=0}^{49}(-1)^k\binom{99}{2k}=\binom{99} ...h> <math>\binom{99}{0}i^0+\binom{99}{1}i^1+\binom{99}{2}i^2+\binom{99}{3}i^3+\binom{99}{4}i^4+\cdots +\binom{99}{98}i^{98}</math>.
  1 KB (173 words) - 19:52, 30 December 2020
• 1990 AHSME Problems/Problem 29
  ...</math> has the property that none of its members is 3 times another. What is the largest number of members such a subset can have? ...s long as no integer between <math>11</math> and <math>33</math> inclusive is within the set. This provides a total of <math>100 - 34 + 1</math> = 67 sol
  2 KB (285 words) - 19:25, 25 September 2020
• 1990 AHSME Problems/Problem 26
  For <math>i\in\{1,2,3,\ldots,10\},</math> suppose Person <math>i</math> picks the number <math>a_ ...sets of equations are independent. The set that involves <math>a_6</math> is
  3 KB (402 words) - 23:17, 23 September 2023
• 1995 IMO Problems/Problem 1
  ...ath>BD</math> is line <math>DN</math>. Since the pairwise radical axes of 3 circles are concurrent, we have <math>AM,DN,XY</math> are concurrent as des .... Thus, <math>N'</math> and <math>N''</math> are distinct, as none of them is <math>N</math>. Hence, by Power of a Point, <cmath>ZM * ZA = ZP * ZT = ZN''
  5 KB (847 words) - 19:03, 12 October 2021
• 2008 IMO Problems/Problem 5
  ...mps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). ...ff, but where none of the lamps <math>n + 1</math> through <math>2n</math> is ever switched on.
  4 KB (677 words) - 01:10, 19 November 2023
• 2008 IMO Problems/Problem 2
  '''(ii)''' Prove that equality is achieved for infinitely many triples of rational numbers <math>x</math>, <m ...ath>\mathbb{Q}/\{1\}</math> to <math>\mathbb{Q}/\{-1\}</math>, the problem is equivalent to showing that
  3 KB (593 words) - 01:09, 19 November 2023
• 2000 USAMO Problems
  ...th>5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},</cmath> where <math>r</math> is the inradius and <math>P, Q, R</math> are the points of tangency of the inc
  3 KB (495 words) - 19:02, 18 April 2014
• 2000 USAMO Problems/Problem 4
  We claim that <math>n = 1999</math> is the smallest such number. For <math>n \le 1998</math>, we can simply color ...i == 0 || j == 9) && !(j-i == 9)) fill(shift(i,j)*unitsquare,rgb(0.3,0.3,0.3));
  2 KB (382 words) - 13:37, 4 July 2013
• 2000 USAMO Problems/Problem 5
  ...s through <math>A_k</math> and <math>A_{k + 1},</math> where <math>A_{n + 3} = A_{n}</math> for all <math>n \ge 1</math>. Prove that <math>\omega_7 = \ <math>O</math> is the intersection of the perpendicular bisectors of <math>\overline{A_1A_2},
  3 KB (609 words) - 09:52, 20 July 2016
• 1986 IMO Problems/Problem 2
  ...hat if <math>P_{1986}=P_0</math>, then the triangle <math>A_1A_2A_3</math> is equilateral. ...math>A_3</math> the rotation sends <math>P</math> to <math>e^{\frac{4i\pi}{3}}(P-a)+a</math>. Thus the result of all three rotations sends <math>P</mat
  2 KB (345 words) - 00:04, 30 January 2021
• 2001 USAMO Problems/Problem 1
  ...ne, with justification, the smallest integer <math>n</math> for which this is possible. We claim that <math>n=23</math> is the minimum. Consider the following construction (replacing colors with num
  5 KB (841 words) - 17:19, 5 May 2022
• 2008 iTest Problems
  ...ates. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are <math>12</math> years old. (Michael is still young enough to get the children's rate), and family memberships
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 94
  Find the largest [[prime]] number less than <math>2008</math> that is a divisor of some integer in the infinite ...r, \left\lfloor \frac{2008^2}{2} \right\rfloor, \left\lfloor \frac{2008^3}{3}\right\rfloor, \left\lfloor \frac{2008^4}{4} \right\rfloor, \cdots</cmath>
  4 KB (571 words) - 21:21, 22 November 2018
• 2008 iTest Problems/Problem 93
  <center><math>{1, 2, 3, . . . , 4n}</math></center> element which is the [[arithmetic mean]] of all the elements in that subset?
  1 KB (232 words) - 21:20, 22 November 2018
• 2008 iTest Problems/Problem 99
  ...difference of the areas of <math>P_{10}</math> and <math>P_{\infty}</math> is written as a fraction <math>\frac{x}{y}</math> in lowest terms, calculate t ...our [[trapezoid]] be <math>P_1 = ABCD</math> (and <math>[ABCD] = \frac{12(3+13)}{2} = 96</math>); then [[without loss of generality]] construct diagona
  4 KB (685 words) - 14:39, 7 October 2017
• 2008 iTest Problems/Problem 80
  ...the polynomial remainder when <math>p(x)</math> is divided by <math>x^4+x^3+2x^2+x+1</math>. Find the remainder when <math>|r(2008)|</math> is divided by <math>1000</math>.
  3 KB (560 words) - 19:49, 23 November 2018
• 2005 PMWC Problems/Problem T7
  ...that Skipper can reach? Calculate an approximate answer by using <math>\pi=3.14</math> or <math>\pi=22/7</math>. ...6}</math>. Adding, we get <math>\frac{8}{3}\pi+\frac{\pi}{6}+\frac{\pi}{6}=3\pi</math>. Then approximate.
  854 bytes (145 words) - 23:28, 21 October 2022
• Asymptote: Macintosh
  ...systems as well as other Unix systems. The third was tested on Mac OS 11.3 "Big Sur"''' .../project/showfiles.php?group_id=120000 here] (at the time of this edit, it is version 2.86) by selecting the latest version and downloading the asymptote
  5 KB (732 words) - 00:47, 13 December 2023
• 1989 IMO Problems/Problem 2
  <math>ABC</math> is a triangle, the bisector of angle <math>A</math> meets the circumcircle of ...ath> and similarly for the other triangles. Thus, the area of the hexagon is equal to <math>[ABC]+\sum\frac{1}{4}a^2\left(\frac{2r}{b+c-a}\right)</math>
  3 KB (568 words) - 11:50, 30 January 2021
• 2007 iTest Problems/Problem 9
  ...>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy the ...>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)
  966 bytes (152 words) - 20:24, 10 January 2019
• 1991 IMO Problems/Problem 4
  ...re edges, the greatest common divisor of the integers labeling those edges is equal to 1. ...s from the set <math>V</math> if their exists an edge sorrounding it which is labelled. Additionally, we shall use up the numbers <math>1,2,...,k</math>
  4 KB (668 words) - 17:45, 30 January 2021
• Fermat's Two Squares Theorem
  ...2</math> or <math>p \equiv 1 \pmod{4}</math>; and that this representation is unique. The theorem is a starting point in the investigation of [[binary quadratic forms]]. Histo
  4 KB (612 words) - 12:10, 30 May 2019
• 1995 AHSME Problems/Problem 15
  ...e circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point <math> \mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 } </math>
  1 KB (183 words) - 21:54, 2 August 2016
• 1995 AHSME Problems/Problem 17
  ...ath> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3)));
  1 KB (201 words) - 16:29, 12 March 2024
• 1995 AHSME Problems/Problem 30
  ...bisects that diagonal. The number of unit cubes that the plane intersects is ...ac 32,\frac 32\right)</math>. The plane that passes through this point and is orthogonal to the diagonal has the equation <math>x+y+z=\frac 92</math>.
  5 KB (828 words) - 05:52, 30 September 2023
• 2007 AMC 12B Problems/Problem 25
  ...<math>\triangle ABC</math> is parallel to <math>\overline{DE}</math>. What is the area of <math>\triangle BDE</math>? <math>\mathrm {(A)} \sqrt{2}\qquad \mathrm {(B)} \sqrt{3}\qquad \mathrm {(C)} 2\qquad \mathrm {(D)} \sqrt{5}\qquad \mathrm {(E)} \sq
  3 KB (470 words) - 19:46, 17 July 2023
• 1973 USAMO Problems/Problem 4
  <center><math>x+y+z=3</math>, <math>x^2+y^2+z^2=3</math>,
  5 KB (888 words) - 08:18, 22 April 2024
• 1976 USAMO Problems/Problem 3
  ...c_1</math>. Thus <math>a_1^2+b_1^2+c_1^2=4a_1^2b_1^2</math>. Since the LHS is divisible by four, all the variables are divisible by 4, and we must do thi ...}</math>. But for this to be true, <math>c^2\equiv 3\bmod{4}</math>, which is an impossibility. Thus there are no non-zero solutions when <math>a^2\equiv
  1 KB (214 words) - 00:00, 10 October 2020
• 1976 USAMO Problems/Problem 1
  fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); draw((3,1)--(3,3)--(7,3)--(7,1)--cycle,black+1);
  4 KB (660 words) - 19:21, 23 November 2016
• 1973 USAMO Problems/Problem 2
  .../math>, <math>X_1=1</math>, <math>X_{n+1}=X_n+2X_{n-1}</math> <math>(n=1,2,3,\dots),</math></center> ...math>, <math>Y_1=7</math>, <math>Y_{n+1}=2Y_n+3Y_{n-1}</math> <math>(n=1,2,3,\dots)</math>.</center>
  2 KB (342 words) - 18:54, 3 July 2013
• 2002 AMC 12A Problems/Problem 3
  ...C 12A Problems|2002 AMC 12A #3]] and [[2002 AMC 10A Problems|2002 AMC 10A #3]]}} If the order in which the exponentiations are performed is changed, how many other values are possible?
  2 KB (224 words) - 12:28, 8 November 2021
• 2002 AMC 12A Problems/Problem 6
  therefore the answer is <math>\boxed{\textbf{(E) } \text{infinitely many}}</math>. ...many numbers <math>m</math> that can satisfy the inequality. So the answer is <math>\boxed{\textbf{(E) } \text{infinitely many}}</math>.
  2 KB (258 words) - 04:58, 21 July 2022
• 2001 IMO Problems/Problem 4
  ...ons <math>a</math>, <math>b</math> the difference <math>f(a) - f(b)</math> is a multiple of <math>m!</math>. ...Using the fact that <math>f(v_t)\equiv t\pmod{m!}</math>, we find the sum is congruent to <math>0+1+\cdots+m!-1=\frac{m!(m!-1)}{2}</math>.
  2 KB (416 words) - 20:01, 19 May 2023
• 2007 AMC 12B Problems/Problem 20
  ...h>b</math>, <math>c</math>, and <math>d</math> are positive integers, what is the smallest possible value of <math>a+b+c+d</math>? real a = 3, b = 1, c = 9, d = 3;
  7 KB (1,143 words) - 21:25, 20 December 2020
• 2001 IMO Problems/Problem 5
  ...th> and <math>BY</math> bisects angle <math>B</math>. Angle <math>A</math> is <math>60^{\circ}</math>. <math>AB + BX = AY + YB</math>. Find all possible dotfactor = 3;
  4 KB (692 words) - 15:01, 15 May 2024
• 1998 AJHSME Problems/Problem 19
  ...</math> and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? ...\dfrac{1}{2} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math>
  2 KB (242 words) - 19:53, 31 October 2016
• 2005 AMC 8 Problems
  have divided the number by 2 to get the correct answer. What is the correct ==Problem 3==
  13 KB (1,821 words) - 22:18, 5 December 2023
• Spiral similarity
  ...ane having the common center. The order in which the composition is taken is not important. The transformation is linear and transforms any given object into an object homothetic to given.
  28 KB (4,863 words) - 00:29, 16 December 2023
• 2003 AMC 12A Problems/Problem 9
  ...he line <math>y=x</math>. If <math>(2,3)</math> is in <math>S</math>, what is the smallest number of points in <math>S</math>? ...th>8</math> points satisfy all of the symmetry conditions. Thus the answer is <math>\boxed{\mathrm{(D)}\ 8}</math>.
  876 bytes (140 words) - 20:35, 30 December 2014
• 2003 AMC 12A Problems/Problem 10
  ...on they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share ...cation, it does not matter what order the three come in.) Hence the answer is <math>\boxed{\mathrm{(D)}\ \dfrac{5}{18}}</math>.
  1 KB (213 words) - 10:16, 4 July 2013
• Cantor set
  ...ong the simplest examples of a [[fractal]]. [[Topology|Topologically]], it is a [[closed set]], and also a [[perfect set]]. Despite containing an [[uncou ...th>, dividing the [[interval]] into two intervals of length <math>\frac{1}{3}</math>. Then remove the middle third of the two remaining segments, and r
  3 KB (413 words) - 16:32, 18 June 2020
• 2004 IMO Problems/Problem 4
  (''Hojoo Lee'') Let <math>n \geq 3</math> be an integer. Let <math>t_1, t_2, \dots , t_n</math> be positive re For <math>n=3</math>, suppose (for sake of contradiction) that <math>t_3 = t_2 + t_1 + k<
  2 KB (322 words) - 00:54, 19 November 2023
• 1999 AHSME Problems/Problem 2
  Which of the following statements is false? ...are equal. Thus equilateral triangles are equiangular, and <math>C</math> is true.
  2 KB (259 words) - 14:28, 13 February 2019
• Inequality Introductory Problem 2
  <math>2\sum_{k=1}^{n}a_{k}^{2}\ge 2(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}+a_{n}a_{1})</math> Which is always true.
  464 bytes (82 words) - 15:18, 23 May 2009

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