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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• Base numbers
  ...the number 2746. This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...<math>10^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s.
  4 KB (547 words) - 17:23, 30 December 2020
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe * Evaluate: <math>\int_2^5 x^3 dx</math> and <math>\int_{.2}^{.4} \cos(x) dx</math>. (The next few questi
  11 KB (2,082 words) - 15:23, 2 January 2022
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
  9 KB (1,547 words) - 03:04, 13 January 2021
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2003 AMC 12B Problems
  <cmath>\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?</cmath> ...ext {(C) } \frac{2}{3} \qquad \text {(D) } 1 \qquad \text {(E) } \frac{14}{3}
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AIME II Problems/Problem 15
  ...p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> pair A = (-5, 12), B = (5, 12), C = (0, 0);
  12 KB (2,000 words) - 13:17, 28 December 2020
• 1988 AIME Problems
  ...rder -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned == Problem 3 ==
  6 KB (902 words) - 08:57, 19 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the == Problem 3 ==
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  ...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went == Problem 3 ==
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1986 AIME Problems/Problem 14
  ..., and <math>\frac{15}{\sqrt{10}}</math>. Determine the [[volume]] of <math>P</math>. ...en <math>AC</math> and that corner, which is <math>\frac {wl}{\sqrt {w^2 + l^2}}</math>.
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 9
  ...t]] <math>P</math> is then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segmen pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425));
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1989 AIME Problems/Problem 15
  ...ven that <math>AP=6</math>, <math>BP=9</math>, <math>PD=6</math>, <math>PE=3</math>, and <math>CF=20</math>, find the area of <math>\triangle ABC</math> ...>, and line <math>XP</math> intersects line <math>YZ</math> at point <math>L</math>, then <math>\dfrac{XP}{PL} = \frac{[XPY] + [ZPX]}{[YPZ]}.</math>
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1993 AIME Problems/Problem 12
  ...int, then <math>P_{n + 1}^{}</math> is the midpoint of <math>\overline{P_n L}</math>. Given that <math>P_7 = (14,92)\,</math>, what is <math>k + m\,</m ...the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, then <math>u=2r-p</math> and <math>v=2s-q</math>. So we start with the point they gave us and
  4 KB (611 words) - 13:59, 15 July 2023
• 1995 AIME Problems/Problem 12
  // p = point on line triple lineintersectplan(triple p, triple d, triple q, triple n)
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 3
  ...object reaches <math>(2,2)</math> in six or fewer steps. Given that <math>p</math> can be written in the form <math>m/n,</math> where <math>m</math> an ...math>2!</math> ways to determine the order of the remaining two steps, for a total of <math>12</math> sequences that we have to exclude. This gives <mat
  3 KB (602 words) - 23:15, 16 June 2019
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ...;label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><asy>import three; defaul
  7 KB (1,169 words) - 15:28, 13 May 2024
• 2001 AIME I Problems/Problem 12
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2001 AIME I Problems/Problem 6
  A fair die is rolled four times. The [[probability]] that each of the final t ...n the diagram below, the lowest <math>y</math>-coordinate at each of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> corresponds to t
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME II Problems/Problem 15
  ...h>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. ...ection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
  4 KB (518 words) - 15:01, 31 December 2021
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[congruent (geometry)|congr <!-- labeling: for(int i = 0; i < P.length; ++i) label("" + ((string) i), P[i], N); -->
  8 KB (1,168 words) - 22:48, 19 February 2022
• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...width of a particular rectangle is doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle? <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath>
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ==Problem 3==
  8 KB (1,355 words) - 14:54, 21 August 2020
• P versus NP
  ...me, or whether there are problems in <math>NP</math> that are not in <math>P</math>. Since all modern computers (with the exception of a few quantum computers) are deterministic, non-deterministic algorithms are
  6 KB (1,104 words) - 15:11, 25 October 2017
• Integral
  ...ath>c</math> is a [[real]] [[constant]]. This is because the derivative of a constant is <math>0</math>. *The integral of a function <math>f(x)</math> is written as <math>\int f(x)\,dx</math>, where
  5 KB (909 words) - 14:16, 31 May 2022
• Perpendicular bisector
  ...ch that <math>AB</math> and <math>l</math> are [[perpendicular]] and <math>l</math> passes through the [[midpoint]] of <math>AB</math>. ...hat plane. The [[set]] of lines which are perpendicular bisectors of form a plane which is the plane perpendicularly bisecting <math>AB</math>.
  2 KB (367 words) - 15:20, 1 January 2014
• 2007 iTest Problems
  ...p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin <math>\mathrm{(A)}\, 4</math>
  30 KB (4,794 words) - 23:00, 8 May 2024
• Euler line
  ...nts. In particular, <math>\overline{OGNH}</math> and <math>OG:GN:NH = 2:1:3</math> ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2020 AMC 10A Problems
  {{AMC10 Problems|year=2020|ab=A}} <cmath>x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?</cmath>
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two useful functions tha ...ining points (which can be thought of as paths with length 0) or paths <tt>p</tt>, <tt>q</tt> together with one of the following operators:
  7 KB (1,205 words) - 21:38, 26 March 2024
• Mock AIME 1 Pre 2005 Problems
  == Problem 3 == ...h that <math>AB = BC = 1, CD = 2,</math> and <math>DE = 9</math>. If <math>P</math> can be any point in space, what is the smallest possible value of <m
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems
  ...rcles are mutually externally tangent. Two of the circles have radii <math>3</math> and <math>7</math>. If the area of the triangle formed by connecting [[Mock AIME 3 Pre 2005/Problem 1|Solution]]
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 14
  ...math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. then <math>r_1=\frac{|AD| \times |DP| \times |AP|}{4A_1}=\frac{(3)(4)(6)}{4A_1}=\frac{18}{A_1}</math>
  3 KB (563 words) - 02:05, 25 November 2023
• 2007 AIME I Problems/Problem 10
  In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so th Now consider the 3x3 that is next to the 3 boxes we have filled in. We must put one ball in each row (since there mus
  13 KB (2,328 words) - 00:12, 29 November 2023
• 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
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat ...ss than <math>1000</math> contain at least one <math>2</math> but no <math>3</math>?
  11 KB (1,713 words) - 22:47, 13 July 2023
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• Asymptote: Useful functions
  The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < Label ''L''="", int ''n''=1, real ''radius''=0, real ''space''=0,<br/>
  4 KB (646 words) - 21:18, 26 March 2024
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• 2001 USAMO Problems/Problem 2
  ...t <math>AD_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</mat
  11 KB (2,091 words) - 08:35, 16 November 2017
• 2007 AMC 10A Problems
  {{AMC10 Problems|year=2007|ab=A}} ...ath>25\%</math> discount. Pam buys 5 tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1959 AHSME Problems
  Each edge of a cube is increased by <math>50</math>%. The percent of increase of the surfa <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300
  22 KB (3,345 words) - 20:12, 15 February 2023
• Construction
  ...to mark off segments, draw circles and arcs, and draw straight lines) are a branch of [[geometry]] that rely on the use of basic geometrical [[axiom]]s A '''compass''' is a tool that can draw circles and arcs of circles.
  3 KB (443 words) - 20:52, 28 August 2014
• Isogonal conjugate
  ...nal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...ll</math> and a point <math>O</math> lying on <math>\ell</math> be given. A pair of lines symmetric with respect to <math>\ell</math> and containing t
  54 KB (9,416 words) - 08:40, 18 April 2024
• Shoelace Theorem
  The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a simple [[polygon]] given the [[Cartesian coordinate system|coordinates]] of ...math>, listed in clockwise order. Then the area (<math>A</math>) of <math>P</math> is
  8 KB (1,358 words) - 15:32, 22 February 2024
• 2008 USAMO Problems/Problem 3
  (''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) ...math> appears in exactly one of the <math>m</math> paths in <math>\mathcal{P}</math>).
  9 KB (1,585 words) - 01:00, 14 August 2014
• Millennium Problems
  ...set of seven problems for which the [[Clay Mathematics Institute]] offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new mi ...Conjecture]], the [[Hodge Conjecture]], the [[Navier-Stokes Equations]], [[P versus NP]], the [[Poincaré Conjecture]], the [[Riemann Hypothesis]], and
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1995 IMO Problems/Problem 1
  ...h>XY</math> meets <math>BC</math> at <math>Z</math>. Let <math>P</math> be a point on the line <math>XY</math> other than <math>Z</math>. The line <math Think about Radical Axis, Power of a Point and Radical Center.
  5 KB (847 words) - 19:03, 12 October 2021
• 2008 iTest Problems
  ...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2005 AMC 8 Problems
  Connie multiplies a number by 2 and gets 60 as her answer. However, she should <math> \textbf{(A)}\ 7.5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 120\q
  13 KB (1,821 words) - 22:18, 5 December 2023
• 2001 USAMO Problems/Problem 6
  Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its in '''Lemma''': If <math>ABCD</math> is an isosceles trapezoid, then <math>a + c = b + d</math>.
  2 KB (302 words) - 18:11, 22 February 2016
• 1992 USAMO Problems/Problem 4
  ...phere through <math>A'</math>, <math>B'</math>, <math>C'</math>, and <math>P</math>. Prove that <math>AA'=BB'=CC'</math>. ...he base angles are equal. Thus, <math>AP=BP</math>. Similarly, <math>A'P=B'P</math>. Thus, <math>AA'=BB'</math>. By symmetry, <math>BB'=CC'</math> and <
  5 KB (807 words) - 18:37, 25 June 2021
• 1974 USAMO Problems/Problem 4
  A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: 3) Play the first game against the stronger of the other two players (the so
  6 KB (1,037 words) - 23:42, 11 May 2018
• 2004 AMC 10B Problems/Problem 4
  ...at are visible. What is the largest number that is certain to divide <math>P</math>? <math> \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(
  2 KB (348 words) - 23:42, 16 January 2024
• 2009 AMC 10B Problems/Problem 22
  ...pieces as shown in this top view, where <math>M</math> is the midpoint of a top edge. The piece whose top is triangle <math>B</math> contains <math>c</ pair P=foot((1,-1),(1,1),(-1,0));
  12 KB (1,868 words) - 03:36, 30 September 2023
• 2009 AIME I Problems
  ...s <math>3</math> distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest g ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that
  7 KB (1,152 words) - 02:24, 23 July 2021
• 2009 AIME I Problems/Problem 5
  ...erline{CL}</math> is the angle bisector of angle <math>C</math>. Let <math>P</math> be the point of intersection of <math>\overline{BK}</math> and <math pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P;
  8 KB (1,224 words) - 19:52, 7 March 2024
• 2009 AIME I Problems/Problem 15
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c</math>. pair A,B,C,D,Ic,Ib,P;
  6 KB (1,048 words) - 19:35, 2 January 2023
• Radical axis
  The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B
  12 KB (2,125 words) - 08:38, 23 May 2024
• 2009 USAMO Problems/Problem 5
  ...</math> and <math>BG</math> meet <math>\omega</math> again at points <math>P</math> and <math>Q</math>, respectively. Let the line through <math>G</mat dotfactor = 3;
  6 KB (973 words) - 19:24, 18 October 2018
• 2009 IMO Problems/Problem 2
  ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent dot("A", (40, 100), N);
  2 KB (280 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  Let <math>n</math> be a positive integer and let <math>a_1,\ldots,a_k (k\ge2)</math> be distinct in ...ctively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent
  3 KB (509 words) - 09:23, 10 September 2020
• 1992 AJHSME Problems
  <math>\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=</math> <math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad
  17 KB (2,346 words) - 13:36, 19 February 2020
• 2010 AMC 12A Problems/Problem 21
  The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \
  5 KB (969 words) - 21:33, 22 June 2022
• 2010 AMC 12B Problems
  <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ A big <math>L</math> is formed as shown. What is its area?
  12 KB (1,845 words) - 13:00, 19 February 2020
• Characteristic polynomial
  The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...<math>A</math> is defined as <math>P_A(t) = \det(tI - A)</math>, which is a <math>n</math>th degree polynomial in <math>t</math>. Here, <math>I</math>
  19 KB (3,412 words) - 14:57, 21 September 2022
• Proofs without words
  rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2) int n = 6; pair shiftR = ((n+2),0); real r = 0.3;
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems/Problem 11
  ...h>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. real P(real x) { return x/3 + 5; }
  4 KB (636 words) - 16:46, 25 November 2023
• 2007 IMO Problems
  (a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>, ...h> such that <math>ABCD</math> is a parallelogram and <math>BCED</math> is a cyclic quadrilateral.
  3 KB (505 words) - 09:24, 10 September 2020
• 2010 AIME II Problems
  ...<math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <mat == Problem 3 ==
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 9
  ...of the smaller hexagon to the area of <math>ABCDEF</math> be expressed as a fraction <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> pair A,B,C,D,E,F;
  5 KB (857 words) - 22:22, 27 August 2023
• 2010 USAMO Problems/Problem 4
  Let <math>ABC</math> be a triangle with <math>\angle A = 90^{\circ}</math>. Points <math>D</math> ==Video Solution in 3 minutes!!!==
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2010 USAJMO Problems/Problem 1
  ...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...> is closed under multiplication and a non-square times a square is always a non-square.
  12 KB (2,338 words) - 20:30, 13 February 2024
• 2010 USAMO Problems/Problem 1
  Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto
  13 KB (2,271 words) - 14:08, 23 May 2024
• 1995 AJHSME Problems
  <math>\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 4 Jose is <math>4</math> years younger than Zack. Zack is <math>3</math> years older than Inez. Inez is <math>15</math> years old. How old
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2010 IMO Problems/Problem 4
  ...> meet again its circumcircle <math>\Gamma</math> at <math>K</math>, <math>L</math>, respectively <math>M</math>. The tangent line at <math>C</math> to ...thout loss of generality]], suppose that <math>AS > BS</math>. By Power of a Point, <math>SP^2 = SC^2 = SB \cdot SA</math>, so <math>\overline{SP}</math
  6 KB (895 words) - 18:32, 2 April 2021
• 2010 AMC 12B Problems/Problem 21
  Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(1) = P(3) = P(5) = P(7) = a</math>,
  7 KB (1,166 words) - 00:04, 26 February 2024
• 2010 IMO Problems/Problem 6
  ...ldots</math> be a sequence of positive real numbers, and <math>s</math> be a positive integer, such that So for solving This Problem, we need to take a assumption that,
  4 KB (786 words) - 08:46, 12 March 2024
• 2010 AMC 10B Problems/Problem 7
  ...le has side lengths <math>10</math>, <math>10</math>, and <math>12</math>. A rectangle has width <math>4</math> and area equal to the <math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\
  2 KB (318 words) - 09:00, 1 September 2022
• 2010 IMO Problems
  ...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math> ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE
  4 KB (603 words) - 09:22, 10 September 2020
• 2006 AMC 8 Problems
  <math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qqu <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad
  16 KB (2,215 words) - 19:18, 10 April 2024
• 1998 AJHSME Problems
  <math>\text{(A)}\ \dfrac{6}{x} \qquad \text{(B)}\ \dfrac{6}{x+1} \qquad \text{(C)}\ \dfrac ...xt{b}\cdot \text{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?
  14 KB (1,920 words) - 19:31, 31 January 2024
• 1999 AMC 8 Problems/Problem 15
  ...ee letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qq
  3 KB (516 words) - 14:50, 21 December 2022
• 2021 AIME II Problems/Problem 5
  ...tau(s)</math> are congruent, is an interval <math>[a,b)</math>. Find <math>a^2+b^2</math>. ...angles. That means that the two small sides squared is less than the <math>3</math>rd side. So the triangles' sides are between <math>6</math> and <math
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2011 AIME I Problems/Problem 4
  ...ngle <math>A</math> intersects <math> \overline{BC} </math> at point <math>L</math>, and the angle bisector of angle <math>B</math> intersects <math> \o ...CN}</math> such that they intersect line <math>{AB}</math> at points <math>P</math> and <math>Q</math>, respectively.
  6 KB (1,068 words) - 18:52, 2 August 2023
• 2011 AIME I Problems
  ...olution in jar <math>C</math> is added to jar B. At the end both jar <math>A</math> and jar <math>B</math> contain solutions that are <math>50\%</math> ...>n</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
  10 KB (1,634 words) - 22:21, 28 December 2023
• 2011 AIME I Problems/Problem 3
  ...nt <math>B</math> is on the positive <math>y</math>-axis. The point <math>P</math> with coordinates <math>(-14,27)</math> in the original system has co ...nt <math>A=(24,-1)</math>, we may write the point-slope equation for <math>L</math> as <math>y+1=\frac{5}{12}(x-24)</math>.
  4 KB (699 words) - 16:01, 13 April 2024
• 1999 AMC 8 Problems
  ...math> To make this statement true, the question mark between the 6 and the 3 should be replaced by <math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2011 USAJMO Problems/Problem 3
  ...nate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>. Consider the set of triangles with verti ...h>w_n^3</math> is constant, say it is <math>k</math>. Then for <math>n=1,2,3</math>,
  15 KB (2,593 words) - 13:37, 29 January 2021
• Block walking
  label("X", (0,3), NW); label("Y", (3,0), SE);
  9 KB (1,228 words) - 21:47, 26 February 2022
• Mock Amc 12
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.
  18 KB (2,206 words) - 19:41, 24 December 2020
• 2003 AMC 8 Problems
  Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of <math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22
  16 KB (2,236 words) - 12:02, 19 February 2024
• 2011 IMO Problems
  ...inct positive integers which achieve the largest possible value of <math>n(A)</math>. ...d the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.
  4 KB (682 words) - 00:12, 18 February 2021
• 1992 AJHSME Problems/Problem 11
  The bar graph shows the results of a survey on color preferences. What percent preferred blue? for (int a = 1; a <= 6; ++a)
  2 KB (226 words) - 00:09, 5 July 2013
• 1997 AHSME Problems/Problem 5
  A rectangle with perimeter <math>176</math> is divided into five congruent re draw(origin--(0,3)--(4,3)--(4,0)--cycle);
  1 KB (231 words) - 14:12, 5 July 2013
• 1993 AJHSME Problems/Problem 13
  ...block letters is painted in black with strokes <math>1</math> unit wide on a <math>5</math> by <math>15</math> rectangular white sign with dimensions as fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black);
  1 KB (177 words) - 04:55, 25 November 2019
• 2003 AMC 8 Problems/Problem 18
  ...one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
  2 KB (250 words) - 22:17, 5 January 2024

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