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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 

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

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