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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

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