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Community Service Scholarships
...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using * Kohl's Kids Who Care Scholarship Program [http://www.kohlscorporation.com/communit

3 KB (350 words) - 01:18, 19 June 2016
Fermat's Little Theorem
...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.

16 KB (2,658 words) - 16:02, 8 May 2024
Divisibility rules
...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b ...its of the number are divisible by <math>2^n</math>. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2,

8 KB (1,315 words) - 18:18, 2 March 2024
2011 AMC 10B Problems/Problem 22
...ength <math>1</math> and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py ...- 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad\textbf{(E)}\ \frac{\sqrt{3}}

4 KB (691 words) - 18:38, 19 September 2021
Geometric sequence
...'geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between any two consecutive ter ...ric sequence with common ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as

4 KB (644 words) - 12:55, 7 March 2022
2021 JMC 10
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2

12 KB (1,784 words) - 16:49, 1 April 2021
2006 AMC 12B Problems
\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) } 24 \qquad \text {(E) } 72</mat

13 KB (2,058 words) - 12:36, 4 July 2023
2006 AMC 12A Problems
{{AMC12 Problems|year=2006|ab=A}} Sandwiches at Joe's Fast Food cost <math>3</math> dollars each and sodas cost <math>2</math> do

15 KB (2,223 words) - 13:43, 28 December 2020
2005 AMC 12A Problems
{{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})

13 KB (1,971 words) - 13:03, 19 February 2020
2004 AMC 12A Problems
{{AMC12 Problems|year=2004|ab=A}} ...</math> is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?

13 KB (1,953 words) - 00:31, 26 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
2000 AMC 12 Problems
<math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{

13 KB (1,948 words) - 12:26, 1 April 2022
2003 AMC 12B Problems
\text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?

13 KB (1,987 words) - 18:53, 10 December 2022
2004 AMC 12B Problems
<math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and

13 KB (2,049 words) - 13:03, 19 February 2020
2006 AMC 12B Problems/Problem 16
Regular hexagon <math>ABCDEF</math> has vertices <math>A</math> and <math>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, res \mathrm{(A)}\ 20\sqrt {3}

1 KB (203 words) - 16:36, 18 September 2023
1991 AJHSME Problems
<math>\text{(A)}\ 222,222,222,222 \qquad <math>\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qqua

17 KB (2,246 words) - 13:37, 19 February 2020
2005 AIME II Problems/Problem 2
...<math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </math> ...We need only calculate the probability the first and second person all get a roll of each type, since then the rolls for the third person are determined

4 KB (628 words) - 11:28, 14 April 2024
2005 AIME II Problems/Problem 10
Given that <math> O </math> is a regular [[octahedron]], that <math> C </math> is the [[cube (geometry) | cu ...rac {s^3}{3\sqrt2}</math> and the whole octahedron has volume <math>\frac {s^3\sqrt2}3</math>.

3 KB (436 words) - 03:10, 23 September 2020
2005 AIME I Problems
...ngent to two circles adjacent to it. All circles are internally tangent to a circle <math> C </math> with radius 30. Let <math> K </math> be the area of ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi

6 KB (983 words) - 05:06, 20 February 2019
2005 AIME I Problems/Problem 4
...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat

8 KB (1,248 words) - 11:43, 16 August 2022
2005 AIME I Problems/Problem 13
A particle moves in the [[Cartesian plane]] according to the following rules: ...> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME II Problems/Problem 14
pair A = origin; pair C = rotate(15,A)*(A+dir(-50));

13 KB (2,129 words) - 18:56, 1 January 2024
2004 AIME II Problems/Problem 6
...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...fraction is equal to <math>8</math>, and the solution is <math>8(11 + 13 + 27) = \boxed{408}</math>.

6 KB (950 words) - 14:18, 15 January 2024
1989 AIME Problems
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if

7 KB (1,045 words) - 20:47, 14 December 2023
1997 AIME Problems
...</math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho

7 KB (1,098 words) - 17:08, 25 June 2020
1999 AIME Problems
...> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</math> A line through the origin cuts this figure into two congruent polygons. The ...all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.

7 KB (1,094 words) - 13:39, 16 August 2020
2000 AIME I Problems
...Let <math>A = (u,v)</math>, let <math>B</math> be the reflection of <math>A</math> across the line <math>y = x</math>, let <math>C</math> be the reflec ...icients of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.

7 KB (1,204 words) - 03:40, 4 January 2023
2001 AIME I Problems
...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.

7 KB (1,212 words) - 22:16, 17 December 2023
2002 AIME I Problems
...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r

8 KB (1,374 words) - 21:09, 27 July 2023
1984 AIME Problems/Problem 10
...ems and that one's score, <math>s</math>, is computed by the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and < Let Mary's score, number correct, and number wrong be <math>s,c,w</math> respectively. Then

7 KB (1,163 words) - 23:53, 28 March 2022
1984 AIME Problems/Problem 14
Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be ex ...prime quintuplet is <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answ

8 KB (1,346 words) - 01:16, 9 January 2024
1984 AIME Problems/Problem 15
...4,16,36,64</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,16,36,64</math>, so ...+ 3^2 + 5^2 + 7^2 + x^2 + y^2 + z^2 + w^2)t^3 \dots = 0.</cmath> By Vieta's, we know that the sum of the roots of this equation is <cmath>1^2 + 3^2 + 5

6 KB (1,051 words) - 04:52, 8 May 2024
1985 AIME Problems/Problem 12
...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of ...> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <

17 KB (2,837 words) - 13:34, 4 April 2024
1986 AIME Problems/Problem 10
...rticipants to think of a three digit number <math>(abc)</math> where <math>a</math>, <math>b</math>, and <math>c</math> represent digits in base <math>1 ...math> be the number <math>100a+10b+c</math>. Observe that <math>3194+m=222(a+b+c)</math> so

3 KB (565 words) - 16:51, 1 October 2023
1986 AIME Problems/Problem 7
...th>1,3,4,9,10,12,13\cdots</math> consists of all those positive [[integer]]s which are [[exponent|powers]] of 3 or sums of distinct powers of 3. Find th ...powers of 3, in base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Therefore, we can

5 KB (866 words) - 00:00, 22 December 2022
1988 AIME Problems/Problem 12
...ments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. Call the [[cevian]]s AD, BE, and CF. Using area ratios (<math>\triangle PBC</math> and <math>\tr

4 KB (727 words) - 23:37, 7 March 2024
1988 AIME Problems/Problem 3
Raise both as [[exponent]]s with base 8: ...ge of base formula]], which states <math>\log_a b = \frac{\log_k b}{\log_k a}</math> for arbitrary <math>k</math>.

3 KB (481 words) - 21:52, 18 November 2020
1989 AIME Problems/Problem 15
=== Solution 1 (Ceva's Theorem, Stewart's Theorem) === ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math

13 KB (2,091 words) - 00:20, 26 October 2023
1989 AIME Problems/Problem 12
...e. Let <math>d</math> be the distance between the [[midpoint]]s of [[edge]]s <math>AB</math> and <math>CD</math>. Find <math>d^{2}</math>. pair A,B,C,D,M,P,Q;

2 KB (376 words) - 13:49, 1 August 2022
1989 AIME Problems/Problem 9
...they showed there was a positive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>. By either <b>Fermat's Little Theorem (FLT)</b> or inspection, we get

6 KB (874 words) - 15:50, 20 January 2024
1989 AIME Problems/Problem 4
...t <math>b+c+d</math> is a [[perfect square]] and <math>a+b+c+d+e</math> is a [[perfect cube]], what is the smallest possible value of <math>c</math>? ...re is a cubed term in <math>5^2 \cdot y^3</math>, <math>3^3</math> must be a factor of <math>c</math>. <math>3^35^2 = \boxed{675}</math>, which works as

3 KB (552 words) - 12:41, 3 March 2024
1990 AIME Problems/Problem 2
...d <math>b</math> equal one of <math>\pm3, \pm1</math>. The possible [[set]]s are <math>(3,1)</math> and <math>(-3,-1)</math>; the latter can be discarde ...qrt{43} + 3)^3 - (\sqrt{43} - 3)^3</math>. Using the difference of [[cube]]s, we get that <math>[\sqrt{43} + 3\ - \sqrt{43} + 3]\ [(43 + 6\sqrt{43} + 9)

5 KB (765 words) - 23:00, 26 August 2023
1991 AIME Problems/Problem 10
...ath>S_b^{}</math> in alphabetical order. When <math>p</math> is written as a [[fraction]] in [[irreducible fraction|lowest terms]], what is its [[numera ...tring comes from <math>aaa</math> and <math>bbb</math> multiplied by <math>27</math>, and <math>S_b</math> denotes the partial sums of <math>P_b</math> (

5 KB (813 words) - 06:10, 25 February 2024
1992 AIME Problems/Problem 5
...^{}_{}</math> are not necessarily distinct. To write the elements of <math>S^{}_{}</math> as fractions in lowest terms, how many different numerators ar ...ultiples of <math>3</math>. We have to count these since it will reduce to a multiple of <math>3</math> which we have removed from <math>999</math>, but

2 KB (277 words) - 20:45, 4 March 2024
1992 AIME Problems/Problem 4
In Pascal's Triangle, each entry is the sum of the two entries above it. The first few In which row of [[Pascal's Triangle]] do three consecutive entries occur that are in the ratio <math>3

3 KB (476 words) - 14:13, 20 April 2024
1993 AIME Problems/Problem 11
...Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the fir The probability that the <math>n</math>th flip in each game occurs and is a head is <math>\frac{1}{2^n}</math>. The first person wins if the coin lands

7 KB (1,058 words) - 20:57, 22 December 2020
1955 AHSME Problems/Problem 27
== Problem 27== ...nd <math>s</math> are the roots of <math>x^2-px+q=0</math>, then <math>r^2+s^2</math> equals:

600 bytes (108 words) - 10:47, 15 February 2021
1996 AIME Problems/Problem 13
...th> [[bisect]]s <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio <math>\frac{[ADB]}{[ABC]}</math> can be written in t pair B=(0,0), C=(15^.5, 0), A=IP(CR(B,30^.5),CR(C,6^.5)), E=(B+C)/2, D=foot(B,A,E);

3 KB (521 words) - 01:18, 25 February 2016
1996 AIME Problems/Problem 5
...x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</math>. Find <math>t</math>. ...) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11</math>. Then

3 KB (585 words) - 22:08, 19 November 2022
1997 AIME Problems/Problem 10
...e deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the foll i. Either each of the three cards has a different shape or all three of the cards have the same shape.

3 KB (585 words) - 19:37, 25 April 2022
1998 AIME Problems/Problem 4
...h>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> In order for a player to have an odd sum, he must have an odd number of odd tiles: that is

5 KB (917 words) - 02:37, 12 December 2022
1999 AIME Problems/Problem 12
...[[radius]] is <math>21</math>. Given that <math>AP=23</math> and <math>PB=27,</math> find the [[perimeter]] of the triangle. pair A=(0,0),B=(50,0),C=IP(circle(A,23+245/2),circle(B,27+245/2)), I=incenter(A,B,C);

3 KB (472 words) - 15:59, 25 February 2022
2001 AIME I Problems/Problem 2
...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...{S}</math>. Let <math>a</math> be the number of elements in <math>\mathcal{S}</math>.

1 KB (225 words) - 07:57, 4 November 2022
2002 AIME I Problems/Problem 13
...ath> and <math>\overline{CE}</math> have lengths <math>18</math> and <math>27</math>, respectively, and <math>AB=24</math>. Extend <math>\overline{CE}</m ...,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C));

6 KB (974 words) - 13:01, 29 September 2023
2002 AIME I Problems/Problem 10
...3</math> and <math>AF=10</math>. Given that <math>EB=9</math> and <math>FC=27</math>, find the integer closest to the area of quadrilateral <math>DCFG</m ...> is <math>10/3</math> that of triangle <math>AEG</math>, since they share a common side and angle, so the area of triangle <math>AGF</math> is <math>10

4 KB (643 words) - 22:44, 8 August 2023
2003 AIME I Problems/Problem 7
...of all possible [[perimeter]]s of <math> \triangle ACD</math>. Find <math> s. </math> pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8);

3 KB (490 words) - 18:13, 13 February 2021
2003 AIME II Problems/Problem 12
...tee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidat ...number of votes cast. Our goal is to determine the smallest possible <math>s</math>.

4 KB (759 words) - 13:00, 11 December 2022
2006 AMC 10B Problems
<math> \mathrm{(A) \ } -2006\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{( <math> \mathrm{(A) \ } -72\qquad \mathrm{(B) \ } -27\qquad \mathrm{(C) \ } -24\qquad \mathrm{(D) \ } 24\qquad \mathrm{(E) \ } 72

14 KB (2,059 words) - 01:17, 30 January 2024
Mock AIME 1 2006-2007 Problems/Problem 7
...lic, since <math>\angle AEB=\angle AFB</math>. We then have, from Power of a Point, that <math>CE\cdot CA=CF\cdot CB</math>. In other words, <math>1\cdo ...math>[CFD]=\frac{CF}{CB}\cdot [CBD]=\frac{2}{3}\cdot \frac{x}{9}=\frac{2x}{27}</math>. The desired ratio is then

3 KB (518 words) - 16:54, 25 November 2015
2007 iTest Problems
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>

30 KB (4,794 words) - 23:00, 8 May 2024
2005 AMC 10A Problems
{{AMC10 Problems|year=2005|ab=A}} <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \

14 KB (2,026 words) - 11:45, 12 July 2021
2003 AMC 10A Problems
{{AMC10 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \

13 KB (1,900 words) - 22:27, 6 January 2021
2003 AMC 10A Problems/Problem 14
<math> \mathrm{(A) \ } 12\qquad \mathrm{(B) \ } 15\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) ...e digits of <math>n</math> is <math>1+5+3+3=12 \Rightarrow \boxed{\mathrm{(A)}\ 12}</math> ~ MathGenius_ (Edited by Sophia866)

2 KB (336 words) - 15:49, 19 August 2023
2020 AMC 10A Problems
{{AMC10 Problems|year=2020|ab=A}} <math>\textbf{(A)}\ {-}\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{

13 KB (1,968 words) - 18:32, 29 February 2024
2007 AIME I Problems/Problem 6
...ending with 39. For example, <math>0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39</math> is a move sequence. How many move sequences are possible for the frog? Let us keep a careful tree of the possible number of paths around every multiple of <math

10 KB (1,519 words) - 00:11, 29 November 2023
2007 AIME I Problems/Problem 11
...<math>S = \sum_{p=1}^{2007} b(p),</math> find the [[remainder]] when <math>S</math> is divided by 1000. ...e are <math>2007 - 1981 + 1 = 27</math> (1 to be inclusive) of them. <math>27*45=1215</math>, and <math>215+740=

3 KB (562 words) - 20:02, 30 December 2023
1961 IMO Problems/Problem 2
...>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''. Prove that <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>

5 KB (860 words) - 13:12, 13 February 2024
2007 AMC 12A Problems
{{AMC12 Problems|year=2007|ab=A}} ...h> discount. Pam buys <math>5</math> tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan?

11 KB (1,750 words) - 13:35, 15 April 2022
Chicken McNugget Theorem
...ritten in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. A consequence of the theorem is that there are exactly <math>\frac{(m - 1)(n

17 KB (2,748 words) - 19:22, 24 February 2024
2007 AMC 12A Problems/Problem 14
Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> <math>(6-a)(6-b)(6-c)(6-d)(6-e)=45</math>

2 KB (278 words) - 02:10, 16 February 2021
2007 AMC 12A Problems/Problem 22
...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 \

15 KB (2,558 words) - 19:33, 4 February 2024
2006 AIME II Problems/Problem 3
...the first <math>100</math> [[positive integer | positive]] [[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divi ...ight\rfloor+\left\lfloor\frac{200}{9}\right\rfloor+\left\lfloor \frac{200}{27}\right\rfloor+\left\lfloor\frac{200}{81}\right\rfloor =66+22+7+2=97</math>

4 KB (562 words) - 18:37, 30 October 2020
2006 AIME II Problems/Problem 11
A [[sequence]] is defined as follows <math> a_1=a_2=a_3=1, </math> and, for a Define the sum as <math>s</math>. Since <math>a_n\ = a_{n + 3} - a_{n + 2} - a_{n + 1} </math>, the s

3 KB (417 words) - 10:07, 12 October 2023
1997 PMWC Problems/Problem I7
...nly <math>12</math> students in the class who got <math>\mathrm{'A'}</math>s and the ratio of boys and girls in the class is <math>4:5</math>, how many ...2 \Longrightarrow g = 15</math>. So there are 12 boys, and <math>12 + 15 = 27</math> students in the class.

750 bytes (111 words) - 14:23, 20 April 2014
2007 AMC 12B Problems
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 fee <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{

12 KB (1,814 words) - 12:58, 19 February 2020
1999 AHSME Problems
<math> \mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm <math>\text{(A) All equilateral triangles are congruent to each other.}</math>

13 KB (1,945 words) - 18:28, 19 June 2023
2001 IMO Problems/Problem 2
...ath>a,b,c</math> be positive real numbers. Prove that <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ...{2}+2bc)+6bc \leq^{AG} (a^{2}+b^{2}+c^{2})+6bc=S+6bc</math> (where <math>S=a^{2}+b^{2}+c^{2}</math>) and its cyclic variations.

7 KB (1,174 words) - 12:14, 3 August 2023
2004 AMC 12A Problems/Problem 13
...h>. How many distinct [[line]]s pass through at least two members of <math>S</math>? ...{(A)}\ 8 \qquad \text {(B)}\ 20 \qquad \text {(C)}\ 24 \qquad \text {(D)}\ 27\qquad \text {(E)}\ 36</math>

2 KB (381 words) - 00:58, 19 October 2020
1995 AHSME Problems
...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq

17 KB (2,387 words) - 22:44, 26 May 2021
1997 PMWC Problems
Evaluate <math>29 \dfrac{27}{28} \times 27 \frac{14}{15}</math> ...represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{

15 KB (2,057 words) - 19:13, 10 March 2015
1951 AHSME Problems
<math> \textbf{(A) \ } \frac {100(M - N)}{M} \qquad \textbf{(B) \ } \frac {100(M - N)}{N} \qq A rectangular field is half as wide as it is long and is completely enclosed

23 KB (3,641 words) - 22:23, 3 November 2023
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
1966 AHSME Problems
<math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t

19 KB (3,159 words) - 22:10, 11 March 2024
1991 IMO Problems/Problem 1
...isectors of the angles <math>A,B,C</math> meet the opposite sides in <math>A^\prime,B^\prime,C^\prime</math> respectively. Prove that ...\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}</math>

2 KB (311 words) - 09:40, 19 November 2023
2007 AMC 8 Problems
Theresa's parents have agreed to buy her tickets to see her favorite band if she spen <math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \

12 KB (1,800 words) - 20:01, 8 May 2023
1998 AHSME Problems/Problem 26
...\circ}</math>, angles <math>B</math> and <math>D</math> are [[right angle]]s, <math>AB = 13</math>, and <math>AD = 46</math>. Then <math>AC=</math> <math>\mathrm{(A)}\ 60

3 KB (391 words) - 14:30, 5 July 2013
2008 AMC 10A Problems
{{AMC10 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

14 KB (2,138 words) - 15:08, 18 February 2023
2008 AMC 12A Problems
{{AMC12 Problems|year=2008|ab=A}} ...}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

13 KB (2,025 words) - 13:56, 2 February 2021
2007 AMC 12B Problems/Problem 13
...ath>3</math> seconds, and then red for <math>30</math> seconds. Leah picks a random three-second time interval to watch the light. What is the probabili <math>\mathrm{(A)}\ \frac{1}{63} \qquad \mathrm{(B)}\ \frac{1}{21} \qquad \mathrm{(C)}\ \fra

1 KB (166 words) - 01:03, 19 October 2020
2008 AIME I Problems/Problem 9
a + b + c &= 10\end{align*}</cmath> ...>. The last doesn't work, obviously. This gives the three solutions <math>(a,b,c) = (5,2,3),(3,5,2),(1,8,1)</math>. In terms of choosing which goes wher

6 KB (909 words) - 15:39, 8 August 2022
2008 AIME I Problems/Problem 11
...math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? ...math>n-2</math> ending in a <math>B</math>. Thus, we have the [[recursion]]s

7 KB (1,173 words) - 22:39, 28 November 2023
2008 AIME I Problems/Problem 14
...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...= O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);

8 KB (1,333 words) - 00:18, 1 February 2024
Mock AIME 1 2007-2008 Problems
A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange ...1+i)^{17} - (1-i)^{17}</math>, where <math>i=\sqrt{-1}</math>. Find <math>|S|</math>.

6 KB (992 words) - 14:15, 13 February 2018
2008 AIME II Problems/Problem 12
The well known problem of ordering <math>x</math> elements of a string of <math>y</math> elements such that none of the <math>x</math> elem ...uch that none of them are next to each other. We would also like to choose a place where we can divide this string into two strings such that the left o

10 KB (1,550 words) - 12:58, 15 July 2023
1997 PMWC Problems/Problem T7
...Take out all the smaller cubes which have at least one red surface and fix a cuboid, <b>keeping the surfaces of the cuboid red</b>. Now what is the maxi ...'' 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>98</mat

2 KB (283 words) - 21:47, 13 July 2023
2006 Alabama ARML TST Problems/Problem 13
<center><math>\sqrt[4]{x+27}+\sqrt[4]{55-x}=4.</math></center> <math>\sqrt[4]{x+27}=0\Rightarrow x=-27, \sqrt[4]{55-x}=\sqrt[4]{82}</math>

1 KB (167 words) - 09:56, 15 December 2018
2008 AIME II Problems/Problem 13
...where <math>a</math> and <math>b</math> are positive integers. Find <math>a + b</math>. ...is)! If you're familiar with inversion you can go plot the inverted figure's Cartesian Plane Equivalent. Then simply continue on with the figure shown i

6 KB (894 words) - 18:56, 25 December 2022
Legendre's Formula
'''Legendre's Formula''' states that ...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]] <math>p</math>.

4 KB (699 words) - 17:55, 5 August 2023
1990 AHSME Problems/Problem 26
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. T <math>\textbf{(A) } 1 \qquad

3 KB (402 words) - 23:17, 23 September 2023
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
1973 USAMO Problems/Problem 4
...oot]]s, [[real]] or [[complex]], of the system of simultaneous [[equation]]s ...</math>, and <math>S_3+aS_2+bS_1+3c=0</math>. Solving each of these, <math>a=-3</math>, <math>b=3</math>, and <math>c=-1</math>. Thus <math>x</math>, <m

5 KB (888 words) - 08:18, 22 April 2024
2002 AMC 10A Problems
{{AMC10 Problems|year=2002|ab=A}} <math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qq

11 KB (1,733 words) - 11:04, 12 October 2021
2002 AMC 10A Problems/Problem 18
A 3x3x3 cube is made of <math>27</math> normal dice. Each die's opposite sides sum to <math>7</math>. What is the smallest possible sum of <math>\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qq

999 bytes (153 words) - 20:43, 28 May 2023
2004 AMC 10B Problems
...math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club? <math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm

13 KB (1,988 words) - 20:19, 15 May 2024
2000 AMC 10 Problems
<math>\textbf{(A)}\ 23 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 99 \qquad\textbf{(D)}\ 111 <math>\textbf{(A)}\ {2000}^{2001} \qquad \textbf{(B)}\ {4000}^{2000} \qquad \textbf{(C)}\ {2

14 KB (2,035 words) - 21:57, 2 May 2024
1976 USAMO Problems/Problem 4
.../math> (i.e., <math>\angle APB=\angle BPC=\angle CPA=90^o</math>) is <math>S</math>, determine its maximum volume. ...pectively. Therefore <math>S=a+b+c+\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}</math>. Let the volume of the tetrahedron be <math>V</math>. Therefore <

2 KB (358 words) - 23:15, 18 July 2016
1999 AHSME Problems/Problem 28
Let <math>x_1, x_2, \ldots , x_n</math> be a sequence of integers such that <math> \mathrm{(A) \ }3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ }5 \qquad \mathrm{(D) \

4 KB (607 words) - 12:25, 3 October 2023
2000 AMC 10 Problems/Problem 18
Charlyn walks completely around the boundary of a square whose sides are each <math>5</math> km long. From any point on her p <math>\textbf{(A)} 24 \qquad\textbf{(B)}\ 27 \qquad\textbf{(C)}\ 39 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 42</math>

2 KB (274 words) - 10:26, 8 November 2021
1986 AJHSME Problems
<math>\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad <math>\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ 1 \qquad \

14 KB (2,054 words) - 15:41, 8 August 2020
2004 AMC 10B Problems/Problem 2
How many two-digit positive integers have at least one <math>7</math> as a digit? <math> \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(

1 KB (208 words) - 19:10, 24 January 2015
2004 AMC 12B Problems/Problem 15
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill <math> \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 </math>

2 KB (326 words) - 09:47, 17 October 2020
1987 AJHSME Problems
<math>\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ . <math>\text{(A)}\ .008 \qquad \text{(B)}\ .08 \qquad \text{(C)}\ .8 \qquad \text{(D)} 1.25

12 KB (1,568 words) - 09:35, 31 October 2021
2009 AMC 12A Problems
{{AMC12 Problems|year=2009|ab=A}} Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Bot

13 KB (2,105 words) - 13:13, 12 August 2020
2009 AMC 10A Problems
{{AMC10 Problems|year=2009|ab=A}} ...math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?

14 KB (2,130 words) - 11:32, 7 November 2021
2002 AMC 12A Problems/Problem 24
...d pairs of real numbers <math>(a,b)</math> such that <math>(a+bi)^{2002} = a-bi</math>. \text{(A) }1001

4 KB (743 words) - 19:54, 14 March 2024
1988 AJHSME Problems
The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of <math>\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)

14 KB (1,872 words) - 15:23, 17 January 2023
2009 AIME I Problems/Problem 6
...are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>? Because <math>{\lfloor x\rfloor}</math> must be an integer, let’s do case work based on <math>{\lfloor x\rfloor}</math>:

4 KB (595 words) - 16:38, 15 February 2021
2009 AIME II Problems
...ng a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finis ...nd <math>c</math> are positive real numbers such that <math>a^{\log_3 7} = 27</math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_{11}25} = \sqrt{

8 KB (1,366 words) - 21:33, 3 January 2021
2000 AMC 8 Problems
<math>\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 21\qqu <math>\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qqua

15 KB (2,165 words) - 03:32, 13 April 2024
1988 AJHSME Problems/Problem 24
for (int a=0; a<6; ++a) label("1.",(0,-2),S);

6 KB (470 words) - 23:56, 4 July 2013
2009 IMO Problems/Problem 2
Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <m dot("A", (40, 100), N);

2 KB (280 words) - 01:16, 19 November 2023
2003 AMC 12A Problems/Problem 11
...math> the area of the circle circumscribed around the triangle. Find <math>A/B</math>. ...\frac{9}{16}\qquad \mathrm{(B) \ } \frac{3}{4}\qquad \mathrm{(C) \ } \frac{27}{32}\qquad \mathrm{(D) \ } \frac{3\sqrt{6}}{8}\qquad \mathrm{(E) \ } 1 </ma

2 KB (297 words) - 13:31, 22 July 2017
1992 AJHSME Problems
<math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad <math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfra

17 KB (2,346 words) - 13:36, 19 February 2020
1993 AJHSME Problems
Which pair of numbers does NOT have a product equal to <math>36</math>? <math>\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \df

14 KB (1,797 words) - 11:13, 28 December 2022
2002 AMC 10B Problems/Problem 25
When <math>15</math> is appended to a list of integers, the mean is increased by <math>2</math>. When <math>1</ma <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \

4 KB (679 words) - 19:11, 17 September 2023
2010 AMC 10A Problems
{{AMC10 Problems|year=2010|ab=A}} Mary’s top book shelf holds five books with the following widths, in centimeters:

13 KB (1,902 words) - 11:20, 5 March 2023
2010 AMC 10B Problems
\textbf{(A)}\ -20,000 <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\

12 KB (1,817 words) - 22:44, 22 December 2020
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
2010 AIME I Problems/Problem 3
...and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>. ...4} = \Bigg(\frac43\Bigg)^4 = \frac {256}{81}</math>, <math>y = \frac {64}{27}</math>, <math>x + y = \frac {256 + 192}{81} = \frac {448}{81}</math>, yiel

3 KB (465 words) - 22:07, 17 December 2021
2010 AIME I Problems/Problem 12
...ath> into two subsets, at least one of the subsets contains integers <math>a</math>, <math>b</math>, and <math>c</math> (not necessarily distinct) such ...ath>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>.

2 KB (284 words) - 21:56, 25 November 2023
2010 AIME I Problems/Problem 13
[[Rectangle]] <math>ABCD</math> and a [[semicircle]] with diameter <math>AB</math> are coplanar and have nonoverl label("$A$",(-16.43287,-9.3374),NE/2);

10 KB (1,418 words) - 23:05, 20 October 2021
Mock AIME 2 2010 Answer Key
...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a

36 KB (6,214 words) - 20:22, 13 July 2023
1994 AJHSME Problems
<math>\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3 <math>\text{(A)}\ 4\dfrac{1}{2} \qquad \text{(B)}\ 6.4 \qquad \text{(C)}\ 9 \qquad \text{(

16 KB (2,292 words) - 13:36, 19 February 2020
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 (2,178 words) - 14:14, 11 September 2021
2010 AIME II Problems/Problem 15
...ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> pair A,B,C,D,E,M,N,P,Q;

9 KB (1,523 words) - 15:24, 21 November 2023
1995 AJHSME Problems
<math>\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 4 <math>\text{(A)}\ 8 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 16 \qqu

14 KB (2,096 words) - 18:29, 2 January 2023
2010 AMC 12B Problems/Problem 23
<math>\textbf{(A)}\ -100 \qquad \textbf{(B)}\ -82 \qquad \textbf{(C)}\ -73 \qquad \textbf{(D ...</math>. Doing something similar for <math> Q(P(x))</math> gives us <math> a = - 54</math>.

6 KB (1,137 words) - 15:22, 1 November 2023
1996 AJHSME Problems
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare

13 KB (1,880 words) - 13:35, 19 February 2020
2004 AMC 8 Problems
...ter length represents <math>72</math> kilometers. How many kilometers does a <math>17</math>-centimeter length represent? <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\q

13 KB (1,860 words) - 19:58, 8 May 2023
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 If <math>\begin{tabular}{r|l}a&b \\ \hline c&d\end{tabular} = \text{a}\cdot \text{d} - \text{b}\cdot \text{c}</math>, what is the value of <math>

14 KB (1,920 words) - 19:31, 31 January 2024
2010 AMC 10B Problems/Problem 20
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 108</m A good diagram is very helpful.

3 KB (483 words) - 18:41, 4 May 2024
2011 AMC 12A Problems
{{AMC12 Problems|year=2011|ab=A}} A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</m

13 KB (1,994 words) - 13:52, 3 July 2021
2011 AMC 12A Problems/Problem 13
\textbf{(A)}\ 27 \qquad ...Assign a mass of 24 to <math>A</math>, a mass of 18 to <math>B</math>, and a mass of 12 to <math>C</math>. Then <math>D</math> has mass 30, so <math>\fr

4 KB (683 words) - 03:12, 23 January 2023
2011 AMC 12A Problems/Problem 14
...independently and at random. What is the probability that the point <math>(a,b)</math> lies above the parabola <math>y=ax^2-bx</math>? \textbf{(A)}\ \frac{11}{81} \qquad

2 KB (343 words) - 23:06, 14 April 2022
2011 AMC 12A Problems/Problem 17
\textbf{(A)}\ \frac{3}{5} \qquad pair A=(0,0), B=(3,0), C=(0,4);

3 KB (462 words) - 17:49, 3 February 2024
1958 AHSME Problems
<math> \textbf{(A)}\ 5\qquad <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq

25 KB (3,872 words) - 14:21, 20 February 2020
1993 AHSME Problems
...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \

20 KB (2,814 words) - 08:15, 27 June 2021
2011 AMC 10A Problems/Problem 23
<math> \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\ ...math>, Barbara skips <math>5 \pmod 9</math>, Candice skips <math>14 \pmod {27}</math>, Debbie skips <math>41 \pmod {81}</math>, Eliza skips <math>122 \pm

6 KB (926 words) - 23:38, 8 April 2024
2011 AMC 12B Problems
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \ Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>,

13 KB (1,978 words) - 16:28, 12 July 2020
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> ...<math>y</math>-axis. The point <math>P</math> with coordinates <math>(-14,27)</math> in the original system has coordinates <math>(\alpha,\beta)</math>

10 KB (1,634 words) - 22:21, 28 December 2023
2011 AIME I Problems/Problem 3
...<math>y</math>-axis. The point <math>P</math> with coordinates <math>(-14,27)</math> in the original system has coordinates <math>(\alpha,\beta)</math> ...>L</math> has slope <math>\frac{5}{12}</math> and contains the point <math>A=(24,-1)</math>, we may write the point-slope equation for <math>L</math> as

4 KB (699 words) - 16:01, 13 April 2024
2011 AIME I Problems/Problem 7
...^i}</math>s into <math>m(m-1)</math> <math>\text{ }\frac{1}{m^{i+1}}</math>s since <math>(m-1)\cdot\frac{1}{m^{i}} = m(m-1)\cdot\frac{1}{m^{i+1}}</math> ...2^{2008} = 2^{2009} + 2^{2009} = 2^{2010}</cmath>. Thus <math>2</math> is a possible value of <math>m</math>. For other values, for example <math>m = 3

10 KB (1,673 words) - 19:27, 8 April 2024
2011 AIME I Problems/Problem 8
...math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <ma pair[] A, B, C, U, V, W, X, Y, Z;

6 KB (1,077 words) - 21:47, 12 April 2022
2011 AIME II Problems/Problem 12
...ates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the [[probability]] that each deleg ...country sit together. This comes to <cmath>\frac{27\cdot6\cdot5\cdot4}6 = 27\cdot 20.</cmath>

5 KB (848 words) - 19:15, 30 April 2023
2011 AIME II Problems/Problem 14
There are <math>N</math> [[permutation]]s <math>(a_{1}, a_{2}, ... , a_{30})</math> of <math>1, 2, \ldots, 30</math> ...that doesn't mean that the number 1 has to have <math>1 \mod 2</math>! It's that the POSITION which NUMBER 1 occupies has <math>1 \mod 2</math>!

10 KB (1,581 words) - 22:09, 27 August 2023
2011 AIME II Problems/Problem 15
...ath>, <math>d</math>, and <math>e</math> are positive integers. Find <math>a + b + c + d + e</math>. ...th> must be an integer and hence <math>P(\lfloor x \rfloor)</math> must be a perfect square. This limits <math>x</math> to <math>5 \le x < 6</math> or <

8 KB (1,273 words) - 14:03, 7 January 2023
2011 AIME II Problems/Problem 9
...ast <math>\frac{1}{540}</math> the desired maximum is at most <math>\frac1{27} - \frac1{540} =\frac{19}{540}</math>. It is easy to see that this upper bo ..._4 = x_6</math>, at which point we know that <math>a+b = 1/3</math>, <math>a^3+b^3 \geq 1/540</math>, and we are trying to maximize <math>3a^2b+3ab^2</m

3 KB (466 words) - 15:06, 16 January 2023
2011 AMC 12B Problems/Problem 18
...ith side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py ...7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3

3 KB (508 words) - 19:26, 7 August 2023
1999 AMC 8 Problems
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ What is the degree measure of the smaller angle formed by the hands of a clock at 10 o clock?

17 KB (2,394 words) - 19:51, 8 May 2023
2011 AMC 10B Problems
<math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad Josanna's test scores to date are <math>90, 80, 70, 60,</math> and <math>85</math>. H

13 KB (2,090 words) - 18:05, 7 January 2021
2003 AMC 10B Problems
<math>\textbf{(A)} -1 \qquad\textbf{(B)} -\frac{2}{3} \qquad\textbf{(C)} \frac{2}{3} \qquad\ ...<math> \ </math><math>1</math> more than a pink pill, and Al's pills cost a total of <math> \ </math><math>546</math> for the two weeks. How much does

15 KB (2,166 words) - 21:17, 16 February 2021
1984 AHSME Problems/Problem 9
...\qquad \mathrm{(C) \ } 29 \qquad \mathrm{(D) \ }28 \qquad \mathrm{(E) \ } 27 </math> ...final number is <math> 2^7 </math> with <math> 25 </math> <math> 0 </math>s annexed onto it. <math> 2^7=128 </math>, which has <math> 3 </math> digits,

917 bytes (126 words) - 12:49, 5 July 2013
1999 AMC 8 Problems/Problem 14
label("$A$",(0,0),SW); label("$16$",(8,0),S);

2 KB (253 words) - 21:16, 24 March 2022
1967 AHSME Problems
...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E

20 KB (3,108 words) - 14:14, 20 February 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
1998 AJHSME Problems/Problem 23
draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); <math> \text{(A)}\ \frac{3}{8}\qquad\text{(B)}\ \frac{5}{27}\qquad\text{(C)}\ \frac{7}{16}\qquad\text{(D)}\ \frac{9}{16}\qquad\text{(E)

2 KB (279 words) - 15:23, 29 May 2021
2005 CEMC Gauss (Grade 7) Problems
= Part A: Each correct answer is worth 5 points = <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad

16 KB (2,317 words) - 03:54, 24 October 2014
1999 AMC 8 Problems/Problem 13
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adult <math>\text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 29 \qquad \text{(E)}\ 30</math>

874 bytes (143 words) - 00:32, 21 November 2019
2009 AMC 8 Problems
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\

18 KB (2,551 words) - 18:46, 27 February 2024
2000 AMC 8 Problems/Problem 25
...\overline{BC} </math> and <math> \overline{CD} </math> are joined to form a triangle, the area of that triangle is pair A,B,C,D;

3 KB (484 words) - 13:59, 22 October 2023
1996 AJHSME Problems/Problem 14
label("$23$",(0.5,0),S); <math>\text{(A)}\ 27 \qquad \text{(B)}\ 29 \qquad \text{(C)}\ 31 \qquad \text{(D)}\ 33 \qquad \t

2 KB (254 words) - 00:24, 5 July 2013
1995 AJHSME Problems/Problem 4
A teacher tells the class, ''"Think of a number, add 1 to it, and double the result. Give the answer to your partne

986 bytes (149 words) - 00:15, 5 July 2013
1996 AHSME Problems/Problem 30
A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive s <math>\textbf{(A)}\ 309 \qquad \textbf{(B)}\ 349 \qquad \textbf{(C)}\ 369 \qquad \textbf{(D)

10 KB (1,515 words) - 13:09, 20 December 2023
1997 AHSME Problems
If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </m

17 KB (2,590 words) - 13:38, 19 February 2020
2008 AMC 8 Problems/Problem 20
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and <math>\frac{3}{4}</math> of the <math>\textbf{(A)}\ 12\qquad

1 KB (196 words) - 15:56, 11 January 2023
1996 AHSME Problems
<math> \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E ...After <math>10</math> days of doing his chores daily, Walter has received a total of <math>36</math> dollars. On how many days did Walter do them excep

15 KB (2,343 words) - 13:39, 19 February 2020
1996 AHSME Problems/Problem 12
A function <math>f</math> from the integers to the integers is defined as fol Suppose <math>k</math> is odd and <math>f(f(f(k))) = 27</math>. What is the sum of the digits of <math>k</math>?

2 KB (354 words) - 14:07, 5 July 2013
1996 AHSME Problems/Problem 28
...math> 4\times 4\times 3 </math> rectangular parallelepiped, vertices <math>A</math>, <math>B</math>, and <math>C</math> are adjacent to vertex <math>D</ <math>A</math>, <math>B</math>, and <math>C</math> is closest to

3 KB (545 words) - 10:21, 16 September 2022
2003 AMC 8 Problems/Problem 25
...gle ABC</math> is folded over side <math>\overline{BC}</math>, point <math>A</math> coincides with <math>O</math>, the center of square <math>WXYZ</math pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5);

2 KB (387 words) - 19:58, 15 April 2023
1950 AHSME Problems
<math> \textbf{(A)}\ 5\frac{1}{3}\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 10\frac{2}{3}\qqua ...math> R=gS-4 </math>. When <math>S=8</math>, <math>R=16</math>. When <math>S=10</math>, <math>R</math> is equal to:

22 KB (3,306 words) - 19:50, 3 May 2023
2009 PUMaC Problems
'''Algebra A''' Let <math>x_1,x_2,\dots,x_n</math> be a sequence of integers, such that <math>-1\leq x_i\leq 2</math>, for <math>i=

25 KB (4,154 words) - 16:27, 2 September 2011
1985 AHSME Problems
<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{( In an arcade game, the "monster" is the shaded sector of a circle of radius <math>1</math> cm, as shown in the figure. The missing pie

17 KB (2,488 words) - 03:26, 20 March 2024
1985 AHSME Problems/Problem 28
...\triangle ABC</math>, we have <math>\angle C = 3\angle A</math>, <math>a = 27</math> and <math>c = 48</math>. What is <math>b</math>? pair A=origin, B=(14,0), C=(10,6);

4 KB (657 words) - 02:54, 20 March 2024
2010 AMC 8 Problems
...students in Mr. Newton's class, and <math>9</math> students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid M <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math>

18 KB (2,768 words) - 21:05, 9 January 2024
2011 AMC 8 Problems
...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{

16 KB (2,371 words) - 17:34, 9 January 2024
Mathtime Version 1 Issue 2
...e the positive integer <math>n</math> such that <math>n^2+20n+40</math> is a perfect square. (Proposed by djmathman) ...umber of ordered integer pairs <math>(a,b)</math> such that <math>a^2+b^2|(a+b)^2</math>. (Proposed by djmathman)

15 KB (2,444 words) - 21:46, 1 January 2012
Mock Geometry AIME 2011 Problems/Problem 11
<math>C</math> is on a semicircle with diameter <math>AB</math> and center <math>O.</math> Circle label("$A$",(-12,0),WSW);

3 KB (432 words) - 14:12, 2 January 2012
2011 AMC 8 Problems/Problem 11
label("M",(30,-5),S); label("Tu",(70,-5),S);

2 KB (346 words) - 15:00, 17 December 2023
Mock Geometry AIME 2011 Problems/Problem 9
...dge length 6, and <math>PA=PB=PC=PD=6\sqrt{2}.</math> The probability that a randomly selected point inside the pyramid is at least <math>\frac{\sqrt{6} ...riangles are equilateral; <math>BD</math> is the diagonal of a square with a side length of <math>6</math> and so <math>BD=6\sqrt{2}=PB=PD</math>.

3 KB (501 words) - 17:54, 16 February 2018
1989 AHSME Problems
<math> \textrm{(A)}\ -7\qquad\textrm{(B)}\ -2\qquad\textrm{(C)}\ 0\qquad\textrm{(D)}\ 1\qquad <math> \textrm{(A)}\ \frac{1}5\qquad\textrm{(B)}\ \frac{1}4\qquad\textrm{(C)}\ \frac{2}7\qqua

15 KB (2,247 words) - 13:44, 19 February 2020
2012 AMC 10A Problems/Problem 19
...ed the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break? <math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qqu

3 KB (422 words) - 17:11, 21 August 2021
2012 AMC 12A Problems
{{AMC12 Problems|year=2012|ab=A}} A bug crawls along a number line, starting at <math>-2</math>. It crawls to <math>-6</math>, the

14 KB (2,197 words) - 13:34, 12 August 2020
2012 AMC 12A Problems/Problem 18
Triangle <math>ABC</math> has <math>AB=27</math>, <math>AC=26</math>, and <math>BC=25</math>. Let <math>I</math> be <math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26

4 KB (717 words) - 19:07, 28 July 2021
2012 AMC 12A Problems/Problem 22
...igcup_{j=1}^{k}p_{j} </math>. The intersection of <math>P</math> and <math>S</math> consists of the union of all segments joining the midpoints of every <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua

5 KB (940 words) - 17:13, 4 April 2020
1989 AHSME Problems/Problem 28
<math> \mathrm{(A) \frac{\pi}{2} } \qquad \mathrm{(B) \pi } \qquad \mathrm{(C) \frac{3\pi}{2 Because <math>\tan(x_1) \times \tan(x_2) = 1</math> by Vieta's formula, <math>x_1 + x_2 = 0.5\pi</math>.

2 KB (282 words) - 21:14, 2 March 2019
2012 AMC 12B Problems
<math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qqu A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width i

20 KB (2,681 words) - 09:47, 29 June 2023
2010 AMC 8 Problems/Problem 1
...students in Mr. Newton's class, and <math>9</math> students in Mrs. Young's class taking the AMC <math>8</math> this year. How many mathematics student <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math>

883 bytes (134 words) - 19:58, 5 February 2023
2012 AIME I Problems/Problem 3
Call a beef meal <math>B,</math> a chicken meal <math>C,</math> and a fish meal <math>F.</math> Now say the nine people order meals <math>\text{B ...h>B</math>'s can either both get <math>C</math>'s, both get <math>F</math>'s, or get one <math>C</math> and one <math>F.</math> We proceed with casework

3 KB (572 words) - 18:56, 13 June 2023
2012 AIME II Problems
...>b_1, b_2, b_3, \ldots</math> have the same common ratio, with <math>a_1 = 27</math>, <math>b_1=99</math>, and <math>a_{15}=b_{11}</math>. Find <math>a_9 ...science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and mus

7 KB (1,228 words) - 12:16, 13 March 2020
Mock AIME II 2012 Problems
...ng a positive integer greater than <math>1</math> and <math>m</math> being a positive integer greater than 2, find the smallest possible value of <math> ...as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the <math>\textit{digital root}</math> o

7 KB (1,274 words) - 21:16, 8 March 2021
2017 AMC 8 Problems
<math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo

12 KB (1,771 words) - 21:13, 20 January 2024
Mock AIME II 2012 Problems/Problem 9
...e any perfect squares greater than <math>1</math> as divisors. Find <math>a+b+c+d+e</math>. Here is a diagram (note that D should be on AC and F should be on BC):

7 KB (1,069 words) - 14:04, 27 December 2012
2012 USAJMO Problems/Problem 4
...rest adjacent points on either side of <math>P_n</math>. Prove that <math>a + b \le n</math>. ...h>, so that <math>c</math> or <math>d</math> equals to <math>1</math>. Let's consider the following two cases.

4 KB (697 words) - 09:23, 16 April 2017
1988 USAMO Problems/Problem 5
...mial <math>(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k</math>, where <math>a, b, \cdots, k</math> are integers. When expanded in powers of <math>x</math ...reverse the order of the coefficients of each factor, then we will obtain a polynomial whose coefficients are exactly the coefficients of <math>p(x)</m

8 KB (1,348 words) - 09:44, 25 June 2022
1974 AHSME Problems
<math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }

15 KB (2,151 words) - 14:04, 19 February 2020
2006 SMT/Algebra Problems/Problem 8
...ging in <math> x=-1, 0, </math> and <math> 1 </math>, we find that <math> (A,B,C)=(1,-2,1) </math>. ...-\frac{2}{2}, </math> and <math> \frac{1}{2} </math> in the beginning, and a <math> \frac{1}{10}, -\frac{2}{10} </math>, and <math> \frac{1}{11} </math>

1 KB (192 words) - 09:52, 28 May 2012
1952 AHSME Problems
If the radius of a circle is a rational number, its area is given by a number which is: ...ational} \qquad \textbf{(C)\ } \text{integral} \qquad \textbf{(D)\ } \text{a perfect square }\qquad \textbf{(E)\ } \text{none of these} </math>

23 KB (3,556 words) - 15:35, 30 December 2023
1954 AHSME Problems
<math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\

23 KB (3,535 words) - 16:29, 24 April 2020
1955 AHSME Problems
<math> \textbf{(A)}\ 3.75\times 10^{-7}\qquad\textbf{(B)}\ 3\frac{3}{4}\times 10^{-7}\qquad\t The smaller angle between the hands of a clock at <math>12:25</math> p.m. is:

22 KB (3,509 words) - 21:29, 31 December 2023
1980 AHSME Problems
<math>\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 The degree of <math>(x^2+1)^4 (x^3+1)^3</math> as a polynomial in <math>x</math> is

15 KB (2,302 words) - 10:47, 30 April 2021
1981 USAMO Problems/Problem 3
Given three angles that add to 180°, one can construct a triangle from them. This is true even if the angles are negative; however, ...rse triangle are: <math>0\degree\leq A,B,C\leq180\degree</math>, and <math>A+B+C=180\degree</math>.

5 KB (982 words) - 12:13, 17 December 2023
2008 AMC 8 Problems
<math>\textbf{(A)}\ 12 \qquad <math>\textbf{(A)}\ 8671 \qquad

14 KB (2,035 words) - 15:23, 26 January 2024
2012 AMC 12B Problems/Problem 20
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible are <math>\textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qqu

4 KB (670 words) - 07:14, 27 December 2022
Lagrange Multipliers
This article discusses Lagrange multipliers, a topic of multivariable calculus. ...e minimum or maximum satisfying both requirements (<math>\lambda</math> is a constant):

5 KB (791 words) - 21:06, 30 November 2020
2004 AMC 8 Problems/Problem 14
for(int a=0; a<=10; ++a) dot((a,b));

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2013 AMC 10A Problems/Problem 25
All <math>20</math> diagonals are drawn in a regular octagon. At how many distinct points in the interior <math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qqu

6 KB (1,054 words) - 09:21, 28 December 2021
2013 AMC 12A Problems/Problem 13
...th>, where these fractions are in lowest terms. What is <math> p + q + r + s </math>? <math> \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D

3 KB (447 words) - 19:29, 4 September 2022
2013 AMC 10A Problems/Problem 18
...t)</math>, where these fractions are in lowest terms. What is <math>p+q+r+s</math>? <math> \textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qqu

4 KB (592 words) - 22:19, 2 November 2023
2013 AMC 10B Problems/Problem 16
pair A,B,C,D,E,P; A=(0,0);

5 KB (761 words) - 19:33, 11 January 2024
2013 AMC 12B Problems
On a particular January day, the high temperature in Lincoln, Nebraska, was <mat <math>\textbf{(A)}\ -13 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\

16 KB (2,459 words) - 02:46, 30 January 2021
2013 AMC 12B Problems/Problem 19
...bf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30</math> pen s = linewidth(0.8)+fontsize(8);

6 KB (1,004 words) - 22:38, 18 June 2023
2013 AMC 12B Problems/Problem 21
...and <math>b\in \{-3,-2,-1,1,2,3\}</math>. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas? <math> \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D)}\ 840

2 KB (268 words) - 14:12, 11 August 2023
2013 AMC 10B Problems
<math> \textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{5}{36} \qquad\textbf{(C)}\ \frac{7}{12} \q ...f Mr. Green's steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of pota

12 KB (1,926 words) - 21:54, 6 October 2022
2013 AIME I Problems/Problem 5
...ath>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. ...omial <math>Q(x)=-x^3-3x^2-3x+8</math>. Note that <math>1/r</math> must be a root of <math>Q</math>. However we can simplify <math>Q</math> as <math>Q(x

3 KB (573 words) - 21:11, 27 August 2023
2013 AIME I Problems/Problem 11
...dergarten class has <math>16</math> registered students. The classroom has a very large number, <math>N</math>, of play blocks which satisfies the condi (a) If <math>16</math>, <math>15</math>, or <math>14</math> students are prese

9 KB (1,539 words) - 12:48, 20 December 2022
1991 AHSME Problems/Problem 10
Point <math>P</math> is <math>9</math> units from the center of a circle of radius <math>15</math>. How many different chords of the circle c (A) 11 (B) 12 (C) 13 (D) 14 (E) 29

2 KB (387 words) - 14:27, 23 June 2021
1991 AHSME Problems
...math>, <math>b</math>, and <math>c</math> we define <math>f(a,b,c)=\frac{c+a}{c-b}</math>, then <math>f(1,-2,-3)</math> is <math> \textbf{(A) } -2 \qquad \textbf{(B) } -\frac{2}{5} \qquad \textbf{(C) } -\frac{1}{4} \

16 KB (2,451 words) - 04:27, 6 September 2021
1991 AHSME Problems/Problem 11
...km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr d <math>\text{(A) } \frac{5}{4}\quad

2 KB (259 words) - 17:32, 23 February 2018
2013 AIME II Problems/Problem 5
...is an integer that is not divisible by the square of any prime. Find <math>a+b+c</math>. pair A = (1, sqrt(3)), B = (0, 0), C= (2, 0);

4 KB (617 words) - 19:46, 2 January 2024
2013 AIME II Problems/Problem 11
...set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <ma Any such function can be constructed by distributing the elements of <math>A</math> on three tiers.

3 KB (574 words) - 01:22, 1 February 2024
2003 AMC 12B Problems/Problem 11
...er watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her ...uad \text {(B) 10:24 PM} \qquad \text {(C) 10:25 PM} \qquad \text {(D) 10:27 PM} \qquad \text {(E) 10:30 PM}

1 KB (168 words) - 00:54, 5 January 2014
2014 AIME II Problems/Problem 11
...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. ...3}{22} = \frac{14 - 6\sqrt{3}}{22}</math>, and <math>AE = \frac{7 - \sqrt{27}}{22}</math>, so the answer is <math>\boxed{056}</math>.

6 KB (1,059 words) - 18:24, 20 January 2024
1967 AHSME Problems/Problem 16
...s=12+15+16</math>, each term expressed in base <math>b</math>. Then <math>s</math>, in base <math>b</math>, is <math>\textbf{(A)}\ 43\qquad

1 KB (176 words) - 01:39, 16 August 2023
2016 AMC 10A Problems
{{AMC10 Problems|year=2016|ab=A}} <math>\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\

14 KB (2,104 words) - 22:26, 16 September 2022
1962 AHSME Problems
<math>\textbf{(A)}\ \frac{4y-1}{8} \qquad <math>\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad

17 KB (2,459 words) - 22:40, 10 April 2023
2013 AMC 8 Problems
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \ ...lf-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)

15 KB (2,162 words) - 20:05, 8 May 2023
2013 AMC 8 Problems/Problem 15
...1</math>, what is the product of <math>p</math>, <math>r</math>, and <math>s</math>? <math>\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 70 \qq

2 KB (295 words) - 03:58, 29 December 2022
1994 AHSME Problems
<math> \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\t A large rectangle is partitioned into four rectangles by two segments paralle

14 KB (2,124 words) - 13:39, 19 February 2020
1981 AHSME Problems
<math> \textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\ <math> \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ \sqrt{5}\qquad\textbf{(C)}\ 3\qquad\textbf{(

17 KB (2,633 words) - 15:44, 16 September 2023
1968 AHSME Problems
Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter <math>\text{(A) } \frac{1}{\pi}\quad\text{(B) } \pi\quad\text{(C) } \frac{\pi^2}{2}\quad\t

16 KB (2,571 words) - 14:13, 20 February 2020
2014 AMC 10A Problems
{{AMC10 Problems|year=2014|ab=A}} <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}

15 KB (2,190 words) - 15:21, 22 December 2020
1994 AHSME Problems/Problem 26
A regular polygon of <math>m</math> sides is exactly enclosed (no overlaps, n <math> \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \q

1 KB (221 words) - 17:32, 9 January 2021
1960 AHSME Problems
If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad

21 KB (3,242 words) - 21:27, 30 December 2020
2014 AMC 12A Problems
{{AMC12 Problems|year=2014|ab=A}} <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}

12 KB (1,863 words) - 19:04, 11 April 2024
2014 AMC 12A Problems/Problem 20
<math>\textbf{(A) }6\sqrt 3+3\qquad \textbf{(B) }\dfrac{27}2\qquad

4 KB (665 words) - 04:35, 22 January 2024
2014 AMC 12A Problems/Problem 15
...sum of all five-digit palindromes. What is the sum of the digits of <math>S</math>? <math>\textbf{(A) }9\qquad

4 KB (612 words) - 22:42, 2 August 2021
2014 AMC 12B Problems
...d have the same number of pennies and nickels. In cents, how much are Leah's coins worth? <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qqu

13 KB (2,066 words) - 14:08, 1 November 2022
2014 AMC 10B Problems
...ld have the same number of pennies as nickels. In cents, how much are Leah's coins worth? <math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D)

13 KB (2,011 words) - 21:54, 8 November 2022
2014 AMC 10B Problems/Problem 11
For the consumer, a single discount of <math>n\%</math> is more advantageous than any of the fo (3) a <math>25\%</math> discount followed by a <math>5\%</math> discount

3 KB (414 words) - 18:03, 11 August 2023

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