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• 2016 AMC 10A Problems/Problem 12
  ...Which of the following is a correct statement about the probability <math>p</math> that the product of the three integers is odd? ...{1}{8}<p<\dfrac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}</math>
  2 KB (297 words) - 14:54, 25 June 2023
• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...3265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Joining an ARML team
  The SD team is entering just its third year of participation, so a permanent process of selecting team members has yet to be decided upon. The A student who wishes to attend practice should try to take the SDMO. If a student wishes to speak with one of the coaches for the team, they may do s
  21 KB (3,435 words) - 00:56, 23 May 2024
• Factorial
  ...!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n ...</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
  10 KB (809 words) - 16:40, 17 March 2024
• 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 ...ibility|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
• Parabola
  ...[[locus]] of points that are equidistant from a point (the [[focus]]) and a line (the [[directrix]]). ...la. The first is [[polynomial]] form: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial
  3 KB (551 words) - 16:22, 13 September 2023
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Divisibility rules
  ...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, A number is divisible by 3 or 9 if and only if the sum of its digits is divis
  8 KB (1,298 words) - 15:07, 23 May 2024
• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. ...</math> must [[#Importance of Primes|have]] a prime factor, which leads to a direct contradiction.
  6 KB (985 words) - 12:38, 25 February 2024
• Complementary counting
  ...ting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Combinatorial identity
  ...binatorially by noting that any combination of <math>r</math> objects from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> o if(n==8){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);}
  12 KB (1,996 words) - 12:01, 18 May 2024
• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
  5 KB (892 words) - 21:52, 1 May 2021
• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
  8 KB (1,469 words) - 21:11, 16 September 2022
• Goldbach Conjecture
  The '''Goldbach Conjecture''' is a yet unproven [[conjecture]] stating that every [[even integer]] greater tha <math>14 = 3 + 11 = 7 + 7</math>
  7 KB (1,201 words) - 16:59, 19 February 2024
• 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
• 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
• 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)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 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>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  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
• 2002 AMC 12A Problems
  {{AMC12 Problems|year=2002|ab=A}} <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mat
  12 KB (1,792 words) - 13:06, 19 February 2020
• 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
• 2001 AMC 12 Problems
  <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D) Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectivel
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc <math>\mathrm{(A)}\ 0
  10 KB (1,547 words) - 04:20, 9 October 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
• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 12
  ...(p,p)</math> and <math>y</math>-intercept <math>(0,-p)</math>, where <math>p\ne 0</math>. What is <math>b</math>? ...t {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) } p</math>
  1 KB (239 words) - 22:20, 17 October 2020
• 2006 AMC 12B Problems/Problem 13
  <asy> defaultpen(linewidth(0.7)+fontsize(11)); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/
  3 KB (447 words) - 03:49, 16 January 2021
• 2006 AMC 12B Problems/Problem 15
  ...s <math> C</math> and <math> D</math> are on the circle centered at <math> P</math>, such that <math> \overline{AD}</math> and <math> \overline{BC}</mat unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12B Problems/Problem 23
  ...re <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? pair A = (B.y,0);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2005 AMC 12B Problems/Problem 11
  {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #11]] and [[2005 AMC 10B Problems|2005 AMC 10B #15]]}} <math>\textbf{(A) }\ {{{\frac{1}{4}}}} \qquad \textbf{(B) }\ {{{\frac{2}{5}}}} \qquad \textb
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 12
  ...<math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>? <math>\textbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qq
  2 KB (317 words) - 12:27, 16 December 2021
• Set
  ...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one r
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 12
  ...r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 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 3
  ...p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>. ...h>47</math>) so there are <math> {15 \choose 2} =105</math> ways to choose a pair of primes from the list and thus <math>105</math> numbers of the first
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 10
  ...has [[slope]] <math> -5. </math> Find the largest possible value of <math> p+q. </math> pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22);
  5 KB (852 words) - 21:23, 4 October 2023
• 2004 AIME I Problems/Problem 7
  Let our [[polynomial]] be <math>P(x)</math>. ...h>P(x)</math> is <math>-1 + 2 - 3 + \ldots + 14 - 15 = -8</math>, so <math>P(x) = 1 -8x + Cx^2 + Q(x)</math>, where <math>Q(x)</math> is some polynomial
  6 KB (941 words) - 11:37, 27 May 2024
• 2004 AIME I Problems
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...he absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m.
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 14
  Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> ...</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 12
  ...k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...> and <math> q </math> are relatively prime positive integers. Find <math> p+q. </math> A positive integer <math>n</math> has exactly two 1s in its binary representa
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems
  ...a, b, c, d, e, </math> and <math> f </math> are positive integers, <math> a </math> and <math> e </math> are relatively prime, and neither <math> c </m A jar has 10 red candies and 10 blue candies. Terry picks two candies at rand
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  ...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...ue taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>.
  7 KB (1,104 words) - 03:13, 27 May 2024
• 1984 AIME Problems
  ...of <math>\triangle ABC</math> such that when lines are drawn through <math>P</math> parallel to the sides of <math>\triangle ABC</math>, the resulting s pair A=(0,0),B=(12,0),C=(4,5);
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  The pages of a book are numbered <math>1_{}^{}</math> through <math>n_{}^{}</math>. When t ...r point <math>P</math> is then drawn, and segments are drawn through <math>P</math> parallel to the sides of the triangle. If these three segments are o
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  ...er we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the p ...r which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>.
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(11)</math>. Let <math>\frac{m}{n}</math>, in lowest terms, be the probability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer mult
  6 KB (902 words) - 08:57, 19 June 2021
• 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
• 1990 AIME Problems
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 1991 AIME Problems
  Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerat Suppose <math>r^{}_{}</math> is a real number for which
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  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 :(a) the winner caught <math>15</math> fish;
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...\,</math> consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence ...circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...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 ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  ...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 ...ny of which could be zero. It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  ...Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> ...egative]] term encountered. What positive integer <math>x</math> produces a sequence of maximum length?
  7 KB (1,084 words) - 02:01, 28 November 2023
• 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
  A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow ...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- ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>.
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...centric circles with radii <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius 1 is colored red, and each regi ...et <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  ...[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...<math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 4
  ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< defaultpen(linewidth(0.6)+fontsize(11));
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 7
  ...three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? We will place <math>A</math>, <math>B</math>, and <math>C</math> with and without the restriction
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 14
  ...<math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. ...draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy>
  13 KB (2,151 words) - 17:48, 27 May 2024
• 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
• 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 ...when it has crawled exactly <math>k</math> meters. We wish to find <math>p=P(7).</math>
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 11
  ...of points <math>P</math> such that the sum of the distances between <math>P</math> and the two foci is constant. Let <math>F_1 = (9, 20)</math>, <math ...n our original path has shortest length. This occurs exactly when we have a straight-line path, and by the above argument, this path passes through the
  5 KB (932 words) - 17:00, 1 September 2020
• 1987 AIME Problems/Problem 3
  ...ositive]] [[integer|integral]] divisor other than 1 and the number itself. A natural number greater than 1 will be called ''nice'' if it is equal to the ...ath> denote the product of the distinct proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances:
  3 KB (511 words) - 09:29, 9 January 2023
• 1988 AIME Problems/Problem 13
  ...ath> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. 0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1 &\Longrightarrow (a
  10 KB (1,585 words) - 03:58, 1 May 2023
• 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>. <math>\frac {d}{a + d} = \frac {[PBC]}{[ABC]}</math>
  4 KB (727 words) - 23:37, 7 March 2024
• 1989 AIME Problems/Problem 12
  Let <math>ABCD</math> be a [[tetrahedron]] with <math>AB=41</math>, <math>AC=7</math>, <math>AD=18</ma pair A,B,C,D,M,P,Q;
  2 KB (376 words) - 13:49, 1 August 2022
• 1989 AIME Problems/Problem 10
  ...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find pair A = (0,0), B = (3, 0), C = (1, 4);
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 6
  ...the same time Allie leaves <math>A</math>, Billie leaves <math>B</math> at a speed of <math>7</math> meters per second and follows the [[straight]] path pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  6 KB (980 words) - 15:08, 14 May 2024
• 1990 AIME Problems/Problem 7
  .../math> can be written in the form <math>ax+2y+c=0_{}^{}</math>. Find <math>a+c_{}^{}</math>. pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17);
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1991 AIME Problems/Problem 13
  ...s that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\frac{1}{2}</math> that both are red or both a ...e socks, respectively. Also, let <math>t=r+b</math>. The probability <math>P</math> that when two socks are drawn randomly, without replacement, both ar
  7 KB (1,328 words) - 20:24, 5 February 2024
• 1991 AIME Problems/Problem 12
  ...in [[rectangle]] <math>ABCD^{}_{}</math> so that [[vertex|vertices]] <math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{} ...el("\(Q\)",Q,E);label("\(R\)",R,SW);label("\(S\)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW); </asy></center>
  8 KB (1,270 words) - 23:36, 27 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 Let us make a chart of values in alphabetical order, where <math>P_a,\ P_b</math> are the
  5 KB (813 words) - 06:10, 25 February 2024
• 1993 AIME Problems/Problem 12
  ...ing the rule: If the die shows label <math>L\,</math>, where <math>L \in \{A, B, C\}</math>, and <math>P_n\,</math> is the most recently obtained point, ...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
• 1993 AIME Problems/Problem 10
  ...of its <math>V</math> vertices, <math>T</math> triangular faces and <math>P</math> pentagonal faces meet. What is the value of <math>100P+10T+V</math>? ...+p=F=32</math> faces. In each vertex, <math>T=2</math> triangles and <math>P=2</math> pentagons are concurrent. Now, the number of edges <math>E</math>
  4 KB (716 words) - 20:50, 17 April 2022
• 1994 AIME Problems/Problem 9
  ...> and <math>q\,</math> are relatively prime positive integers. Find <math>p+q.\,</math> *Case 1. We draw a pair on the first two cards. The second card is the same as the first with
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 5
  ...of <math>n\,</math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let <center><math>S=p(1)+p(2)+p(3)+\cdots+p(999)</math></center>.
  2 KB (275 words) - 19:27, 4 July 2013
• 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
• 1996 AIME Problems/Problem 12
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. ...math>5</math> of these pairs / differences, our answer is just <math>\frac{11}{3} \cdot 5 = {\frac{55}{3}}</math>.
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 11
  ...> that have a positive [[imaginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</mat ...e left with <math>\mathrm{cis}\ 60, 72, 144</math>; their product is <math>P = \mathrm{cis} (60 + 72 + 144) = \mathrm{cis} \boxed{276}</math>.
  6 KB (1,022 words) - 20:23, 17 April 2021
• 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>. ...>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 15
  ...th>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 12
  ...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which r
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1998 AIME Problems/Problem 14
  ...and <math>m\le n\le p.</math> What is the largest possible value of <math>p</math>? <cmath>2mnp = (m+2)(n+2)(p+2)</cmath>
  2 KB (390 words) - 21:05, 29 May 2023
• 1998 AIME Problems/Problem 12
  ...h>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}</math> so <math>a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = \boxed{083}</math>.
  5 KB (876 words) - 20:27, 9 June 2022
• 1999 AIME Problems/Problem 12
  pair A=(0,0),B=(50,0),C=IP(circle(A,23+245/2),circle(B,27+245/2)), I=incenter(A,B,C); path P = incircle(A,B,C);
  3 KB (472 words) - 15:59, 25 February 2022
• 1999 AIME Problems/Problem 1
  ...However, if the common difference is <math>6</math>, we find that <math>5,11,17,23</math>, and <math>29</math> form an [[arithmetic sequence]]. Thus, th ...terms of the sequence are all prime numbers. The sequence becomes <math>5, 11, 17, 23, 29</math>, so the answer is <math>029</math>.
  2 KB (332 words) - 13:22, 3 August 2020
• 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
• 2003 AIME I Problems/Problem 12
  ...f <math> ABCD </math> is <math> 640 </math>. Find <math> \lfloor 1000 \cos A \rfloor. </math> (The notation <math> \lfloor x \rfloor </math> means the g pair A=(0,0),B=(1.8,0),D=IP(CR(A,x),CR(B,BD)),C=OP(CR(D,1.8),CR(B,2.80 - x));
  3 KB (487 words) - 22:14, 24 November 2019
• 2003 AIME II Problems/Problem 11
  ...h>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> <math>DM=\sqrt{15^2-\left(\frac{25} {2}\right)^2}=\frac{5} {2} \sqrt{11}.</math>
  5 KB (772 words) - 19:47, 1 August 2023
• 2003 AIME II Problems/Problem 9
  ...oots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math> When we use long division to divide <math>P(x)</math> by <math>Q(x)</math>, the remainder is <math>x^2-x+1</math>.
  3 KB (475 words) - 21:53, 6 May 2024
• 2002 AIME II Problems/Problem 13
  ...ath>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>P.</math> Points <math>Q</math> and <math>R</math> lie on <math>\overline{AB} pair A,B,C,D,E,X,P,Q,R;
  6 KB (935 words) - 13:23, 3 September 2021
• 2002 AIME II Problems/Problem 12
  ...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...ne <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2002 AIME II Problems/Problem 10
  ..., <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>. ...pi - \frac{\pi x}{180} \Rightarrow x = \frac{180\pi}{180+\pi} \Rightarrow (p,q) = (180,180)</math>.
  2 KB (336 words) - 19:30, 24 June 2020
• 2002 AIME II Problems/Problem 7
  ...ve integer <math>k</math> such that <math>1^2+2^2+3^2+\ldots+k^2</math> is a multiple of <math>200</math>. ...}</math> is a multiple of <math>200</math> if <math>k(k+1)(2k+1)</math> is a multiple of <math>1200 = 2^4 \cdot 3 \cdot 5^2</math>.
  3 KB (403 words) - 12:10, 9 September 2023
• 2002 AIME II Problems/Problem 2
  ...[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? <math>PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}</math>
  709 bytes (103 words) - 03:32, 6 December 2019
• 2001 AIME II Problems/Problem 10
  ...{3} - 1)</math>, we see that <math>j-i = 6</math> works; also, <math>a-b | a^n - b^n</math> implies that <math>10^{6} - 1 | 10^{6k} - 1</math>, and so a ...an easily verify that <math>10^6 - 1 \nmid 10^a - 1</math> for <math>1 \le a \le 5</math>.
  4 KB (549 words) - 23:16, 19 January 2024
• 2000 AIME II Problems/Problem 10
  ...al]] <math>ABCD</math>, [[tangent]] to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <mat Take the <math>\tan</math> of both sides and use the identity for <math>\tan(A+B)</math> to get <cmath>\tan(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))
  2 KB (399 words) - 17:37, 2 January 2024
• 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{
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...t a price of five for &#36;<math>2</math>. They sell all the candy bars at a price of two for &#36;<math>1</math>. What was the profit, in dollars? <math>\mathrm{(A)} 100 \qquad \mathrm{(B)} 200 \qquad \mathrm{(C)} 300 \qquad \mathrm{(D)} 4
  12 KB (1,874 words) - 21:20, 23 December 2020
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[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
• 2000 AMC 12 Problems/Problem 8
  draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); <math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{
  8 KB (1,202 words) - 16:17, 10 May 2024
• 2000 AMC 12 Problems/Problem 6
  ...0 AMC 12 Problems|2000 AMC 12 #6]] and [[2000 AMC 10 Problems|2000 AMC 10 #11]]}} <math>\textbf{(A)}\ 22 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 119 \qquad\textbf{(D)}\ 18
  1 KB (228 words) - 19:31, 29 April 2024
• Angle bisector theorem
  ...$",D,(0,-1)); dot(A^^B^^C^^D,blue);label("$b$",(A+C)/2,(1,0));label("$c$",(A+B)/2,(0,1));label("$m$",(B+D)/2,(0,-1));label("$n$",(D+C)/2,(0,-1)); </asy> # Let ABC be a triangle with angle bisector AD with D on line segment BC. If <math> BD =
  3 KB (438 words) - 14:20, 4 March 2023
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 5
  For a [[prime number]] <math>p</math>, define the [[function]] <math>f_p(n)</math> as follows: If there exists <math>y</math>, <math>0 \leq y < p</math>, such that
  2 KB (340 words) - 15:52, 3 April 2012
• 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 ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems
  ...r positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. ...ll positive integers <math>n</math> such that <math>n^2+12n-2007</math> is a perfect square. Find the remainder when <math>S</math> is divided by <math>
  5 KB (848 words) - 23:49, 25 February 2017
• Expected value
  Given an event with a variety of different possible outcomes, the '''expected value''' is what on A video that goes over the type of Expected value, practical examples, and pr
  5 KB (789 words) - 20:56, 10 May 2024
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2005 AMC 10A Problems/Problem 24
  ...th>n</math> is it true that both <math> P(n) = \sqrt{n} </math> and <math> P(n+48) = \sqrt{n+48} </math>? <math> \textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qqua
  6 KB (895 words) - 13:52, 4 April 2024
• 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
• 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
• 2001 AMC 8 Problems
  ...class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ball. If it takes him <math>2</math> seconds to paint one dimple, how <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D)
  13 KB (1,994 words) - 13:04, 18 February 2024
• 1985 AIME Problems
  When a right triangle is rotated about one leg, the volume of the cone produced is ...>b</math>, and <math>c</math> are positive integers which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>.
  7 KB (1,071 words) - 19:24, 23 February 2024
• 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
• Asymptote: Advanced Configuration
  ...rs commonly would like to do with their figures: make the images output in a different format, and include Asymptote code directly into LaTeX files. ...ly make your images in pdf format rather than eps format, with the help of a program called '''GhostScript'''.
  12 KB (1,931 words) - 13:53, 26 January 2020
• 2002 AMC 12P Problems/Problem 3
  The dimensions of a rectangular box in inches are all positive integers and the volume of the b <math>\text{(A) }36 \qquad \text{(B) }38 \qquad \text{(C) }42 \qquad \text{(D) }44 \qquad
  1 KB (197 words) - 22:07, 30 December 2023
• 2007 AIME II Problems/Problem 9
  ...iangle]] <math>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to ...thing for <math>F</math>, we find that <math>FD = 105</math> as well. Draw a line through <math>E,F</math> parallel to the sides of the rectangle, to in
  5 KB (818 words) - 11:05, 7 June 2022
• 2007 AIME II Problems/Problem 10
  .../math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) **<math>A</math> must have either 0 or 6 elements, probability: <math>\frac{2}{2^6} =
  8 KB (1,367 words) - 11:48, 23 October 2022
• Mock AIME 1 Pre 2005 Problems
  ...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 ...th>1 + 7 + 7^2 + \cdots + 7^{2004}</math> is divided by <math>1000</math>, a remainder of <math>N</math> is obtained. Determine the value of <math>N</ma
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems
  A function <math>f(x)</math> is defined for all real numbers <math>x</math>. ...are consonants. A string of <math>M's, O's,</math> and <math>P's</math> is a word in Zuminglish if and only if between any two <math>O's</math> there ap
  7 KB (1,135 words) - 23:53, 24 March 2019
• 1960 IMO Problems
  ...g the property that <math>N</math> is divisible by 11, and <math>\dfrac{N}{11}</math> is equal to the sum of the squares of the digits of <math>N</math>. ...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma
  3 KB (511 words) - 21:21, 20 August 2020
• 2007 AIME I Problems
  ...ly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the oth ...mber <math>z</math> is equal to <math>9+bi</math>, where <math>b</math> is a positive real number and <math>i^{2}=-1</math>. Given that the imaginary p
  7 KB (1,218 words) - 15:28, 11 July 2022
• 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 ...nother row that is complementary to the first. We remove those two and use a similar argument again to show that every group of <math>6</math> rows can
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2007 AIME I Problems/Problem 12
  ...{6} + s</math>, where <math>p,q,r,s</math> are integers. Find <math>\frac{p-q+r-s}2</math>. ...sectorpoint(A,B)), Cp=rotate(theta,A)*C, Bp=rotate(theta,A)*B, X=extension(A,Bp,B,C), Y=extension(B,C,Bp,Cp);
  10 KB (1,458 words) - 20:50, 3 November 2023
• 2007 AIME II Problems
  ...rs among the four letters in AIME or the four digits in <math>2007</math>. A set of plates in which each possible sequence appears exactly once contains ...of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 1989 USAMO Problems
  ...tegers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>T_{1988} = a S_{1989} - b</math> and <math>U_{1988} = c S_{1989} - d</math>. ...laying in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
  2 KB (326 words) - 18:52, 18 July 2016
• 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
• 2006 AIME II Problems/Problem 3
  ...[[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math> ...oor \frac{100}{27}\right\rfloor+\left\lfloor\frac{100}{81}\right\rfloor=33+11+3+1=48 </math>
  4 KB (562 words) - 18:37, 30 October 2020
• 2006 AIME II Problems/Problem 12
  ...| divisible]] by the [[perfect square | square]] of any prime, find <math> p+q+r. </math> pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D;
  6 KB (1,033 words) - 02:36, 19 March 2022
• 2006 AIME II Problems/Problem 13
  ...c{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)</math>. Thus, <math>j</math> is a factor of <math>N</math>. <cmath>(3^2\cdot5^2),\ (3^2\cdot7^2),\ (3^4\cdot5),\ (3^4\cdot7),\ (3^4\cdot 11)</cmath>
  4 KB (675 words) - 10:40, 14 July 2022
• 2006 AIME II Problems/Problem 10
  ...</math> beats team <math> B. </math> The [[probability]] that team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, ...We let this probability be <math>p</math>; then the probability that <math>A</math> and <math>B</math> end with the same score in these five games is <m
  6 KB (983 words) - 13:42, 8 December 2021
• 2006 AIME II Problems
  ...vex hexagon <math>ABCDEF</math>, all six sides are congruent, <math>\angle A</math> and <math>\angle D</math> are right angles, and <math>\angle B, \ang ...\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>.
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1998 PMWC Problems
  triangle((11,0),4); squ((11,-6),4);
  11 KB (1,738 words) - 19:25, 10 March 2015
• 1999 PMWC Problems
  label("$A$", (3.9,6.75), NE * labelscalefactor); pair P,Q,R,S;
  6 KB (703 words) - 21:21, 21 April 2014
• 2000 PMWC Problems
  <math>y</math> is a number that has <math>8</math> different factors (including the number <mat Given that <math>A^4=75600\times B</math>. If <math>A</math> and <math>B</math> are positive integers, find the smallest value of
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2001 PMWC Problems
  void wdot(pair p) fill(circle(p,.025),white);
  4 KB (641 words) - 21:24, 21 April 2014
• 2003 PMWC Problems
  pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); draw((M--N)^^(origin--a)^^(origin--b));
  7 KB (918 words) - 16:15, 22 April 2014
• 2007 AMC 12B Problems
  <math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{ A college student drove his compact car 120 miles home for the weekend and av
  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
• 2003 Polish Mathematical Olympiad Problems/Problem 3
  ..."natural number" is ambiguous, but this is irrelevant, as every integer is a divisor of <math>2^0 -1 = 0</math>.'' ..., <math>2^q \equiv 2 \pmod{q}</math>, so <math>2 \equiv 1 \pmod{q}</math>, a contradiction.
  5 KB (919 words) - 23:29, 20 January 2016
• 2000 AMC 12 Problems/Problem 10
  The point <math>P = (1,2,3)</math> is reflected in the <math>xy</math>-plane, then its image \text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1,
  1 KB (173 words) - 08:56, 21 August 2023
• 2008 AMC 12B Problems/Problem 23
  <math>\text{(A)}\ 11\qquad \text{(B)}\ 12\qquad \text{(C)}\ 13\qquad \text{(D)}\ 14\qquad \text{ ...h> is a perfect square. When these are multiplied, they equal <math>2^{a+n-a} \times 5^{b+n-b} = 10^n</math>. <math>\log 10^n=n</math> so the number of
  5 KB (814 words) - 18:02, 17 January 2023
• 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
• 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
  ...represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{ ...\hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array} </cmath>
  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
• 2006 Alabama ARML TST Problems/Problem 10
  Let <math>p</math> be the probability that Scooby Doo solves any given mystery. The pro ...m{2006}{1800}\cdot p^{1800}(1-p)^{206}=\binom{2006}{1801}\cdot p^{1801} (1-p)^{205}</cmath>
  1,003 bytes (140 words) - 20:29, 20 May 2012
• 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
• 2002 AMC 10A Problems/Problem 22
  ...fied repeatedly by the following operation: remove all tiles numbered with a [[perfect square]], and renumber the remaining tiles consecutively starting <math>\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20</math>
  3 KB (430 words) - 23:13, 13 September 2023
• 2008 AMC 10A Problems
  {{AMC10 Problems|year=2008|ab=A}} ...nut machine at <math>\text{8:30}\ {\small\text{AM}}</math>. At <math>\text{11:10}\ {\small\text{AM}}</math> the machine has completed one third of the da
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 5
  <math>\begin{matrix} a_{11}x_{1} + \ldots + a_{1q}x_{q} = 0, \\ ...om the set <math>\{ - 1,0,1\}</math><math>.</math> Prove that there exists a solution <math>x_{1}, \ldots,x_{q}</math> for the system with the propertie
  2 KB (377 words) - 16:28, 29 January 2021
• Partition (combinatorics)
  ...partitions of 3: <math>3 = 2+1 =1+1+1</math>. Each of the [[summand]]s is a ''part'' of the partition. ...ence of a formula! No simpler formula is known, and the existence of such a formula is doubtful.
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2008 AMC 12A Problems
  {{AMC12 Problems|year=2008|ab=A}} ...nut machine at <math>\text{8:30}\ {\small\text{AM}}</math>. At <math>\text{11:10}\ {\small\text{AM}}</math> the machine has completed one third of the da
  13 KB (2,025 words) - 13:56, 2 February 2021
• 1992 USAMO Problems/Problem 3
  ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible ...ts Y\subseteq Z</math>, <math>\sigma(Y)=x</math>, <math>\nexists \sigma(Y)=p+1</math>.
  5 KB (858 words) - 07:52, 19 July 2016
• 2008 AMC 12B Problems
  A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2006 Alabama ARML TST Problems
  River draws four cards from a standard 52 card deck of playing cards. Exactly 3 of them are 2’s. Find t There exist positive integers <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> with no common f
  5 KB (769 words) - 20:56, 24 March 2015
• 2008 AMC 12B Problems/Problem 25
  ...Bisectors of <math>\angle A</math> and <math>\angle D</math> meet at <math>P</math>, and bisectors of <math>\angle B</math> and <math>\angle C</math> me <math>\textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}
  12 KB (2,015 words) - 20:54, 9 October 2022
• 1985 AJHSME Problems
  <math>\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=</math> <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)} \fr
  12 KB (1,670 words) - 17:42, 24 November 2021
• 2008 AIME I Problems
  Of the students attending a school party, <math>60\%</math> of the students are girls, and <math>40\%</ ...e without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. Th
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  ...te which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin ...by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> cm and is cut parallel to one face of the cheese. T
  7 KB (1,167 words) - 21:33, 12 August 2020
• Mock AIME 5 Pre 2005 Problems
  &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}</cmath>
  6 KB (909 words) - 07:27, 12 October 2022
• 2005 USAMO Problems/Problem 2
  ...th>157 \equiv -147 \equiv 5 \pmod{19}</math>. For reference, we construct a table of powers of five: Evidently, the order of 5 is 9. Hence 5 is the square of a multiplicative generator of the nonzero integers mod 19, so this table show
  7 KB (1,053 words) - 10:38, 12 August 2015
• Cone
  ...three-dimensional [[solid]]. It consists of a [[circle | circular]] base, a [[point]] (called the ''vertex''), and all the points that lie on [[line se path3 rightanglemark(triple A, triple B, triple C, real s=8) { // olympiad package
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2008 AMC 10B Problems/Problem 10
  Points <math>A</math> and <math>B</math> are on a circle of radius <math>5</math> and <math>AB=6</math>. Point <math>C</math> <math>\mathrm{(A)}\ \sqrt{10}\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ \sqrt{14}\qq
  1 KB (247 words) - 12:36, 7 June 2021
• 2008 Mock ARML 2 Problems
  <math>ABCD</math> is a convex quadrilateral such that <math>|AB| = 5</math>, <math>|BC| = 17</math ...which is less than <math>30</math> (a proper divisor of <math>n</math> is a positive integer that divides but is not equal to <math>n</math>).
  4 KB (582 words) - 21:57, 8 May 2019
• Sylow Theorems
  ...nd are one of the most important results in the field. They are named for P. Ludwig Sylow, who published their proof in 1872. Throughout this article, <math>p</math> will be an arbitrary prime.
  11 KB (2,071 words) - 12:25, 9 April 2019
• 2003 AMC 12B Problems/Problem 18
  ...math>x</math> has a prime factorization <math>a^cb^d.</math> What is <math>a + b + c + d?</math> <math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\
  6 KB (914 words) - 11:07, 7 September 2023
• KGS math club
  A group of people on [http://www.gokgs.com Kiseido Go Server] Mathematics roo ...our, or of a less-than-or-equal rank. What is the greatest possible length a parade can reach?
  22 KB (3,358 words) - 15:17, 18 July 2017
• 1999 AHSME Problems/Problem 29
  ...cted at random inside the circumscribed sphere. The probability that <math>P</math> lies inside one of the five small spheres is closest to <math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D
  3 KB (522 words) - 11:39, 3 October 2023
• 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
• 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
• 2008 iTest Problems/Problem 99
  ...the areas of <math>P_{10}</math> and <math>P_{\infty}</math> is written as a fraction <math>\frac{x}{y}</math> in lowest terms, calculate the number of .../9 -- (A+D)/3+2*((A+B)/3 - (A+D)/3)/3, d); D(4D/9 -- (A+D)/3 + ((A+B)/3 - (A+D)/3)/3, d);
  4 KB (685 words) - 14:39, 7 October 2017
• 2008 iTest Problems/Problem 48
  A repunit is a natural number whose digits are all <math>1</math>. For instance, <cmath>1,11,111,1111, \ldots</cmath>
  988 bytes (145 words) - 17:14, 14 July 2018
• 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
• 2002 AMC 12A Problems/Problem 16
  ...from the set <math>\{ 1, 2, 3, 4, 5 \}</math>, and Sergio randomly selects a number from the set <math>\{ 1, 2, ..., 10 \}</math>. What is the probabili ...A)}\ 2/5 \qquad \text{(B)}\ 9/20 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 11/20 \qquad \text{(E)}\ 24/25</math>
  6 KB (927 words) - 02:00, 1 February 2024
• 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
• 1973 USAMO Problems
  Two points <math>P</math> and <math>Q</math> lie in the interior of a regular tetrahedron <math>ABCD</math>. Prove that angle <math>PAQ<60^o</mat <center> <math>X:1, 1, 3, 5, 11, 21, \dots</math>,</center>
  2 KB (273 words) - 18:53, 3 July 2013
• 2001 AMC 10 Problems
  ...f{(B) } 6 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 10 \qquad\textbf{(E) } 11</math> A number <math>x</math> is <math>2</math> more than the product of its recipr
  14 KB (1,983 words) - 16:25, 2 June 2022
• 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 2
  ...>AX</math> and <math>BY</math>. You may assume that <math>AB</math> is not a diameter. ...(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1;
  5 KB (848 words) - 23:41, 6 July 2020
• 1976 USAMO Problems
  ...or white. Prove that with <i>any</i> such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as * (b) Exhibit a black-white coloring of a <math>4\times 6</math> board in which the four corner squares of every rect
  3 KB (510 words) - 19:01, 3 July 2013
• 1985 AJHSME Problems/Problem 18
  ...en copies of the same pamphlet (at the same price) cost more than <math>\$ 11.00</math>. How much does one copy of this pamphlet cost? <math>\text{(A)}</math> <math>\$1.07</math>
  1 KB (235 words) - 09:03, 22 January 2023
• 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 13
  ...5</math> mm; dime, <math>1.35</math> mm; quarter, <math>1.75</math> mm. If a stack of these coins is exactly <math>14</math> mm high, how many coins are ...\qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 11 </math>
  3 KB (454 words) - 17:27, 8 January 2024
• 2001 AMC 12 Problems/Problem 19
  ...cients are all equal. The <math>y</math>-intercept of the graph of <math>y=p(x)</math> is 2. What is <math>b</math>? <math>(\mathrm{A})\ -11 \qquad (\mathrm{B})\ -10 \qquad (\mathrm{C})\ -9 \qquad (\mathrm{D})\ 1 \qq
  1 KB (179 words) - 22:48, 18 August 2023
• 1976 IMO Problems
  ...e area of <math>32 \text{ cm}^{2}</math> the sum of two opposite sides and a diagonal is <math>16 \text{ cm}</math>. Determine all the possible values t A box whose shape is a parallelepiped can be completely filled with cubes of side <math>1.</math>
  2 KB (409 words) - 16:25, 29 January 2021
• 2009 AMC 12A Problems
  {{AMC12 Problems|year=2009|ab=A}} <math>\textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\
  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
• 2001 AMC 12 Problems/Problem 10
  path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; draw(shift(3*i,3*j)*p);
  2 KB (252 words) - 00:11, 15 August 2022
• 2001 AMC 12 Problems/Problem 23
  ...two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? \text{(A) }\frac {1 + i \sqrt {11}}{2}
  6 KB (1,035 words) - 09:18, 3 September 2023
• 2009 AMC 12B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\t
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 25
  path p = scale(.15)*unitcircle; ...if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp);
  15 KB (2,229 words) - 03:36, 4 September 2021
• 2009 AMC 10B Problems
  ...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E
  15 KB (2,262 words) - 00:53, 18 June 2021
• Mock AIME 1 Pre 2005 Problems/Problem 10
  ...P \cdot DP \cdot EP \cdot FP \cdot GP</math>. Determine the value of <math>p^2</math>. {{Mock AIME box|year=Pre 2005|n=1|num-b=9|num-a=11|source=14769}}
  540 bytes (96 words) - 00:28, 23 December 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 12
  ...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat ...<math>P</math>,<math>Q</math> be the two points of tangent such that <math>P</math> is on <math>BI</math> and <math>Q</math> is on <math>AI</math>. We k
  12 KB (1,970 words) - 22:53, 22 January 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>. defaultpen(fontsize(11)+0.8); size(300);
  6 KB (1,048 words) - 19:35, 2 January 2023
• 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 .../math>, <math>b^{\log_7 11} = 49</math>, and <math>c^{\log_{11}25} = \sqrt{11}</math>. Find
  8 KB (1,366 words) - 21:33, 3 January 2021
• Missouri Mathcounts
  ...nce at nationals can be described as spastic. Reaching a high of 2nd, with a people coming in second or making the semifinals of the countdown, to team * 2006 - George Yu (11), Amadeus Zhu, Michael Zhao, Owen Hill, Coach: Mike Rowson
  4 KB (624 words) - 09:40, 29 April 2024
• 2009 AIME II Problems/Problem 8
  ...for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let <math>m</math> and <math>n</math> be re ...that Dave will make at least two more throws than Linda. Obviously, <math>p</math> is then also the probability that Linda will make at least two more
  5 KB (859 words) - 19:54, 27 January 2024
• 2009 AIME II Problems/Problem 7
  ...hen <math>\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}</math> is expressed as a fraction in lowest terms, its denominator is <math>2^ab</math> with <math>b ...terms, its denominator will be of the form <math>2^a</math> for some <math>a\leq 2\cdot 2009</math>.
  8 KB (1,312 words) - 16:23, 30 March 2024
• 2009 AIME II Problems/Problem 10
  ...and <math>r</math> is not divisible by the square of any prime. Find <math>p</math> + <math>q</math> + <math>r</math>. ...),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30));
  9 KB (1,415 words) - 13:56, 18 December 2022
• Determinant
  For an <math>n\times n</math> matrix <math>a = (a_{ij})</math>, the determinant is defined by \det a &= \sum_{\sigma \in S_n} \text{sign} (\sigma) a_{1\sigma(1)}
  8 KB (1,345 words) - 00:31, 9 May 2020
• 1990 AJHSME Problems
  <math>\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1 <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad
  15 KB (2,059 words) - 15:03, 6 October 2021
• 1990 AJHSME Problems/Problem 10
  ...ext{C}</math>. If this sum equals the sum of the dates behind <math>\text{A}</math> and <math>\text{B}</math>, then the letter is for(int a=0; a<6; ++a)
  1 KB (231 words) - 10:51, 8 September 2019
• 1965 IMO Problems
  <cmath>a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0</cmath> (a) <math>a_{11}</math>, <math>a_{22}</math>, <math>a_{33}</math> are positive numbers;
  3 KB (497 words) - 12:39, 29 January 2021
• 2003 AMC 12A Problems/Problem 11
  == Problem 11 == ...math> the area of the circle circumscribed around the triangle. Find <math>A/B</math>.
  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
• 2010 AMC 12A Problems
  {{AMC12 Problems|year=2010|ab=A}} <math>\textbf{(A)}\ -4020 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 12A Problems/Problem 10
  The first four terms of an arithmetic sequence are <math>p</math>, <math>9</math>, <math>3p-q</math>, and <math>3p+q</math>. What is t <math>\textbf{(A)}\ 8041 \qquad \textbf{(B)}\ 8043 \qquad \textbf{(C)}\ 8045 \qquad \textbf{
  1 KB (178 words) - 20:47, 27 October 2022
• 2010 AMC 12A Problems/Problem 23
  <math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ ...(if in doubt, look at the answer choices), we only need to consider <math>P \mod 25</math>.
  10 KB (1,525 words) - 09:44, 24 April 2024
• 2010 AMC 10A Problems
  {{AMC10 Problems|year=2010|ab=A}} \mathrm{(A)}\ 1
  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
• 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
• Mock AIME 1 2010 Problems
  ...>20 \sqrt{2}</math>, and let <math>S</math> be the set of all points <math>P</math> such that <math>m \angle APB \geq 45^{\circ}</math>. Find the last t ...ment in which every player plays every other player exactly once. There is a round robin tournament with 2010 people. In each match, the winner scores o
  7 KB (1,297 words) - 01:29, 25 November 2016
• Proofs without words
  Engel 313 E12, 316 6, 11, 323 4, 326/7 rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2)
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems
  ...ly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form <math>\frac {m}{n}</math>, where <math> ...math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 6
  ...l numbers <math>x</math>, and suppose <math>P(11) = 181</math>. Find <math>P(16)</math>. real P(real x) { return 8*(x-1)^2/5+1; }
  6 KB (1,019 words) - 20:39, 20 November 2023
• 2010 AIME I Problems/Problem 15
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. label("$A$",(-1.04408,-0.60958),NE);
  14 KB (2,210 words) - 13:14, 11 January 2024
• Mock AIME 2 2010 Problems
  ...where <math>a, b</math> are relatively prime positive integers. Find <math>a+b</math>. ...ath> and <math>b</math> are relatively prime positive integers, find <math>a + b</math>.
  7 KB (1,150 words) - 09:10, 8 October 2018
• 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
• 2010 AIME II Problems
  ...losest side of <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</ ...where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such t
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 11
  Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties: ...he long diagonals are <math>\{a_{13},a_{22},a_{31}\}</math> and <math>\{a_{11},a_{22},a_{33}\}</math>, no more than one of the eight has all three entrie
  6 KB (1,057 words) - 01:58, 8 January 2023
• 1992 USAMO Problems
  Find, as a function of <math>\, n, \,</math> the sum of the digits of ...integer <math>n \le 1500</math>, there is a subset <math>S</math> of <math>A</math> for which <math>\sigma(S) = n</math>. What is the smallest possible
  2 KB (310 words) - 20:54, 3 July 2013
• 2010 USAMO Problems/Problem 3
  If we can demonstrate a sequence in which for all <math>1 \le i \le 1005</math> the product a_4 = 7/a_3 = (3\cdot 7)/(5a_1) \ldots.</math> And as a result:
  11 KB (1,889 words) - 13:45, 4 July 2013
• 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 \qquad \text{(E)}\ 22</math>
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2010 AMC 12B Problems/Problem 16
  ..., 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>? ...ac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math>
  3 KB (419 words) - 13:42, 11 July 2021
• 2010 AMC 12B Problems/Problem 23
  ...math> and <math>-49</math>. What is the sum of the minimum values of <math>P(x)</math> and <math>Q(x)</math>? <math>\textbf{(A)}\ -100 \qquad \textbf{(B)}\ -82 \qquad \textbf{(C)}\ -73 \qquad \textbf{(D
  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
• 2010 AMC 10A Problems/Problem 11
  == Problem 11 == ...lity <math>a \le 2x + 3 \le b</math> is <math>10</math>. What is <math>b - a</math>?
  2 KB (259 words) - 19:54, 31 August 2022
• 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
• 2011 AMC 12A Problems/Problem 12
  ...<math>A.</math> How many hours did it take the power boat to go from <math>A</math> to <math>B</math>? \textbf{(A)}\ 3 \qquad
  5 KB (928 words) - 23:01, 14 April 2022
• 2011 AMC 12A Problems/Problem 20
  Let <math>f(x)=ax^2+bx+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Suppose that <math \textbf{(A)}\ 1 \qquad
  4 KB (724 words) - 17:36, 9 January 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 12B Problems
  <math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \ <math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2011 AMC 12B Problems/Problem 25
  ...<math>\frac{m}{k}</math>. For every odd integer <math>k</math>, let <math>P(k)</math> be the probability that ...math>1 \leq n \leq 99!</math>. What is the minimum possible value of <math>P(k)</math> over the odd integers <math>k</math> in the interval <math>1 \leq
  8 KB (1,339 words) - 10:42, 24 September 2022
• 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 6
  ...th> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. .../math>, the equation of the parabola can be expressed in the form <cmath>y=a\left(x-\frac{1}{4}\right)^2-\frac{9}{8}.</cmath>
  4 KB (661 words) - 01:18, 11 December 2023
• 2002 AMC 8 Problems
  A [[circle]] and two distinct [[Line|lines]] are drawn on a sheet of paper. What is the largest possible number of points of intersecti <math>\text {(A)}\ 2 \qquad \text {(B)}\ 3 \qquad {(C)}\ 4 \qquad {(D)}\ 5 \qquad {(E)}\ 6<
  15 KB (2,102 words) - 09:58, 5 May 2024
• 2011 AIME I Problems/Problem 12
  ...one other man. Find the least number of women in the line such that <math>p</math> does not exceed 1 percent. ...Thus, there are five cases to consider, where <math>(k)</math> refers to a consecutive group of <math>k</math> men:
  2 KB (407 words) - 11:46, 17 February 2020
• 2011 AIME II Problems/Problem 4
  ...h>, and point <math>M</math> is the midpoint of <math>AD</math>. Let <math>P</math> be the point of the intersection of <math>AC</math> and <math>BM</ma ...B,20/31*abs(B-C))), M = (A+D)/2, P = IP(M--2*M-B, A--C), D2 = IP(D--D+P-B, A--C);
  6 KB (944 words) - 21:31, 14 January 2024
• 2011 AIME II Problems
  Gary purchased a large beverage, but only drank <math>m/n</math> of it, where <math>m</math> The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer
  8 KB (1,301 words) - 08:43, 11 October 2020
• 2011 AIME II Problems/Problem 10
  ...and chord <math>\overline{CD}</math> of length 14 intersect at point <math>P</math>. The distance between the [[midpoint]]s of the two chords is 12. The The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma
  11 KB (1,720 words) - 03:12, 18 December 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>. Table of values of <math>P(x)</math>:
  8 KB (1,273 words) - 14:03, 7 January 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
• Mock Geometry AIME 2011 Problems
  ...th> shares no perfect square common factor with <math>c</math>. Find <math>a+b+c.</math> Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. H
  8 KB (1,349 words) - 19:10, 14 June 2022
• 2011 USAMO Problems
  ...<math>c</math> be positive real numbers such that <math>a^2 + b^2 + c^2 + (a + b + c)^2 \le 4</math>. Prove that ...h>\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.</cmath>
  3 KB (484 words) - 12:26, 17 April 2016
• 2001 AMC 8 Problems/Problem 19
  Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the s label("s",(0,7),W); label("p",(0,6),W); label("e",(0,5),W); label("e",(0,4),W); label("d",(0,3),W);
  2 KB (397 words) - 23:05, 13 August 2019
• 2007 AMC 10B Problems
  <math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) } Define the operation <math>\star</math> by <math>a \star b = (a+b)b.</math> What is <math>(3 \star 5) - (5 \star 3)?</math>
  15 KB (2,297 words) - 12:57, 19 February 2020
• 1999 AHSME Problems/Problem 12
  ...of the graphs of two different fourth degree polynomial functions <math> y=p(x)</math> and <math> y=q(x)</math>, each with leading coefficient 1? <math> \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \
  884 bytes (146 words) - 19:50, 6 March 2016
• 2007 AMC 10B Problems/Problem 14
  Some boys and girls are having a car wash to raise money for a class trip to China. Initially <math>40\%</math> of the group are girls. Sh <math>\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 10 \qq
  2 KB (297 words) - 23:19, 29 September 2023
• 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
• 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
• 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
• 2011 IMO Problems/Problem 1
  ...which <math>a_i +a_j</math> divides <math>s_A</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possibl ...+a_3)+(a_2+a_1) = s_A \ge 0</math>, so again <math>(a_2, a_4)</math> isn't a good couple. We have in total 6 couples. So <math>n_A \leq 6-2=4</math>.
  9 KB (1,718 words) - 23:08, 26 June 2014
• 2009 AMC 8 Problems
  Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The ...extbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14 </math>
  18 KB (2,551 words) - 18:46, 27 February 2024
• 2002 AMC 8 Problems/Problem 11
  A sequence of squares is made of identical square tiles. The edge of each squ path p=origin--(1,0)--(1,1)--(0,1)--cycle;
  2 KB (327 words) - 23:33, 18 January 2024
• 2002 AMC 8 Problems/Problem 21
  Harold tosses a coin four times. The probability that he gets at least as many heads as tai ...text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{5}{8}\qquad\text{(E)}\ \frac{11}{16} </math>
  3 KB (421 words) - 14:50, 21 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
• 1997 AHSME Problems/Problem 27
  ...a least common positive period <math>p</math> for all of them. Find <math>p</math>. <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qqua
  3 KB (456 words) - 14:14, 5 July 2013
• 2009 AMC 8 Problems/Problem 4
  path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p);
  5 KB (633 words) - 01:56, 26 November 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 25
  <math> \text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\tex The first equation is a [[circle]], so we find its center and [[radius]] by [[completing the square
  9 KB (1,441 words) - 17:51, 22 October 2023
• 1996 AHSME Problems/Problem 29
  If <math>n</math> is a positive integer such that <math>2n</math> has <math>28</math> positive div <math> \text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\tex
  5 KB (828 words) - 05:52, 26 October 2023
• 2010 PUMaC Problems
  == Division A == Assume that <math>f(a+b) = f(a) + f(b) + ab</math>, and that <math>f(75) - f(51) = 1230</math>. Find <mat
  22 KB (3,694 words) - 23:58, 3 June 2022
• 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((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black);
  1 KB (177 words) - 04:55, 25 November 2019
• 2009 PUMaC Problems
  '''Algebra A''' Given that <math>P(x)</math> is the least degree polynomial with rational coefficients such th
  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
• 2010 AMC 8 Problems
  ...ics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are <math>11</math> students in Mrs. Germain's class, <math>8</math> students in Mr. New <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq
  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
• 2006 AMC 8 Problems/Problem 17
  Jeff rotates spinners <math>P</math>, <math>Q</math> and <math>R</math> and adds the resulting numbers. W label("$P$", (-2,2));
  2 KB (330 words) - 05:08, 22 July 2018
• 1973 Canadian MO Problems
  ...inequalities, <math>x<\frac{1}{4x}</math> and <math>x<0</math>; i.e. find a single inequality equivalent to the two simultaneous inequalities. <math>\text{(iii)}</math> Give a rational number between <math>11/24</math> and <math>6/13</math>.
  4 KB (540 words) - 18:23, 8 October 2014
• 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
• 2012 AMC 10A Problems/Problem 25
  ...xtbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math> ...0</math>. Then the set of points <math>(x,y,z)</math> is a tetrahedron, or a triangular pyramid. The point <math>(x,y,z)</math> distributes uniformly in
  13 KB (2,133 words) - 01:22, 6 February 2024
• 2012 AMC 10A Problems/Problem 19
  ...esday Paula worked by herself and finished 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 16
  ..._1</math> lies on circle <math>C_2</math> and <math>XZ=13</math>, <math>OZ=11</math>, and <math>YZ=7</math>. What is the radius of circle <math>C_1</mat <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AMC 10B 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
  18 KB (2,350 words) - 18:48, 9 July 2023
• 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
• 2012 AMC 10B Problems/Problem 18
  ...esult actually has the disease. Which of the following is closest to <math>p</math>? ...}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}</math>
  2 KB (379 words) - 14:00, 22 August 2022
• 2015 AIME I Problems
  ...successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. ...uch that <math>16p+1</math> is the cube of a positive integer. Find <math>p</math>.
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  ...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and ...h> miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are <math>n</math> miles from Dodge, and they have been trav
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 12
  ...positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> pair A,B,C,D,E0,F;
  9 KB (1,523 words) - 12:23, 7 September 2022
• 1968 IMO Problems/Problem 2
  ...e only need to check these possibilities. It is easy to rule out 1 through 11, since <math>x^2-10x-22<0</math> for those values. However, <math>12^2-10\c ...where <math>a</math> and <math>b</math> are digits satisfying <math>0 \leq a, b < 10</math>. Then, we can use SFFT:
  3 KB (556 words) - 23:51, 6 December 2022
• 2012 AIME II Problems
  ...atio, with <math>a_1 = 27</math>, <math>b_1=99</math>, and <math>a_{15}=b_{11}</math>. Find <math>a_9</math>. ...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
• 2012 AIME II Problems/Problem 12
  ...to <math>10,000</math> which are simultaneously <math>7</math>-safe, <math>11</math>-safe, and <math>13</math>-safe. ...sidues <math>\mod 13</math>. The Chinese Remainder Theorem states that for a number <math>x</math> that is
  2 KB (351 words) - 18:03, 27 May 2024
• Mock AIME II 2012 Problems
  Given that <cmath>\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right) ...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>
  7 KB (1,274 words) - 21:16, 8 March 2021
• Mock AIME II 2012 Problems/Problem 6
  ...e positive integers, and <math> \text{gcd}(m,p)=1 </math>. Find <math>m+n+p</math>. label("$y=\frac{1}{2}x$",(11,5),S);
  3 KB (478 words) - 03:06, 5 April 2012
• Mock AIME II 2012 Problems/Problem 14
  ...f the <math>\textit{near Carmichael number}</math>, <math>n</math>, <math>(p-1)</math> divides <math>(n-1)</math> and <math>n</math> is not prime. Find ...d{a}\equiv 1^n\pmod{a}</math>, and therefore we get <math>1-1\equiv 0\pmod{a}</math>. Therefore, some two digit <math>\textit{near Carmichael numbers}<
  4 KB (789 words) - 03:23, 5 April 2012
• Mock AIME I 2012 Problems
  ...and <math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? A permutation of the numbers <math>\{2</math>, <math>3</math>, <math>4</math>
  7 KB (1,309 words) - 11:13, 8 April 2012
• Mock AIME I 2012 Problems/Problem 3
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. ...rt{195}</math>. Also note from the trisection that <math>[NME]=[NEF]=[NFO]=A/3</math>. Now <math>ME'\parallel NF,ME=EF\Longrightarrow NE=EE'</math>. Sim
  1 KB (208 words) - 17:31, 7 April 2012
• 2002 Romanian NMO Problems
  Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.
  10 KB (1,695 words) - 10:03, 10 May 2012
• 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 ...tain a polynomial whose coefficients are exactly the coefficients of <math>p(x)</math> in reverse order. Therefore, if
  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/Team Problems
  Given <math> \triangle ABC </math>, where <math> A </math> is at <math> (0,0) </math>, <math> B </math> is at <math> (20,0) </ ...would be <math> 1\cdot7+2\cdot3+3\cdot2=19 </math>, and so on. Given that a sequence of integers having this form starts with <math> 2 </math>, and the
  6 KB (910 words) - 17:32, 27 May 2012
• 1974 AHSME Answer Key
  #A #A
  2 KB (233 words) - 06:08, 20 January 2023
• 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
• 1953 AHSME Problems
  ...ll sell them at <math>5</math> for <math>20</math> cents. In order to make a profit of \$ <math>1.00</math>, he must sell: <math>\textbf{(A)}\ 67 \text{ oranges} \qquad
  21 KB (3,123 words) - 14:24, 20 February 2020
• 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
• 1989 AHSME Problems/Problem 5
  Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpi for(int a:x){
  2 KB (295 words) - 07:43, 22 October 2014
• 1986 USAMO Problems
  ...le by one or more primes <math>p</math> from the interval <math>2\le p \le 11</math>? ...ivisible by one or more primes <math>p</math> from the interval <math>2\le p \le 13</math>?
  3 KB (500 words) - 19:00, 23 October 2019
• 2020 Mock Combo AMC 10
  Here are the problems from the 2020 Mock Combo AMC 10 II, a mock contest created by the AoPS user fidgetboss_4000. ...friend, Sara, along with eight other classmates, are randomly seated along a row of ten chairs. What is the probability that Fred sits next to Sara?
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2008 AMC 8 Problems
  <math>\textbf{(A)}\ 12 \qquad <math>\textbf{(A)}\ 8671 \qquad
  14 KB (2,035 words) - 15:23, 26 January 2024
• Viviani's theorem
  ...the altitudes from any point in the triangle is equal to the altitude from a vertex of the triangle to the other side. ...math>\triangle ABC</math> be an equilateral triangle and <math>P</math> be a point inside the triangle.
  4 KB (420 words) - 14:13, 4 June 2021
• 2012 AMC 8 Problems
  ...many pounds of meat does she need to make <math> 24 </math> hamburgers for a neighborhood picnic? <math> \textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}6\frac{2}3\qquad\textbf{(C)
  13 KB (1,835 words) - 08:51, 8 March 2024
• 2012 AMC 12B Problems/Problem 24
  ...ositive numbers is unbounded if for every integer <math>B</math>, there is a member of the sequence greater than <math>B</math>. <math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qqu
  6 KB (969 words) - 10:06, 5 November 2021
• 2004 AMC 8 Problems/Problem 15
  ...e used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will path p=rotate(30)*polygon(6);
  1 KB (215 words) - 20:20, 12 October 2020
• 2013 AMC 12A Problems/Problem 24
  ...ity that the lengths of these three segments are the three side lengths of a triangle with positive area? <math> \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C
  3 KB (395 words) - 15:54, 8 November 2022
• 2013 AMC 12A Problems
  {{AMC12 Problems|year=2013|ab=A}} <math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \
  14 KB (2,206 words) - 19:31, 15 May 2024
• 2013 AMC 12A Problems/Problem 19
  ...> AB = 86 </math>, and <math> AC = 97 </math>. A circle with center <math> A </math> and radius <math> AB </math> intersects <math> \overline{BC} </math <math> \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \
  3 KB (546 words) - 15:24, 19 September 2021
• 2013 AMC 10A Problems/Problem 23
  ...<math>AB = 86</math>, and <math>AC=97</math>. A circle with center <math>A</math> and radius <math>AB</math> intersects <math>\overline{BC}</math> at <math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 61\qquad\tex
  5 KB (846 words) - 23:02, 21 August 2023
• 2013 AMC 10A Problems
  {{AMC10 Problems|year=2013|ab=A}} A taxi ride costs \$1.50 plus \$0.25 per mile traveled. How much does a 5-mile taxi ride cost?
  12 KB (1,894 words) - 15:59, 3 January 2024
• 2013 AMC 12B Problems
  On a particular January day, the high temperature in Lincoln, Nebraska, was <mat ...}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\ -3 \qquad \textbf{(E)}\ 11</math>
  16 KB (2,459 words) - 02:46, 30 January 2021
• 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
  ...les twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling? (a) the number <math>n</math> is divisible by <math>5,</math>
  9 KB (1,580 words) - 13:07, 24 February 2024
• 2013 AIME I Problems/Problem 15
  Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions (a) <math>0\le A<B<C\le99</math>,
  4 KB (661 words) - 23:14, 26 May 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
• 2013 AIME I Problems/Problem 10
  ...sum of the <math>{p}_{a,b}</math>'s for all possible combinations of <math>a</math> and <math>b</math>. Since <math>r+si</math> is a root, by the Complex Conjugate Root Theorem, <math>r-si</math> must be the
  2 KB (389 words) - 12:05, 10 August 2021
• 2013 AIME I Problems/Problem 12
  ...ively prime, and c is not divisible by the square of any prime. Find <math>a+b+c+d</math>. ...\sqrt{3}) * \frac{\sqrt{3} + 3}{2} = \frac{5\sqrt{3} + 9}{4}</math>. <math>a + b + c + d = 9 + 5 + 3 + 4 = \boxed{021}</math>
  9 KB (1,490 words) - 02:25, 2 May 2024
• 1991 AHSME Problems/Problem 10
  ...ius <math>15</math>. How many different chords of the circle contain <math>P</math> and have integer lengths? (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
• 2013 AIME II Problems
  ...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. Positive integers <math>a</math> and <math>b</math> satisfy the condition
  8 KB (1,402 words) - 12:17, 13 March 2020
• 2013 AIME II Problems/Problem 8
  ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. real a;
  8 KB (1,357 words) - 09:23, 11 March 2024
• 2013 AIME II Problems/Problem 12
  ...all polynomials of the form <math>z^3 + az^2 + bz + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Find the number of ..., if <math>\omega+\omega^*=2\Re{(\omega)}</math> is an integer, then <math>a,b,</math> and <math>c</math> are clearly integers. Therefore <math>2\Re{(\o
  8 KB (1,393 words) - 19:00, 24 May 2023
• Rotation
  A rotation of a planar figure is a transformation that preserves area and angles, but not orientation. The res ...rotate triangle <math>ABC</math> <math>60^{\circ}</math> clockwise around a point <math>O</math>, also known as the '''center of rotation'''.
  3 KB (432 words) - 23:22, 13 January 2021
• Mock AIME 6 2006-2007 Problems
  Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degree Alvin, Simon, and Theodore are running around a <math>1000</math>-meter circular track starting at different positions. Al
  7 KB (1,173 words) - 21:04, 7 December 2018
• 2014 AIME II Problems/Problem 11
  ==Problem 11== ...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>.
  6 KB (1,059 words) - 18:24, 20 January 2024
• 2014 AIME II Problems/Problem 10
  ...ac{1}{z+w}=\frac{1}{z}+\frac{1}{w}</math>. Then the area enclosed by <math>P</math> can be written in the form <math>n\sqrt{3}</math>, where <math>n</ma Thus <math>P</math> is an isosceles triangle with area <math>\frac{1}{2}(2014 - (-1007))
  6 KB (1,045 words) - 13:08, 21 January 2024
• 2014 AIME II Problems/Problem 15
  ...<math>p(k)</math> if <math>p(k)>2</math>, and <math>X(k)=1</math> if <math>p(k)=2.</math> Let <math>\{x_n\}</math> be the sequence defined by <math>x_0= ...d x_i</math>. This provides motivation to translate <math>x_i</math> into a binary sequence <math>y_i</math>.
  4 KB (770 words) - 17:44, 11 October 2023
• 2014 AIME II Problems/Problem 5
  ...yinclude>Real numbers <math>r</math> and <math>s</math> are roots of <math>p(x)=x^3+ax+b</math>, and <math>r+4</math> and <math>s-3</math> are roots of ...</math> in <math>p(x)</math> and <math>q(x)</math> are both equal to <math>a</math>, and equating the two coefficients gives
  6 KB (1,069 words) - 15:51, 31 December 2023
• 1986 USAMO Problems/Problem 1
  ...le by one or more primes <math>p</math> from the interval <math>2\le p \le 11</math>? ...ivisible by one or more primes <math>p</math> from the interval <math>2\le p \le 13</math>?
  4 KB (668 words) - 15:49, 20 August 2016
• Northeastern WOOTers Mock AIME I Problems
  Let <math> \triangle ABC </math> be a triangle with <math>AC=15</math>, <math>BC=112</math>, and <math>AB=113</ma ...</math>, for some primes <math>p</math> and <math>q</math>. Find <math>ab+p+q</math>.
  8 KB (1,336 words) - 09:10, 30 May 2020
• ASIA TEAM Problems
  Let the order of a lily pad number <math>x</math>, denoted <math>o(x)</math>, be the minimum n ...first fly, he places it on <math>a_1</math>. After that, when he catches a fly, he places it on <math>a_i</math> for which <math>i</math> is the least
  10 KB (1,710 words) - 23:23, 10 January 2020
• Mock AIME I 2015 Problems
  ...ath>. Finally, Palmer announces the product of the numbers in boxes D and A. If <math>k</math> is the number that Palmer says, what is <math>20k</math ...th> and <math>b</math> are positive relatively prime integers. Find <math>a+b</math>.
  9 KB (1,463 words) - 14:48, 12 February 2017
• 2016 AMC 10A Problems
  {{AMC10 Problems|year=2016|ab=A}} What is the value of <math>\dfrac{11!-10!}{9!}</math>?
  14 KB (2,104 words) - 22:26, 16 September 2022
• 1962 AHSME Problems/Problem 24
  ...her, can do a job in <math>x</math> hours. When working alone, <math>\text{P}</math> needs an additional <math>6</math> hours to do the job; <math>\text <math> \textbf{(A)}\ \frac{2}3\qquad\textbf{(B)}\ \frac{11}{12}\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3
  1 KB (220 words) - 23:01, 16 April 2014
• 1962 AHSME Problems/Problem 38
  ..., with an additional increase of <math>100</math>, the population is again a perfect square. The original population is a multiple of:
  4 KB (570 words) - 16:52, 28 June 2023
• 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 18
  Isabella uses one-foot cubical blocks to build a rectangular fort that is <math>12</math> feet long, <math>10</math> feet wi triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
  3 KB (412 words) - 22:30, 18 December 2023
• 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
• 1958 AHSME Problems/Problem 12
  If <math> P = \frac{s}{(1 + k)^n}</math> then <math> n</math> equals: <math> \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 + k)}}\qquad
  763 bytes (128 words) - 23:08, 13 March 2015
• 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
  ...ing from an increase in <math>\pi</math> units in the diameter. Then <math>P</math> equals: <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
• 2014 AMC 10A Problems/Problem 22
  ...ine{AB}=20</math> and <math>\overline{BC}=10</math>. Let <math>E</math> be a point on <math>\overline{CD}</math> such that <math>\angle CBE=15^\circ</ma ...rt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 </math>
  12 KB (1,821 words) - 18:16, 29 October 2023
• 2018 AIME I Problems/Problem 14
  Let <math>SP_1P_2P_3EP_4P_5</math> be a heptagon. A frog starts jumping at vertex <math>S</math>. From any vertex of the heptag ...th>. We can count the number of left/right (L/R) paths of length <math>\le 11</math> that start at <math>S</math> and end at either <math>P_4</math> or <
  4 KB (649 words) - 17:29, 22 December 2023
• 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
• 2007 iTest Problems/Problem 12
  <math>\text{(A) }\frac{2}{3}\qquad ...<math>W</math> be a game where the winning team won and <math>L</math> be a game where the losing team lost. Note that for all the possible outcomes,
  3 KB (500 words) - 04:44, 10 June 2018
• 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 12B Problems
  <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qqu ...ale on balloons: buy <math> 1 </math> balloon at the regular price and get a second at <math> \frac{1}{3} </math> off the regular price. What is the gre
  13 KB (2,066 words) - 14:08, 1 November 2022
• 2014 AMC 10B Problems
  <math> \textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) <math>\textbf {(A) } 16 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 32 \qquad \textbf {(D)
  13 KB (2,011 words) - 21:54, 8 November 2022
• 2014 AIME II Problems
  ...s Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is pai ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  8 KB (1,410 words) - 00:04, 29 December 2021
• 2014 AIME I Problems
  ...each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. ...nd urn contains <math>16</math> green balls and <math>N</math> blue balls. A single ball is drawn at random from each urn. The probability that both bal
  9 KB (1,472 words) - 13:59, 30 November 2021
• 1993 AHSME Problems/Problem 24
  ...rd shiny penny appears and <math>a/b</math> is in lowest terms, then <math>a+b=</math> <math>\text{(A) } 11\quad
  3 KB (558 words) - 00:25, 23 December 2020
• 1973 AHSME Problems
  A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1992 AHSME Problems
  If <math>3(4x+5\pi)=P</math> then <math>6(8x+10\pi)=</math> <math>\text{(A) } 2P\quad
  16 KB (2,548 words) - 13:40, 19 February 2020
• 1990 AHSME Problems
  <math>\textbf{(A)}\ \pm\frac{1}{2}\qquad\textbf{(B)}\ \pm 1\qquad\textbf{(C)}\ \pm 2\qquad\t <math>\textbf{(A) } -16\qquad
  14 KB (2,099 words) - 01:15, 10 September 2021
• 1969 AHSME Problems
  <math>\frac{a}{b},a \ne b,b \ne 0</math>, the value of the fraction is changed to <math>\frac{c <math>\text{(A) } \frac{1}{c-d}\quad
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1970 AHSME Problems
  <math>\textbf{(A) }\sqrt{2}+\sqrt{3}\qquad A square and a circle have equal perimeters. The ratio of the area of the circle to the ar
  15 KB (2,366 words) - 07:52, 26 December 2023
• 1988 AHSME Problems
  <math>\text{(A)}\ \sqrt{20} \qquad Triangles <math>ABC</math> and <math>XYZ</math> are similar, with <math>A</math> corresponding to <math>X</math> and <math>B</math> to <math>Y</math>
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1987 AHSME Problems
  <math>\text{(A)}\ 1 - x^5\qquad A triangular corner with side lengths <math>DB=EB=1</math> is cut from equila
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1986 AHSME Problems
  <math>\textbf{(A)}\ 2y \qquad <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
  17 KB (2,512 words) - 18:30, 12 October 2023
• 1983 AHSME Problems
  <math>\textbf{(A)}\ 8\qquad ...lane. At most how many points on <math>C</math> are <math>3</math> cm from P?
  15 KB (2,309 words) - 23:43, 2 December 2021
• 1982 AHSME Problems
  <math>\textbf{(A)} \ 2 \qquad If a number eight times as large as <math>x</math> is increased by two, then one
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1979 AHSME Problems
  label("$A$",(-2,1),W); <math>\textbf{(A) }8\qquad
  17 KB (2,664 words) - 01:34, 19 March 2022
• 1978 AHSME Problems
  <math>\textbf{(A) }-1\qquad If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
  15 KB (2,432 words) - 01:06, 22 February 2024
• 2007 iTest Problems/Problem 14
  <math>\text{(A) } 0 \quad \text{(L) } 11 \quad
  2 KB (265 words) - 21:23, 4 February 2023
• 1977 AHSME Problems
  <math>\text{(A)}\ x \qquad <math>\textbf{(A)}\ \text{ equiangular}\qquad
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1976 AHSME Problems
  <math>\textbf{(A) }-2\qquad For how many real numbers <math>x</math> is <math>\sqrt{-(x+1)^2}</math> a real number?
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1975 AHSME Problems
  <math>\textbf{(A)}\ 3/4 \qquad <math>\textbf{(A)}\ \text{all }m\qquad
  16 KB (2,512 words) - 04:48, 27 November 2021
• 1972 AHSME Problems
  <math>\text{(A) I and II}\qquad If a dealer could get his goods for <math>8</math>% less while keeping his selli
  17 KB (2,725 words) - 00:21, 26 May 2024
• 2007 iTest Problems/Problem 21
  James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad
  2 KB (247 words) - 03:34, 14 June 2018
• 2007 iTest Problems/Problem 22
  <math>\text{(A) } 0 \quad \text{(L) } 11 \quad
  929 bytes (137 words) - 22:05, 10 January 2019
• 2007 iTest Problems/Problem 25
  <math>\text{(A) } 0\qquad \text{(L) } 11\qquad
  2 KB (270 words) - 14:35, 29 July 2018
• 2007 iTest Problems/Problem 24
  ...such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube. <math>\text{(A) }0 \quad
  2 KB (340 words) - 19:49, 30 June 2018
• 1971 AHSME Problems
  <math>\textbf{(A) }9\qquad \textbf{(C) }11\qquad
  18 KB (2,703 words) - 20:50, 11 September 2023
• 1965 AHSME Problems
  <math>\textbf{(A)}\ 0 \qquad A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1964 AHSME Problems
  <math>\textbf{(A)}\ \log_{10}50 \qquad <math>\textbf{(A)}\ \text{a parabola} \qquad
  19 KB (2,907 words) - 14:16, 20 February 2020
• 1963 AHSME Problems
  <math>\textbf{(A)}\ (0,0)\qquad <math>\textbf{(A)}\ -14 \qquad
  20 KB (3,122 words) - 14:17, 20 February 2020
• 1961 AHSME Problems
  <math>\textbf{(A)}\ \frac{1}{25} \qquad An automobile travels <math>a/6</math> feet in <math>r</math> seconds. If this rate is maintained for <ma
  18 KB (2,905 words) - 18:33, 5 April 2023
• 1957 AHSME Problems
  ...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua
  26 KB (3,950 words) - 21:09, 31 August 2020
• 1956 AHSME Problems
  <math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 36 <math> \textbf{(A)}\ \text{broke even}\qquad
  20 KB (3,039 words) - 22:44, 12 February 2021
• 2011 UNCO Math Contest II Problems
  pair P,R; P=unit(A1-D1)+D1;
  5 KB (846 words) - 22:10, 7 November 2014
• 2010 UNCO Math Contest II Problems
  • The positive integers are <math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,\cdots</math> • The prime numbers are <math>2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \cdots</math>
  5 KB (826 words) - 22:12, 7 November 2014
• 2009 UNCO Math Contest II Problems
  ...ve <math>3</math>-digit numbers <math>abc</math> are there such that <math>a+b=c</math> For example, <math>202</math> and <math>178</math> ...but <math>245</math> and <math>317</math> do not have that property. Find A-B*c+3 square.
  5 KB (753 words) - 20:35, 8 February 2024
• 2008 UNCO Math Contest II Problems
  Let <math>S = {a,b,c,d}</math> be a set of four positive integers. If pairs of distinct elements of <math>S</ma ...btained: <math>5, 10, 11, 13, 14, 19.</math> Determine the values of <math>a,
  6 KB (871 words) - 22:13, 7 November 2014
• 1993 UNCO Math Contest II Problems
  A student thinks of four numbers. She adds them in pairs to get the six sums ...34.</math> Use the given data to determine the distance between city <math>A</math> and city <math>C</math>.
  4 KB (607 words) - 15:40, 20 October 2014
• 1993 UNCO Math Contest II Problems/Problem 9
  ...<math>ABCD</math>. If <math>AP=5</math> , <math>BP=10</math> and <math>CP=11</math>, find the length of <math>DP</math>. pair P=(2,2);
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• 1986 AHSME Problems/Problem 16
  ...side <math>BC</math> is extended, as shown in the figure, to a point <math>P</math> so that <math>\triangle PAB</math> pair A=origin, P=(1.5,5), B=(8,0), C=P+2.5*dir(P--B);
  1 KB (210 words) - 23:51, 10 February 2018
• 1998 CEMC Gauss (Grade 7) Problems
  = Part A: Each correct answer is worth 5 points = <math>\text{(A)}\ 1 \qquad \text{(B)}\ 1000 \qquad \text{(C)}\ 0.1 \qquad \text{(D)}\ 10 \
  10 KB (1,590 words) - 16:43, 29 January 2021
• 2016 AMC 12A Problems
  {{AMC12 Problems|year=2016|ab=A}} What is the value of <math>\frac{11!-10!}{9!}</math>?
  15 KB (2,348 words) - 17:20, 19 January 2024
• 2015 AMC 10A Problems
  {{AMC10 Problems|year=2015|ab=A}} <math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf
  12 KB (1,887 words) - 18:52, 14 May 2024
• 2015 AMC 12A Problems
  {{AMC12 Problems|year=2015|ab=A}} <math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf
  13 KB (2,117 words) - 12:33, 24 August 2023
• 2015 AMC 10B Problems
  <math> \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \q Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe
  14 KB (2,247 words) - 11:38, 26 March 2024
• 2015 AMC 12B Problems
  <math>\textbf{(A)}\; -2 \qquad\textbf{(B)}\; \dfrac{1}{16} \qquad\textbf{(C)}\; \dfrac{7}{4} Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishe
  13 KB (2,064 words) - 13:39, 1 October 2022
• 2015 AMC 12B Problems/Problem 25
  ...<math>d</math> are not divisible by the square of any prime. What is <math>a+b+c+d</math> ? <math>\textbf{(A)}\; 2016 \qquad\textbf{(B)}\; 2024 \qquad\textbf{(C)}\; 2032 \qquad\textbf{
  13 KB (2,121 words) - 18:16, 19 May 2024
• 2016 AMC 10A Problems/Problem 17
  ...the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>? <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 1
  4 KB (749 words) - 03:26, 29 April 2024
• 2015 AIME I Problems/Problem 12
  ...th> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. ...{a} + \binom{a+1}{a} + \binom{a+2}{a} + \dots + \binom{b}{a} = \binom{b+1}{a+1}.</cmath>
  5 KB (683 words) - 23:01, 9 August 2021
• 2015 AIME I Problems/Problem 7
  ...and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</m pair A,B,C,D,E,F,G,H,J,K,L,M,N;
  6 KB (1,105 words) - 21:02, 9 November 2023
• 2015 AIME I Problems/Problem 11
  ...<math>a+b=\sqrt{x^2-y^2}</math>. Solving for <math>a</math>, we get <math>a=\frac{\sqrt{x^2-y^2}}{1+\frac{x}{y}}</math>. Then, using Pythagorean Theore ...oot of perpendicular from <math>I</math> to <math>BC</math> is point <math>P</math>.
  5 KB (906 words) - 17:43, 27 September 2023
• 2015 AIME II Problems
  ...ect to take Latin. The probability that a randomly chosen Latin student is a sophomore is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</ma ...where <math>p</math> and <math>q</math> are positive integers. Find <math>p + q</math>.
  8 KB (1,326 words) - 19:15, 13 January 2024
• 2015 AIME II Problems/Problem 6
  ...ers <math>a</math> and <math>c</math>. Can you tell me the values of <math>a</math> and <math>c</math>?" ...says, "You're right. Here is the value of <math>a</math>." He writes down a positive integer and asks, "Can you tell me the value of <math>c</math>?"
  5 KB (946 words) - 14:06, 14 February 2023
• 2015 AIME II Problems/Problem 11
  ...e{OB}</math> intersects lines <math>AB</math> and <math>BC</math> at <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4 pair A,B,C,O, Q, P, M, N;
  9 KB (1,249 words) - 21:59, 26 November 2023
• 2015 AIME II Problems/Problem 15
  ...A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> aga D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems
  ...and <math>1</math> satisfy <math>S(a)S(-a)=2016</math>. Find <math>S(a)+S(-a)</math>. ...th> is directly proportional to <math>k</math>. The probability of rolling a <math>7</math> with this pair of dice is <math>\frac{m}{n}</math>, where <m
  8 KB (1,360 words) - 12:19, 29 January 2022
• 2016 AMC 10A Problems/Problem 22
  <math>\textbf{(A) }110\qquad\textbf{(B) }191\qquad\textbf{(C) }261\qquad\textbf{(D) }325\qqu ...so we can set <math>n=2^3 \cdot 5</math>, so <math>2^{10} \cdot 5^4 \cdot 11</math> has <math>110</math> factors. Therefore, <math>n=2^3 \cdot 5</math>
  3 KB (429 words) - 19:27, 30 December 2022
• 2017 AIME I Problems
  ...points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on For a positive integer <math>n</math>, let <math>d_n</math> be the units digit of
  7 KB (1,163 words) - 16:43, 2 June 2022
• 2015 IMO Problems/Problem 2
  Determine all triples of positive integers <math>(a,b,c)</math> such that each of the numbers <cmath> ab-c,\; bc-a,\; ca-b </cmath>
  3 KB (610 words) - 22:51, 16 October 2015
• 2015 AMC 8 Problems
  ...feet long and <math>9</math> feet wide? (There are <math>3</math> feet in a yard.) <math>\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad
  16 KB (2,322 words) - 14:04, 2 February 2024
• 2015 AMC 8 Problems/Problem 18
  ...tained by adding a constant to the previous term. For example, <math>2,5,8,11,14</math> is an arithmetic sequence with five terms, in which the first ter <math>\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \
  5 KB (641 words) - 10:28, 13 January 2024
• 2016 AMC 12A Problems/Problem 12
  ...sion(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E, <math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3
  4 KB (682 words) - 18:19, 20 March 2024
• 2019 AMC 12A Problems
  {{AMC12 Problems|year=2019|ab=A}} ...us <math>4</math> inches is <math>N</math> percent larger than the area of a pizza with radius <math>3</math> inches. What is the integer closest to <ma
  14 KB (2,180 words) - 22:25, 25 April 2024
• 2016 AMC 10B Problems
  ...is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \tfrac{1}{2}</math>? <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\
  12 KB (1,868 words) - 17:50, 30 December 2023
• 2018 UNCO Math Contest II Problems
  A prime is an integer strictly greater than one that is evenly divisible by n itself and 1. The primes are 2, 3, 5, 7, 11, 13, 17, . . .
  6 KB (1,055 words) - 12:37, 30 July 2021
• 2018 UNCO Math Contest II Answer Key
  6) <math>C; p(x) = 2x^3-2x + 3</math> a) <math>G(3) = 5, G(4) = 8, G(5) = 13</math>
  777 bytes (122 words) - 04:37, 13 January 2019
• 2018 UNCO Math Contest II Problems/Problem 10
  coins is in the shape of a domino (two adjoining squares) and the other two are in the shape of have announced a contest. Seripian students can win fame and glory and 100 of each of the
  2 KB (332 words) - 11:33, 1 December 2020
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems
  ...ath> to <math>9</math>. If <math>\sqrt{STARS} = SAT</math>, what are <math>A, R, S</math>, and <math>T</math>? (a) Determine <math>f\circ g(x)</math> and <math>g\circ f(x)</math>.
  5 KB (846 words) - 03:36, 19 January 2019
• 2016 AMC 10B Problems/Problem 23
  ...A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}</math> We draw a diagram to make our work easier:
  8 KB (1,368 words) - 16:43, 9 August 2023
• 2016 AMC 10B Problems/Problem 18
  <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\t This simplifies to <math>\frac {n \cdot (2k+n-1)}{2} = 345</math>. This gives a nice equation. We multiply out the 2 to get that <math>n \cdot (2k+n-1)=690
  5 KB (813 words) - 16:55, 9 June 2023
• 2016 AMC 12B Problems/Problem 10
  ...>0</math>. The area of <math>PQRS</math> is <math>16</math>. What is <math>a+b</math>? <math>\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \q
  2 KB (293 words) - 07:08, 19 February 2021
• 2016 AMC 12B Problems
  ...is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \frac{1}{2}</math>? <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\
  14 KB (2,037 words) - 19:09, 29 July 2023
• 2016 AIME I Problems/Problem 6
  real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6; pair A=(0,0),B=(c,0),D=(c/2,-sqrt(25-(c/2)^2));
  14 KB (2,397 words) - 20:04, 27 August 2023
• 2016 AIME I Problems/Problem 12
  ...ast positive integer <math>m</math> such that <math>m^2 - m + 11</math> is a product of at least four not necessarily distinct primes. ...math>. Therefore, all prime factors of <math>m^2-m+11</math> are <math>\ge 11</math>.
  3 KB (498 words) - 20:32, 22 November 2022
• KGS math club/solution pairing
  and pair them up so that the sum of each pair is a perfect square? ...th>a</math> is a '''partner''' of <math>b</math>, if <math>a + b</math> is a perfect square.
  11 KB (1,114 words) - 20:30, 24 March 2016
• 2016 AIME II Problems/Problem 12
  ...ve four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections i Choose a section to start coloring. Assume, WLOG, that this section is color <math>1
  20 KB (3,328 words) - 21:13, 5 April 2024
• 2016 AIME II Problems/Problem 10
  ...<math>ABC</math> is inscribed in circle <math>\omega</math>. Points <math>P</math> and <math>Q</math> are on side <math>\overline{AB}</math> with <math pair A = origin, B = (13,0), P = (4,0), Q = (7,0),
  7 KB (1,090 words) - 22:09, 31 January 2024
• 2016 AIME II Problems
  ...ex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats <math>5</math> of his peanuts, Betty eats ...ath> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
  8 KB (1,312 words) - 21:16, 3 March 2021
• 2006 iTest Problems
  <math>\mathrm{(A)}\, 8</math> <math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>
  31 KB (4,811 words) - 00:02, 4 November 2023
• 1954 AHSME Problems/Problem 37
  ...ath>, <math>PQ</math> extended to <math>D</math> and <math>\angle n</math> a right angle, then: path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)
  3 KB (560 words) - 10:11, 3 May 2020
• 1954 AHSME Problems/Problem 10
  ...sum of the numerical coefficients in the expansion of the binomial <math>(a+b)^6</math> is: <math>\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\
  397 bytes (52 words) - 20:40, 17 February 2020
• 1990 IMO Problems/Problem 6
  Prove that there exists a convex <math>1990\text{-gon}</math> with the following two properties: (a) All angles are equal.
  3 KB (522 words) - 13:54, 30 January 2021
• 2016 AMC 8 Problems
  The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes is that? <math>\textbf{(A) } 605 \qquad\textbf{(B) } 655\qquad\textbf{(C) } 665\qquad\textbf{(D) } 10
  12 KB (1,665 words) - 12:37, 1 April 2024
• 2016 AMC 8 Problems/Problem 24
  ...igit number <math>RST</math> is divisible by <math>3</math>. What is <math>P</math>? <math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \text
  5 KB (781 words) - 22:06, 21 January 2024
• Mock AIME 4 Pre 2005/Problems
  ...> 3</math> and <math>a_{2004} = 7.</math> <math>b_1, b_2, \ldots</math> is a sequence of real numbers in which <math>b_n</math> is the geometric mean of ...nt in its interior such that <math>ABP</math> is equilateral. There exists a unique pair <math>\{m, n\}</math> of relatively prime positive integers suc
  7 KB (1,094 words) - 15:39, 24 March 2019
• Mock AIME 3 2010 Problems
  ...can distribute 1 piece of candy among 243 hungry schoolchildren sitting in a row -3. We have a triangle abc with sides ab, bc, ca with lengths α, β, and γ, respectivel
  14 KB (2,904 words) - 18:24, 16 May 2017
• 2017 AMC 10A Problems/Problem 24
  ...cmath>has three distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>Wh <math>\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{
  10 KB (1,708 words) - 23:16, 7 October 2023
• 2017 AMC 12A Problems/Problem 12
  ...get their names from how many minutes it takes them to run one lap around a circular race track: Horse <math>k</math> runs one lap in exactly <math>k</ <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\t
  2 KB (353 words) - 14:21, 19 December 2020
• 2017 AMC 12A Problems/Problem 23
  ...cmath>has three distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>Wh <math>\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{
  10 KB (1,861 words) - 10:47, 17 October 2021
• 2017 AMC 10A Problems
  {{AMC10 Problems|year=2017|ab=A}} <math>\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\q
  15 KB (2,285 words) - 18:02, 28 October 2023
• 2017 AMC 12A Problems/Problem 25
  The vertices <math>V</math> of a centrally symmetric hexagon in the complex plane are given by ...f the <math>12</math> numbers selected. What is the probability that <math>P=-1</math>?
  18 KB (2,878 words) - 01:47, 16 December 2023
• 2017 AMC 12A Problems/Problem 24
  ...t <math>\frac{DX}{BD} = \frac{1}{4}</math> and <math>\frac{BY}{BD} = \frac{11}{36}</math>. <math>\textbf{(A) }17\qquad\textbf{(B) }\frac{59 - 5\sqrt{2}}{3}\qquad\textbf{(C) }\frac{91
  8 KB (1,322 words) - 03:24, 7 October 2023
• 2017 AMC 12A Problems/Problem 16
  ...the largest semicircle. What is the radius of the circle centered at <math>P</math>? pair P = (-1,0)+(2+6/7)*dir(36.86989);
  13 KB (1,982 words) - 17:12, 20 December 2022
• 2017 AMC 12A Problems
  {{AMC12 Problems|year=2017|ab=A}} <math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15</math>
  15 KB (2,418 words) - 16:58, 7 November 2022
• 2017 AMC 12B Problems/Problem 16
  ...\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{11}{21}</math> ...h>2^{18}</math>, and the only way to have an odd number is if there is not a <math>2</math> in that number's prime factorization. This only happens with
  4 KB (671 words) - 13:55, 10 July 2023
• 2017 AMC 12B Problems/Problem 10
  <math>\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 12\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ ...P(Likes <math>\cap</math> Says Dislike) = .6 <math>\cdot</math> .2 = .12. P(Says Dislike) = (.4 <math>\cdot</math> .9) + (.6 <math>\cdot</math> .2) = .
  1 KB (212 words) - 20:28, 24 February 2022
• 2017 AIME I Problems/Problem 11
  ==Problem 11== ...rangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</ma
  6 KB (1,019 words) - 12:40, 24 January 2024
• 2017 AIME I Problems/Problem 15
  ...<math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);
  22 KB (3,622 words) - 17:11, 6 January 2024
• 2017 AIME II Problems
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  7 KB (1,200 words) - 15:02, 8 September 2020
• 2017 AIME II Problems/Problem 10
  ...of <math>\overline{CM}</math> and <math>\overline{DN}</math>. Point <math>P</math> lies on the quadrilateral <math>BCON</math>, and <math>\overline{BP} pair A,B,C,D,M,n,O,P;
  4 KB (718 words) - 00:21, 27 January 2023
• Circular Inversion
  ...e transformed point on the ray extending from <math>O</math> through <math>A</math>. pair A = (6, 0);
  16 KB (2,516 words) - 23:48, 15 January 2024
• 2017 USAJMO Problems/Problem 3
  ...Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> in pair A = (0, 3sqrt(3)), B = (-3,0), C = (3,0), P = (0, -sqrt(3)), D = (0, 0), E1 = (6, -3sqrt(3)), F = (-6, -3sqrt(3)), O =
  10 KB (1,829 words) - 03:13, 16 March 2022
• 2017 UNCO Math Contest II Problems
  A prime is an integer strictly greater than one that is evenly divisible by n itself and 1. The primes are 2, 3, 5, 7, 11, 13, 17, . . .
  7 KB (1,192 words) - 15:14, 20 August 2020
• Graphgrid.asy
  Graphgrid.asy is a template created by user Lord_Baltimore to allow easy creation of equation ...th>",(0,-11),S);label("<math>x</math>",(11,0),E);label("<math>-x</math>",(-11,0),W);
  4 KB (627 words) - 22:18, 16 November 2017
• 2020 AMC 12A Problems
  {{AMC12 Problems|year=2020|ab=A}} Carlos took <math>70\%</math> of a whole pie. Maria took one third of the remainder. What portion of the whole
  14 KB (2,073 words) - 15:15, 21 October 2021
• 2020 AMC 12B Problems
  <math>\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \q ...^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}</cmath><math>\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780
  16 KB (2,417 words) - 01:03, 28 April 2022
• 2017 AMC 8 Problems/Problem 10
  A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are select <math>\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}
  2 KB (243 words) - 15:35, 26 May 2024
• 2017 IMO Problems/Problem 1
  ...math> such that there exists a number <math>A</math> such that <math>a_n = A</math> for infinitely many values of <math>n</math>. ..., <math>a_8=36</math>, <math>a_9=6</math>, <math>a_{10}=9</math>, <math>a_{11}=3</math> and the pattern <math>3,6,9</math> repeats.
  3 KB (559 words) - 03:04, 19 November 2023
• 2018 AMC 10A Problems/Problem 12
  <math>\textbf{(A) } 1 \qquad ...> or <math>-1</math>. Remember that if <math>P-Q=a</math>, then <math>Q-P=-a</math>. Because we have already taken <math>1</math> and <math>-1</math> in
  7 KB (1,070 words) - 16:49, 26 May 2024
• 2018 AMC 10A Problems/Problem 23
  ...h> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like th pair A, B, C, D, F;
  16 KB (2,454 words) - 13:30, 23 September 2023
• 2018 AMC 10A Problems/Problem 22
  ...0<\gcd(d, a)<100</math>. Which of the following must be a divisor of <math>a</math>? ...bf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}</math>
  10 KB (1,599 words) - 04:51, 6 August 2023
• 2018 AMC 12A Problems
  {{AMC12 Problems|year=2018|ab=A}} A large urn contains <math>100</math> balls, of which <math>36 \%</math> are
  15 KB (2,380 words) - 18:52, 7 April 2022
• 2018 AMC 12B Problems
  Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbrea <math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D)
  14 KB (2,118 words) - 15:36, 28 October 2021
• 2018 AMC 10B Problems/Problem 19
  ...t will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? \textbf{(A) }7 \qquad
  5 KB (888 words) - 05:20, 12 April 2024
• 2018 AMC 10B Problems
  Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbrea \textbf{(A) } 90 \qquad
  15 KB (2,237 words) - 23:08, 15 November 2023
• 2018 AMC 12B Problems/Problem 17
  ...c{4}{7}</cmath>and <math>q</math> is as small as possible. What is <math>q-p</math>? <math>\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 </m
  11 KB (1,937 words) - 00:18, 23 October 2023
• 2018 AMC 12B Problems/Problem 21
  ...th> and <math>I</math> denote the circumcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> an <math>\textbf{(A)}\ \frac52\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ \frac{13}{4}\qquad\textbf{(E)}\
  3 KB (540 words) - 04:12, 30 August 2022
• 2018 AMC 12B Problems/Problem 22
  ..., 2, 3, 4, 5, 6, 7, 8, 9\}</math>. How many such polynomials satisfy <math>P(-1) = -9</math>? <math>\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D)
  3 KB (432 words) - 20:29, 14 November 2022
• 2018 AMC 10B Problems/Problem 9
  ...<math>10</math>. What other sum occurs with the same probability as <math>p</math>? <math>\textbf{(A)} \text{ 13} \qquad \textbf{(B)} \text{ 26} \qquad \textbf{(C)} \text{ 32}
  5 KB (835 words) - 13:48, 16 February 2024
• 1974 IMO Problems/Problem 4
  Consider decompositions of an <math>8\times8</math> chessboard into <math>p</math> non-overlapping rectangles subject to the following conditions: .../math> for which such a decomposition is possible. For this value of <math>p,</math> determine all possible sequences <math>a_1, a_2, \cdots, a_p.</math
  3 KB (440 words) - 16:03, 29 January 2021
• 1963 AHSME Problems/Problem 10
  Point <math>P</math> is taken interior to a square with side-length <math>a</math> and such that is it equally distant from two <math>\textbf{(A)}\ \frac{3a}{5}\qquad
  1 KB (184 words) - 08:22, 5 June 2018
• 1963 AHSME Problems/Problem 12
  ...am <math>PQRS</math> are <math>P(-3,-2), Q(1,-5), R(9,1)</math> with <math>P</math> and <math>R</math> diagonally opposite. <math>\textbf{(A)}\ 13 \qquad
  2 KB (266 words) - 04:04, 7 June 2018
• 2018 AIME I Problems/Problem 4
  ...ath> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. By the Law of Cosines on <math>\triangle ABC</math>, we have: <cmath>\cos(A) = \frac{10^2+10^2-12^2}{2*10*10} = \frac{7}{25}</cmath> By the Law of Cos
  14 KB (2,247 words) - 20:07, 12 January 2024
• 2018 AIME I Problems
  ...e number of ordered pairs of integers <math>(a,b)</math> with <math>1 \leq a \leq 100</math> and <math>b \geq 0</math> such that the polynomial <math>x^ ...e{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }</math>, where <math>a > 0</math>. Find the base-<math>10</math> representation of <math>n</math>.
  8 KB (1,284 words) - 14:35, 9 August 2021
• 2018 AIME II Problems
  ...e at <math>B</math> at the same time. Find the number of meters from <math>A</math> to <math>B</math>. ...inder when <math>4(a_{n-1} + a_{n-2} + a_{n-3})</math> is divided by <math>11</math>. Find <math>a_{2018} \cdot a_{2020} \cdot a_{2022}</math>.
  9 KB (1,385 words) - 00:26, 21 January 2024
• 2018 AIME I Problems/Problem 12
  The question asks us for the probability that a randomly chosen subset of the set of the first 18 positive integers has the ...converted set based on <math>S = s(T')</math>, the sum of the elements of a subset <math>T'</math> of <math>U'</math>.
  26 KB (4,044 words) - 13:58, 24 January 2024

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