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• International Mathematical Olympiad
  ...le (inclusive and integers only). Full credit is only given for complete, correct solutions. Each solution is intended to be in the form of a [[proof writin
  3 KB (490 words) - 03:32, 23 July 2023
• American High School Mathematics Examination
  For every correct answer: 5 points
  3 KB (388 words) - 23:07, 5 February 2024
• AIME Problems and Solutions
  ...inally, additions to and improvements on the solutions in the AoPSWiki are always welcome.
  3 KB (391 words) - 16:00, 21 February 2024
• 2004 AMC 12A Problems
  On the AMC 12, each correct answer is worth <math>6</math> points, each incorrect answer is worth <math
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  ...ssumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  ..., and <math>R</math> the area of the red square. Which of the following is correct?
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2003 AMC 12B Problems
  Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch rea
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2005 AMC 12B Problems/Problem 22
  Inductive Step: Suppose the formula is correct for <math>z_k</math>, then ...z|</math>. Therefore, the magnitude of <math>\frac{iz_n}{|z_n|}</math> is always <math>1</math>, meaning that all of the numbers in the sequence <math>z_k</
  4 KB (660 words) - 17:40, 24 January 2021
• 2004 AIME I Problems/Problem 6
  ...4}</math> But, this over-counts since it counts numbers like 0213. We can correct for this over-counting. Lock the first digit as 0 and permute 3 other chose
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME II Problems/Problem 7
  ...clear from the problem setup that <math>0<\theta<\frac\pi2</math>, so the correct value is <math>\tan(\theta)=\frac53</math>. Next, extend rays <math>\overri
  9 KB (1,501 words) - 05:34, 30 October 2023
• 1984 AIME Problems
  ...the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and <math>w</math> is the number of wrong answers. Students are not
  6 KB (933 words) - 01:15, 19 June 2022
• 1988 AIME Problems
  ...ailable ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as
  6 KB (902 words) - 08:57, 19 June 2021
• 2002 AIME II Problems
  ...bsent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1984 AIME Problems/Problem 10
  ...the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and <math>w</math> is the number of wrong answers. (Students are no Let Mary's score, number correct, and number wrong be <math>s,c,w</math> respectively. Then
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1988 AIME Problems/Problem 8
  Since this is an AIME problem, there is exactly one correct answer, and thus, exactly one possible value of <math>f(14,52)</math>.
  4 KB (538 words) - 13:24, 12 October 2021
• 1988 AIME Problems/Problem 1
  ...available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has <math>\{1,2,3,6,9\}</math> as its
  1 KB (181 words) - 18:23, 26 August 2019
• 2006 AMC 10B Problems/Problem 7
  ...As no other option choice fits, <math>\boxed{\textbf{(A)}-x}</math> is the correct solution.
  1 KB (179 words) - 10:33, 19 August 2022
• 1991 AIME Problems/Problem 6
  THIS SOLUTION IS INCORRECT, PLEASE CORRECT IT IF YOU HAVE TIME!
  3 KB (447 words) - 17:02, 24 November 2023
• 1991 AIME Problems/Problem 1
  ...r of <math>11</math> is <math>(5,11)</math>, and checking shows that it is correct.
  4 KB (628 words) - 22:05, 7 June 2021
• 2006 AMC 10B Problems/Problem 25
  ...way to do the problem is by the process of elimination. The only possible correct choices are the highest powers of each prime, <math>2^3=8</math>, <math>3^2
  5 KB (878 words) - 14:39, 3 December 2023
• 2000 AIME I Problems/Problem 15
  ...th> to <math>96</math> alternate in fake-real-fake-real, where we have the correct order of cards once the first <math>96</math> have moved and we can start p ...the spacing of the cards moved, <math>a</math> is an integer such that the correct first card is moved, and <math>k</math> is an integer greater than or equal
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 5
  ...ing <math>2</math> and <math>7</math> for <math>w_1</math> does not give a correct product. Thus, <math>\frac{27}{50}</math> must be a reduced form of the ac
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2001 AIME I Problems/Problem 6
  ...ination counts only one of the permutations; we can say that it counts the correct (ascending order) permutation. ...sformations of the problem, a recursion formula can be a robust way to the correct answer.
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2003 AIME II Problems/Problem 13
  ...th> and <math>C</math>. Oh wait they are symmetric. So then if this is the correct answer, why am I wrong, or what happened to that factor of <math>3</math>?"
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2002 AIME II Problems/Problem 10
  ...bsent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive
  2 KB (336 words) - 19:30, 24 June 2020
• 2000 AIME II Problems/Problem 13
  ...math>, so no real solution exists for <math>x</math>. Thus our solution is correct.
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2005 AMC 10B Problems/Problem 24
  ...math>11</math> while <math>(a-b)</math> is a factor of nine (1 or 9). The correct guesses are <math>a = 6, b = 5</math> causing <math>x = 65, y = 56,</math>
  5 KB (845 words) - 19:23, 17 September 2023
• 2006 AMC 12A Problems/Problem 3
  ...h>, we see <math>6\cdot8=30+18</math>. Therefore, we can see our answer is correct.
  1 KB (155 words) - 17:30, 16 December 2021
• 2006 AMC 12A Problems/Problem 8
  Thus, the correct answer is <math>\boxed{\textbf{(C) }3}.</math> Note to readers: make sure to always read the problem VERY carefully before attempting; it could mean the differ
  3 KB (450 words) - 02:00, 13 January 2024
• 2006 AMC 12A Problems/Problem 9
  ...= 72</math>. 28 is not divisible by 3, so we know that this number is not correct. Moving on to 7, <math>13 \cdot 7 = 91</math>. We know that 9 is a multiple
  3 KB (429 words) - 18:14, 26 September 2020
• Furman University Wylie Mathematics Tournament
  The test is scored as 1 point for each correct answer, 0 points for blank answers, and -.25 for incorrect answers except f
  2 KB (365 words) - 21:21, 18 March 2017
• University of South Carolina High School Math Contest/1993 Exam/Problem 20
  ...lifying of the brackets. Open the brackets and you should notice why it is correct. If you are also wondering whether or not if we got all the sets in the abo
  2 KB (263 words) - 18:13, 19 October 2021
• Pierre de Fermat
  ...rn techniques. However, a flaw was discovered soon after. Wiles managed to correct the proof by October 1994, thus solving the last of Fermat's problems. It i
  5 KB (860 words) - 17:10, 21 March 2023
• P versus NP
  The [[Clay Mathematics Institute]] has offered a USD \$1,000,000 prize for a correct solution, as it has listed it as one of its [[Millennium Problems]].
  6 KB (1,104 words) - 15:11, 25 October 2017
• Congruent (geometry)
  ...given side of <math>A'</math> on the straight line <math>a'</math>, we can always find one and only one point <math>B'</math> so that the segment <math>AB</m Every segment is congruent to itself; that is, we always have
  10 KB (1,655 words) - 21:43, 24 March 2022
• 2005 AMC 10A Problems/Problem 4
  So our correct answer choice is <math>\boxed{\textbf{(B) }\frac{2}{5}x^2}</math>
  2 KB (265 words) - 19:07, 25 December 2022
• Derangement
  ...c{1}{1} + \frac{1}{2} - \frac{1}{6}\right) = 2</math>, which we know to be correct.
  3 KB (473 words) - 12:57, 20 February 2024
• Cobb County Math Meets
  ...ive minute, twenty-five question multiple-choice test worth six points per correct response, and ten ciphering questions. Each ciphering question is worth te
  2 KB (307 words) - 20:25, 12 March 2012
• United States of America Computing Olympiad
  ...s. Every test case has equal weight within the problem. For each test case correct, you receive the points for the test case. Programs get 4 seconds per test Each test case will give feedback on how well your program did. Correct test cases show how much time and memory used. Incorrect test cases are dif
  2 KB (297 words) - 01:41, 21 January 2023
• Cyprus Mathematical Olympiad
  ...ts must answer 30 multiple choice mathematics problems in 60 minutes. Each correct answer gives 4 points, each wrong answer subtracts 1 point and no answer gi
  2 KB (366 words) - 14:26, 4 September 2017
• 1999 AMC 8 Problems/Problem 16
  Altogether, Tori answered <math>7 + 12 + 21 = 40</math> questions correct. ...e needed to answer <math>45 - 40 = 5</math> more questions to pass, so the correct answer is <math>\boxed{(B) 5}</math>
  1 KB (167 words) - 20:30, 11 January 2024
• Special relativity
  ...ciple of Relativity says it is impossible to ascertain that one of them is correct while the other is wrong.
  3 KB (538 words) - 23:39, 2 March 2008
• 2001 AMC 8 Problems
  On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and eac
  13 KB (1,994 words) - 13:04, 18 February 2024
• Georgia Tech High School Math Competition
  ...ng questions, each of which contestants have three minutes to answer. Each correct answer is worth five points. Each team's score is determined by the top fou
  3 KB (475 words) - 21:51, 31 December 2013
• Georgia Junior Varsity State Tournament
  ...twenty-five question multiple-choice test. Four points are given for each correct answer, and 1.1 points are given for each blank answer. On this round a [[
  2 KB (324 words) - 10:23, 24 April 2007
• Vestavia Hills High School Math Tournament
  ...ears. The tests consist of 25 questions which are worth 4 point each for a correct answer, minus 1 point for an incorrect answer. Ciphering rounds- 4 of them,
  1 KB (160 words) - 12:31, 27 December 2006
• GCTM State Math Tournament
  ...ons and five free-response questions; a student's score is five points per correct questions, plus one point per answer left blank; there is no penalty for in
  2 KB (353 words) - 17:02, 7 June 2009
• Georgia Southern Invitational Mathematics Tournament
  ...nior high division; a student's score is forty points plus four points per correct question minus one point per incorrect question. No calculators are permit
  1 KB (154 words) - 09:45, 21 March 2010
• Mock AIME 4 2006-2007 Problems/Problem 10
  ...o we divide the whole sum by 3 and we add or subtract <math>q(n)</math> to correct for the integer based on the modularity of the sum with 3
  4 KB (595 words) - 12:14, 25 November 2023
• 2007 AIME II Problems/Problem 14
  The answer is clearly correct, but the proof has a gap, i.e. there is no reason that <math>f(-2)\neq1</ma
  7 KB (1,335 words) - 17:44, 25 January 2022
• 1989 USAMO Problems/Problem 2
  All possibilities yield a contradiction, so our assumption can not be correct.
  3 KB (438 words) - 01:19, 27 December 2023
• 2007 Cyprus MO/Lyceum/Problems
  ...ac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}</math>, which of the following is correct?
  13 KB (1,990 words) - 08:29, 19 December 2009
• 2007 Cyprus MO/Lyceum/Problem 11
  ...ac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}</math>, which of the following is correct?
  808 bytes (139 words) - 16:56, 6 May 2007
• 2006 Cyprus MO/Lyceum/Problems
  Which of the following is correct, about the graph of <math>f</math>? ...and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct?
  11 KB (1,672 words) - 10:56, 27 April 2008
• Polish Mathematical Olympiad
  ...s have three problems to solve in five hours. Students with three to four correct problems are usually invited to the final round, which has the same format
  1 KB (223 words) - 22:34, 2 January 2008
• 2007 AMC 12B Problems/Problem 5
  The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each pro
  864 bytes (121 words) - 10:50, 4 July 2013
• 2006 Cyprus MO/Lyceum/Problem 4
  Which of the following is correct, about the graph of <math>f</math>? From above, <math>\mathrm{(A)}</math> is not correct because the graph does not intersect the x-axis (it is tangent to it).
  1 KB (216 words) - 10:46, 27 April 2008
• 2006 Cyprus MO/Lyceum/Problem 9
  ...and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct?
  1 KB (182 words) - 10:40, 27 April 2008
• 2007 AMC 12A Problems/Problem 22
  ...ind out that out of the 9 cases, in 4 the value <math>n_{?}</math> has the correct sum of digits. <br/> ...s of whether the sum carries or not, the modulo 9 of the sum of the digits always increases by <math>1</math>.
  15 KB (2,558 words) - 19:33, 4 February 2024
• PMWC Problems and Solutions
  ...inally, additions to and improvements on the solutions in the AoPSWiki are always welcome.
  924 bytes (121 words) - 00:41, 12 October 2013
• 2007 AMC 12B Problems
  The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each pro
  12 KB (1,814 words) - 12:58, 19 February 2020
• 2002 AMC 12A Problems/Problem 2
  ...; the number that Cindy started with is <math>3(43)+9=138</math>. Now, the correct result is <math>\frac{138-3}{9}=\frac{135}{9}=15</math>. Our answer is <mat ...ath>. Solve for <math>x</math> gives us <math>x=138</math>. Therefore, the correct result is <math>\frac{138-3}{9}=\frac{135}{9}=\boxed{\textbf{(A) }15}</math
  1 KB (163 words) - 12:46, 8 November 2021
• 1999 AHSME Problems
  Which one of the following must necessarily be correct?
  13 KB (1,945 words) - 18:28, 19 June 2023
• 2004 AMC 12A Problems/Problem 2
  On the AMC 12, each correct answer is worth <math>6</math> points, each incorrect answer is worth <math
  837 bytes (122 words) - 20:14, 3 July 2013
• Ideal gas law
  ...-[[density]], low-pressure gases. For higher densities, it is necessary to correct this equation. For greater precision, the [[van der Waals equation]] is ano
  1 KB (225 words) - 08:59, 11 March 2008
• 2008 AMC 12B Problems/Problem 23
  ...whence we immediately obtain <math>\framebox[1.2\width]{(A)}</math> as the correct answer.
  5 KB (814 words) - 18:02, 17 January 2023
• 2007 AMC 10A Problems/Problem 9
  Therefore the correct answer is <math>\mathrm{(E)}</math>
  2 KB (268 words) - 12:40, 3 June 2021
• 1995 AHSME Problems
  The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1951 AHSME Problems
  <math> \textbf{(B)}\ \text{In some cases there is more than one correct order in proving certain propositions.} </math> <math> \textbf{(D)}\ \text{It is not possible to arrive by correct reasoning at a true conclusion if, in the given, there is an untrue proposi
  23 KB (3,641 words) - 22:23, 3 November 2023
• 2002 AMC 12B Problems/Problem 12
  For all integers x, <math>x^2</math> is always a positive integer. So solve for <math>\frac{n}{20-n} = 0</math>, getting < ...hat also yield an integer n value, meaning that there are 4 values, so the correct answer is <math>\boxed{(D)}</math>
  4 KB (579 words) - 05:54, 17 October 2023
• 2004 AMC 12B Problems/Problem 23
  ...e of <math>2004</math>, so there is a very high probability that it is the correct answer.
  3 KB (533 words) - 14:52, 29 October 2023
• 2008 AMC 12B Problems/Problem 22
  Subtracting that from 1 to get the probability she can park, the correct answer is <math>\boxed{E}</math>.
  4 KB (653 words) - 11:06, 15 October 2022
• 2008 AIME I Problems/Problem 6
  ...Indeed, note that <math>a(1,k) = 2^{1-1}(1+2k-2)=2k-1</math>, which is the correct formula for the first row. We claim the result by [[induction]] on <math>n<
  3 KB (509 words) - 17:21, 22 March 2018
• NYSML Problems and Solutions
  ...inally, additions to and improvements on the solutions in the AoPSWiki are always welcome.
  1 KB (115 words) - 13:51, 1 May 2008
• Mathleague.org
  |4 points per correct answer, 0 points per blank answer, -1 points per incorrect answer |1 point per correct answer, 0 points otherwise
  5 KB (801 words) - 12:47, 23 September 2023
• 2003 AMC 12B Problems/Problem 18
  ...<math>a+b+c+d=7+5+11+8=31</math>. <math>\boxed{\textbf{(B)}\ 31}</math> is correct.
  6 KB (914 words) - 11:07, 7 September 2023
• 1995 AHSME Problems/Problem 13
  The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit ...either a <math>5</math> (no carry) or a <math>6</math> (carry) to create a correct statement.
  1 KB (223 words) - 13:59, 5 July 2013
• Millennium Problems
  ...of them as soon as the mathematical community had pronounced his solution correct. All but one of these problems had been solved by the meeting in 2000, and
  13 KB (1,969 words) - 17:57, 22 February 2024
• 2008 iTest Problems
  Michael plays catcher for his school's baseball team. He has always been a great player behind the plate, but this year as a junior, Michael's Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of <math>2008^{2008}</math>)?
  71 KB (11,749 words) - 01:31, 2 November 2023
• Asymptote: Macintosh
  ...e are what you really need to compile it - the first 2 are to get into the correct folder), in the terminal, run:
  5 KB (732 words) - 00:47, 13 December 2023
• 2005 AMC 8 Problems
  have divided the number by 2 to get the correct answer. What is the correct
  13 KB (1,821 words) - 22:18, 5 December 2023
• 2003 AMC 12A Problems/Problem 10
  ...ssumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
  1 KB (213 words) - 10:16, 4 July 2013
• Intermediate Math Open
  ...have given up, they turn them in for 4 more. A scoreboard reports how many correct answers various teams have. There are a total of 36 problems, to be complet
  1 KB (243 words) - 17:53, 1 November 2014
• 2002 AMC 10A Problems
  ...9, but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be? ..., and <math>R</math> the area of the red square. Which of the following is correct?
  11 KB (1,733 words) - 11:04, 12 October 2021
• 2002 AMC 10A Problems/Problem 2
  ...}5.5) + (\text{a little})</math>, and as all the options are integers, the correct one is <math>\boxed{\textbf{(C) }6}</math>.
  1,017 bytes (138 words) - 12:26, 8 November 2021
• 2002 AMC 12A Problems/Problem 8
  ..., and <math>P</math> the area of the red square. Which of the following is correct?
  2 KB (345 words) - 12:50, 8 November 2021
• 1962 IMO Problems/Problem 3
  ...if he clearly understands what he is doing- just a little mistake) Here is correct soluton:
  3 KB (581 words) - 02:00, 6 May 2023
• 1963 IMO Problems/Problem 4
  Can someone change this answer so it's correct?
  3 KB (415 words) - 00:51, 15 September 2023
• 1998 AHSME Problems/Problem 18
  Clearly, <math>M = A+C \Longrightarrow</math> the correct answer is <math>\mathrm{(A)}</math>. ...math> which is negative. Thus <math>\mathrm{(A)}</math> is indeed the only correct answer.
  1 KB (187 words) - 14:29, 5 July 2013
• 2000 AMC 10 Problems/Problem 5
  Only <math>1</math> quantity changes, so the correct answer is <math>\boxed{\text{B}}</math>.
  1 KB (242 words) - 01:24, 27 July 2023
• 1985 AJHSME Problems/Problem 25
  ...if one of these is correct, then the other option in its pair must also be correct. There can't be 2 answers. So, the only remaining answer choice is <math>\b
  2 KB (300 words) - 09:07, 22 January 2023
• 1986 AJHSME Problems/Problem 23
  This is equal to the area of a small circle, hence the correct answer is <math>\boxed{\text{(B)}\ 1}</math>.
  2 KB (277 words) - 21:32, 3 July 2013
• 1987 AJHSME Problems
  ...sts of <math>20</math> questions. The scoring is <math>+5</math> for each correct answer, <math>-2</math> for each incorrect answer, and <math>0</math> for e
  12 KB (1,568 words) - 09:35, 31 October 2021
• 1987 AJHSME Problems/Problem 3
  &= 1800 & \text{ Thus \boxed{\text{E}} is the correct answer}
  1 KB (138 words) - 01:03, 22 January 2020
• 2004 AMC 12B Problems/Problem 9
  The entire situation is in the picture below. The correct answer is <math>\boxed{\mathrm{(E)}\ (3,2)}</math>.
  955 bytes (149 words) - 19:56, 3 July 2013
• 2004 AMC 12B Problems/Problem 14
  ...th>. So we place bets on <math>\boxed{\mathbf{(D)}240/13} </math> which is correct!
  7 KB (1,083 words) - 22:41, 23 November 2020
• 2009 AMC 12A Problems/Problem 15
  ...w verify which of <math>n=96</math> and <math>n=97</math> will give us the correct imaginary part.
  4 KB (634 words) - 16:34, 3 December 2020
• 2001 AMC 12 Problems/Problem 8
  Therefore the correct answer is <math>\boxed{\textbf{(C)} \text{ A cone with slant height of } 10
  2 KB (279 words) - 00:32, 30 December 2023
• 2001 AMC 12 Problems/Problem 23
  Note: I think this solution is not correct. the products of the roots are integers do not mean the product of the two Note: I believe this solution is correct. We know that the two real solutions are integers and that the final produc
  6 KB (1,035 words) - 09:18, 3 September 2023
• 2001 AMC 12 Problems/Problem 16
  ...hings on all legs is <math>\frac{1}{2^{8}}</math>. Therefore the number of correct permutations must be <math>\boxed{\frac {16!}{2^8}}</math>.
  3 KB (418 words) - 23:48, 3 July 2022

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