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• Pythagorean Theorem
  ...a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.
  5 KB (886 words) - 13:51, 15 May 2024
• Inequality
  Linear inequalities can be solved much like linear equations to get implicit restrictions upon a variable. H
  12 KB (1,798 words) - 16:20, 14 March 2023
• AMC 10
  ...ity]] and other secondary school math topics. Problems are designed to be solved by students without any background in calculus or trigonometry.
  4 KB (574 words) - 15:28, 22 February 2024
• United States of America Mathematical Olympiad
  ...his two day, 9 hour exam (two 4.5 hour sessions). All six questions can be solved using precalculus methods, but an extensive understanding of the material a
  6 KB (869 words) - 12:52, 20 February 2024
• Completing the square
  ...ving a lot of equations easy. In fact, all [[quadratic equation]]s can be solved by completing the square. All quadratic equations in the form <math>(x+a)^2=b</math> can be solved by taking the [[square root]] of <math>b</math> and subtracting <math>a</ma
  2 KB (422 words) - 16:20, 5 March 2023
• Factoring
  These factorizations are useful for problems that could otherwise be solved by [[Newton sums]] or problems that give a polynomial and ask a question ab
  3 KB (532 words) - 22:00, 13 January 2024
• Quadratic equation
  ...ath>) and <math>x</math> is the unknown variable. Quadratic equations are solved using one of three main strategies: [[factoring]], [[completing the square]
  2 KB (264 words) - 12:04, 15 July 2021
• Contest strategies
  ...wishful thinking” — trying to make a problem look like a similar one solved before.
  3 KB (538 words) - 13:13, 16 January 2021
• Induction
  ...aren't any problems in other areas, such as [[Inequalities]], that can be solved with induction. ...cs Olympiad | IMO]]. A good example of an upper-level problem that can be solved with induction is [http://www.artofproblemsolving.com/Forum/viewtopic.php?p
  5 KB (768 words) - 20:45, 1 September 2022
• Orthic triangle
  .... Problems written in either excenters or the orthic triangle can often be solved by shifting perspective to the other, via the medium of this lemma. Also, n
  8 KB (1,408 words) - 11:54, 8 December 2021
• Constructive counting
  ...olving ability becomes more advanced, there are fewer problems that can be solved by constructive counting alone; the list of examples terminates at a late-i
  12 KB (1,896 words) - 23:55, 27 December 2023
• Casework
  ...Unlike the selections in this article, most problems cannot be completely solved through casework. However, it is crucial as an intermediate step across all
  5 KB (709 words) - 10:28, 19 February 2024
• Common factorizations
  These factorizations are useful for problem that could otherwise be solved by [[Newton sums]] or problems that give a polynomial, and ask a question a
  2 KB (327 words) - 02:06, 28 April 2024
• Math Bee
  ...in 35 minutes, while the second phase has 20 free-response questions to be solved in 15 minutes.
  1 KB (197 words) - 10:59, 14 April 2024
• Functional equation
  ...n''', roughly speaking, is an equation in which some of the unknowns to be solved for are [[function]]s. For example, the following are functional equations
  2 KB (361 words) - 14:40, 24 August 2021
• Trigonometry
  ...lving the trigonometric functions appear frequently on contests. These are solved by clever usage of the trigonometric functions' countless [[Trigonometric i
  8 KB (1,217 words) - 20:15, 7 September 2023
• Modular arithmetic/Introduction
  ...rogression of more difficult and more interesting problems that are easily solved using modular arithmetic.
  15 KB (2,396 words) - 20:24, 21 February 2024
• 2006 AIME I Problems/Problem 10
  ...ight)</math> and <math>\left(\frac 32,2\right)</math>, which can be easily solved to be <math>6x = 2y + 5</math>. Thus, <math>a^2 + b^2 + c^2 = \boxed{065}</
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AMC 12A Problems/Problem 25
  This question can be solved fairly directly by casework and pattern-finding. We give a somewhat more g
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12B Problems/Problem 24
  The slope <math>BC</math> can also be solved similarly, noting that it makes a <math>120^{\circ}</math> angle with <math
  4 KB (761 words) - 09:10, 1 August 2023
• Variable
  ...ny problems [[algebra]]icly. For instance, [[word problem]]s can often be solved using variables to represent desired but unknown [[expression]]s.
  1 KB (203 words) - 21:35, 15 November 2007
• Problem solving
  ...ften says, sometimes, you just have to do something. A problem can’t be solved by just staring at it!
  6 KB (1,039 words) - 17:43, 30 July 2018
• 2004 AIME II Problems/Problem 14
  ...ct" <math>n</math> by summing all the sub-values from the cases that we've solved.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 1984 AIME Problems
  ...</math>. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over <math>80</mat
  6 KB (933 words) - 01:15, 19 June 2022
• 1984 AIME Problems/Problem 10
  ...</math>. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over <math>80</mat Note that if we have the number <math>25</math>, that can be solved via multiple methods, and if we keep repeating our cycle of base numbers, w
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1986 AIME Problems/Problem 3
  Remark: The quadratic need not be solved. The value of <math>ab</math> can be found through Vieta's.
  3 KB (545 words) - 23:44, 12 October 2023
• 1988 AIME Problems/Problem 10
  .... Plugging in our numbers, we have that <math>V - 72 + 26 = 2</math>. When solved, this gives <math>V = 48</math>. Proceed using any of the methods above.
  6 KB (902 words) - 17:40, 19 May 2024
• 1997 AIME Problems/Problem 12
  ...<math>97</math>, take the average to get <math>\boxed{58}</math>. (Note I solved this problem the solution one way but noticed this and this probably genera
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 9
  ...th> satisfies the equation <math>a^2 = a+1</math> (this is the equation we solved to get it). Then, we can simplify <math>a^{12}</math> as follows using the
  4 KB (586 words) - 21:53, 30 December 2023
• 1998 AIME Problems/Problem 7
  We want <math>x_1 +x_2+x_3+x_4 =98</math>. This seems like it can be solved with stars and bars, however note that the quadruples all need to be odd. T
  5 KB (684 words) - 11:41, 13 August 2023
• 2000 AIME I Problems/Problem 7
  ...is the type of problem that makes you think symmetry, but actually can be solved easily with substitution, and other normal technniques
  5 KB (781 words) - 15:02, 20 April 2024
• 2005 AMC 10B Problems/Problem 20
  ...s <math>2</math> away from each other. The mean of the numbers than can be solved using these digits is <math>55555</math>. The total amount of numbers that
  2 KB (356 words) - 19:15, 17 September 2023
• Furman University Wylie Mathematics Tournament
  ...st has consisted of 32 multiple choice questions and two tiebreakers to be solved in 2 hours. The topics tested are generally the standard topics on non-[[C
  2 KB (365 words) - 21:21, 18 March 2017
• Pierre de Fermat
  ...advances were made over the next few hundred years towards it, yet no one solved it. A prize of 100,000 German marks was offered for the problem (though thi
  5 KB (860 words) - 17:10, 21 March 2023
• P versus NP
  ...omial time. <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic'' algorithm in polynomial time. The <math>P</math
  6 KB (1,104 words) - 15:11, 25 October 2017
• Navier-Stokes Equation
  ...he help of [[calculus]]. In practical terms only the simplest cases can be solved in this way and their exact solution is known. These cases often involve no
  3 KB (553 words) - 22:08, 2 May 2022
• System of equations
  The first equation can be solved for <math>x</math>: <center><math> x = 12y + 2. </math></center> Some systems can be solved by taking advantage of specific forms. Such systems can often seem tough t
  5 KB (784 words) - 23:27, 30 July 2020
• Linear congruence
  ...th>p</math> are [[constant]]s and <math>x</math> is the [[variable]] to be solved for.
  1 KB (225 words) - 15:52, 3 August 2022
• University of Georgia Mathematics Tournament
  ...m rounds. In the ciphering round, ten points are given for each correctly solved problem, and there are two minutes permitted for each problem. The written
  1 KB (210 words) - 18:56, 28 September 2006
• Kennesaw State University Mathematics Competition
  ...' schools. It will consist of twenty-five multiple-choice questions to be solved in ninety minutes. Papers will be mailed to Kennesaw State University (KSU
  1 KB (177 words) - 20:14, 12 March 2012
• Mock MathCounts
  ...and also gives the problem solvers more time to make sure the problems are solved correctly.
  26 KB (3,265 words) - 21:34, 20 March 2024
• Rockdale Mathematics Competition
  ...e individual round consists of five pairs of two problems, each pair to be solved in seven minutes. For the purposes of team awards, each question is worth
  2 KB (267 words) - 21:50, 6 March 2016
• 2003 AMC 12A Problems/Problem 15
  ...xed{\mathrm{(C)}\ \frac{\sqrt{3}}{4}-\frac{1}{24}\pi}</math>. We have thus solved the problem.
  3 KB (380 words) - 21:53, 19 March 2022
• 2007 AIME I Problems/Problem 12
  ...>\triangle ADB'</math> is a <math>45 - 60 - 75 \triangle</math>. It can be solved by drawing an [[altitude]] splitting the <math>75^{\circ}</math> angle into
  10 KB (1,458 words) - 20:50, 3 November 2023
• 2007 AIME II Problems/Problem 6
  This problem can be solved via recursion since we are "building a string" of numbers with some conditi
  2 KB (338 words) - 15:30, 7 August 2022
• 2004 IMO Shortlist Problems/N1
  ...on. Thus <math> \displaystyle m </math> does not exist and the problem is solved.
  7 KB (1,166 words) - 22:14, 10 May 2007
• LaTeX:Help
  ...ccur that don't have error messages in the compiler. If your problem isn't solved by this page, consider posting on the AoPS [http://www.artofproblemsolving.
  7 KB (1,109 words) - 20:01, 8 September 2022
• 2001 IMO Problems/Problem 3
  ...solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys. ...he boys are the pigeons, and the guy/pigeon enters a hole/problem which he solved in common with the girl).
  2 KB (453 words) - 07:27, 26 November 2020
• Factor Theorem
  Here are some problems that can be solved using the Factor Theorem:
  3 KB (508 words) - 12:31, 2 May 2024
• 2000 AMC 12 Problems/Problem 15
  This is quite trivially solved, as <math>3x = \dfrac{9x}{3}</math>, so <math>P(3x) = P(9x/3) = 81x^2 + 9x
  3 KB (441 words) - 21:11, 29 April 2023
• 1959 AHSME Problems
  ...C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} </math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 2001 AMC 12 Problems/Problem 4
  <math>\dfrac{1}{3}[(m-10) + 5 + (m + 15)] = m</math>, which can be solved to get <math>m=10</math>.
  2 KB (311 words) - 21:53, 10 February 2024
• Julia Robinson Mathematics Festival
  ...007-08 season there is a series of problem solving challenges which can be solved individually or in small teams. These problems are structured similarly to
  2 KB (252 words) - 22:19, 8 February 2008
• 2008 AMC 12B Problems/Problem 24
  ...motivates us to look at our discriminants in the quadratic formula that is solved for <math>x</math>.
  9 KB (1,482 words) - 13:52, 4 April 2024
• Asian Pacific Mathematics Olympiad
  ...day before). It consists of five problems, worth seven points each, to be solved in one four-hour session. Each country largely grades its own papers. A c
  1 KB (196 words) - 10:24, 12 October 2020
• 2008 AIME II Problems/Problem 14
  which can be easily solved by moving <math>1-\rho^2</math> to RHS and taking square roots. Final answe
  8 KB (1,387 words) - 11:56, 29 January 2024
• 2006 IMO Problems/Problem 5
  ...2(t) = t</math>, which is a contradiction by Lemma 1. Thus the problem is solved. <math>\blacksquare</math>
  3 KB (704 words) - 14:42, 7 September 2021
• 2008 USAMO Problems/Problem 6
  ...of arrangements for <math>n</math> participants. The problem is completely solved.
  13 KB (2,414 words) - 14:37, 11 July 2016
• 2008 Mock ARML 1 Problems/Problem 4
  .../math> and <math>(2)</math> form a two equation linear system which can be solved to find that <math>f(4) = \boxed{\frac{5}{8}}</math>.
  2 KB (242 words) - 09:03, 1 June 2008
• 2001 IMO Shortlist Problems/C8
  ...solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
  588 bytes (90 words) - 18:36, 20 August 2008
• Millennium Problems
  ...ad pronounced his solution correct. All but one of these problems had been solved by the meeting in 2000, and thus it was natural to create a new set of such ...function of the input, often expressed using big-O notation) can also be ''solved'' in polynomial time. The set <math>P</math> consists of decision problems
  13 KB (1,969 words) - 17:57, 22 February 2024
• Implicitly defined function
  Equations defining functions implicitly can sometimes be solved to give the function explicitly. However, this is not generally true, and ...s <math>y</math> as a function of <math>x</math>. This equation cannot be solved for <math>y</math> as an [[elementary function]] in terms of <math>x</math>
  1 KB (241 words) - 19:48, 8 September 2009
• Distinguishability
  One problem that can be solved by this is finding the number of solutions to <math>a_1 + a_2 + \cdots + a_
  5 KB (806 words) - 17:13, 18 May 2024
• 2009 AMC 12B Problems/Problem 14
  <math>c</math> is now solved to be <math>\frac{2}{3}</math>. <math>\boxed{\mathrm{(C)}}</math>.
  4 KB (695 words) - 21:33, 7 October 2023
• MaCh 8
  ...he 1-hour Mach round. This is a fun round consisting of 100 problems to be solved in the hour allotted; a runner obtains the problems in groups of 10. Later
  2 KB (350 words) - 10:09, 1 April 2009
• Purple Comet! Math Meet
  ...answer to a problem is a nonnegative integer. All of the problems can be solved without a calculator, but the test allows using calculators, referring to r
  2 KB (237 words) - 00:25, 3 September 2018
• 1961 AHSME Problems/Problem 5
  The answer is <math>\boxed{\textbf{(C)}}</math>. This problem can also be solved by traditionally expanding and combining like terms (though it would take m
  723 bytes (104 words) - 19:54, 17 May 2018
• Substitution
  3. Repeat until there is a single equation in one variable, which can be solved by means of other methods.
  3 KB (428 words) - 16:53, 10 April 2020
• 2010 AMC 12A Problems/Problem 11
  This problem is quickly solved with knowledge of the laws of exponents and logarithms.
  1 KB (188 words) - 11:31, 4 October 2023
• 2002 AMC 10B Problems/Problem 25
  Therefore, since we correctly solved it and checked our work multiple ways, we can be confident that the answer
  4 KB (679 words) - 19:11, 17 September 2023
• Characteristic polynomial
  ...</math>), we can find a linear system in the <math>a_i</math>s that can be solved for each constant. Refer to the [[#Introductory|introductory problems]] bel
  19 KB (3,412 words) - 14:57, 21 September 2022
• Pell equation
  ...that if <math>D = d^2</math> is a perfect square, then this problem can be solved using [[difference of squares]]. We would have <math>x^2 - Dy^2 = (x+dy)(x-
  6 KB (1,076 words) - 10:20, 29 October 2023
• 2010 AIME I Problems/Problem 4
  This can be solved quickly and easily with [[generating functions]].
  3 KB (470 words) - 22:15, 27 August 2023
• 2010 AIME II Problems/Problem 15
  This problem can be solved with barycentric coordinates. Let triangle <math>ABC</math> be the referenc
  9 KB (1,523 words) - 15:24, 21 November 2023
• 1966 IMO Problems
  ...one problem, half did not solve problem <math>A</math>. How many students solved only problem <math>B</math>?
  3 KB (509 words) - 12:39, 29 January 2021
• 2010 AMC 12B Problems/Problem 25
  When I solved it, I actually ended up trying 2, 3, and 5 just to be safe. It actually end
  4 KB (579 words) - 21:26, 16 February 2021
• 2011 AIME II Problems/Problem 10
  ...s the diagonal or side of quadrilateral. Equation for wrong case cannot be solved. For example,
  11 KB (1,720 words) - 03:12, 18 December 2023
• Olympiad books
  *''Solved and Unsolved Problems in Number Theory'' - '''Daniel Shanks''' - '''Problem ...onometry Problems'' - '''Titu Andreescu, Zuming Feng''' - '''Both''' book (solved examples and approaches + problems).
  17 KB (2,261 words) - 00:30, 22 April 2024
• ...ne of the seven [[Millennium Problems]], and is the only one that has been solved, in 2003 by [[Grigori Perelman]].
  690 bytes (107 words) - 15:12, 31 March 2017
• 1966 IMO Problems/Problem 1
  ...one problem, half did not solve problem <math>A</math>. How many students solved only problem <math>B</math>? ...math>2x - a</math> students solved B only, and <math>x - a</math> students solved C only. Let <math>2y - 1</math> be the number of students solving A; then <
  2 KB (285 words) - 03:14, 2 January 2023
• 2011 IMO Problems/Problem 1
  ...doesn't actually work. -Someone who didn't write up the above solution but solved the problem in a similar way)
  9 KB (1,718 words) - 23:08, 26 June 2014
• 2000 AMC 8 Problems/Problem 24
  ...f you see <math>\triangle BFD</math>, then you can see that the problem is solved quickly after determining <math>\angle BFD</math>.
  2 KB (331 words) - 15:50, 9 May 2023
• 1950 AHSME Problems/Problem 25
  ...)(5^4)}{5^2}= \log_{5}\frac{(5^7)}{5^2}= \log_{5} 5^5 </math> which can be solved to get <math>\boxed{\mathrm{(D)}\ 5}</math>.
  1,008 bytes (123 words) - 00:58, 12 October 2020
• 1973 Canadian MO Problems/Problem 7
  ...ed the previous identity in part <math>\text{i}</math>. Therefore, we have solved the problem.
  2 KB (351 words) - 18:56, 27 May 2022
• 2012 AMC 10B Problems/Problem 18
  This question can be solved by considering all the possibilities: This question can also be solved by using a <math>\dfrac{probability1}{probability2}</math> solution.
  2 KB (379 words) - 14:00, 22 August 2022
• 2017 AMC 8 Problems
  ...correct. Zoe had correct answers to <math>90\%</math> of the problems she solved alone. What was Zoe's overall percentage of correct answers?
  12 KB (1,771 words) - 21:13, 20 January 2024
• 2010 AMC 8 Problems/Problem 18
  Note that we could have solved this without the measurement of <math>30</math> inches.
  2 KB (301 words) - 09:04, 10 March 2023
• 2006 SMT/Advanced Topics Problems
  ...Stanford have incorrect solutions. This problem was definitely written and solved correctly. What is the probability that Smartie thinks its solution is wron
  4 KB (596 words) - 20:09, 27 May 2012
• 2006 SMT/Geometry Problems
  ...\gamma </math> and tangent to the <math> x </math> axis. Find an equation, solved for <math> y </math> if possible, for the locus of possible centers <math>
  5 KB (837 words) - 12:22, 27 May 2012
• 2006 SMT/Geometry Problems/Problem 3
  ...\gamma </math> and tangent to the <math> x </math> axis. Find an equation, solved for <math> y </math> if possible, for the locus of possible centers <math>
  1 KB (200 words) - 12:22, 28 May 2012
• 1984 USAMO Problems
  ...who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
  2 KB (336 words) - 12:33, 18 July 2016
• 2013 AMC 12B Problems/Problem 17
  ...<cmath> 3c^2 - 4c - 20 \leq 0 </cmath> This quadratic inequality is easily solved, and it is seen that equality holds for <math>c = -2</math> and <math>c = \
  7 KB (1,225 words) - 14:59, 8 August 2021
• 2013 AIME II Problems/Problem 13
  ...we just use Heron's formula for <math>[CEB]</math> after <math>x</math> is solved then noticing that <math>[ABC] = 2 \times [CEB]</math>?
  13 KB (2,116 words) - 23:24, 21 March 2024
• KGS math club/solution 11 22
  The first question has been solved (you can find various papers on the net,
  425 bytes (73 words) - 14:00, 16 April 2013
• 2005 IMO Problems/Problem 6
  ...solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.
  394 bytes (56 words) - 01:00, 19 November 2023
• 2014 AIME II Problems/Problem 8
  This problem can be very easily solved using Descartes' Circle Theorem. It states that if we have 4 circles that a
  6 KB (826 words) - 21:31, 9 January 2024
• 1972 IMO Problems/Problem 6
  ...istence of such a tetrahedron XYZW will be proven, and the problem will be solved. ...eft(p_3;\;p_4\right)</math>. Hence, according to the above, the problem is solved.
  7 KB (1,370 words) - 15:42, 29 January 2021
• 1983 IMO Problems/Problem 2
  It can be easily solved with this lemma:
  7 KB (1,267 words) - 23:35, 29 January 2021
• 2018 AIME I Problems/Problem 14
  This is easily solved by recursion/dynamic programming. To simplify the problem somewhat, let us
  4 KB (649 words) - 17:29, 22 December 2023
• 1960 AHSME Problems
  <math>\textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad \\
  21 KB (3,242 words) - 21:27, 30 December 2020

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