https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Djb86&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T09:55:34ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki_talk:Competition_ratings&diff=69906AoPS Wiki talk:Competition ratings2015-04-12T10:10:35Z<p>Djb86: </p>
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<div>12 April 2015 (djb86): Would it be helpful to have the scale first, and then the competition ratings?</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki_talk:Competition_ratings&diff=69905AoPS Wiki talk:Competition ratings2015-04-12T10:10:06Z<p>Djb86: Created page with "Would it be helpful to have the scale first, and then the competition ratings?"</p>
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<div>Would it be helpful to have the scale first, and then the competition ratings?</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Competition_ratings&diff=69586AoPS Wiki:Competition ratings2015-03-27T12:18:55Z<p>Djb86: /* Putnam */</p>
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<div>{{shortcut|[[A:CR]]}}<br />
<br />
This page contains an approximate estimation of the difficulty level of various [[List of mathematics competitions|competitions]]. It is designed with the intention of introducing contests of similar difficulty levels (but possibly different styles of problems) that readers may like to try to gain more experience.<br />
<br />
Each entry groups the problems into sets of similar difficulty levels and suggests an approximate difficulty rating, on a scale from 1 to 10 (from easiest to hardest). Note that many of these ratings are not directly comparable, because the actual competitions have many different rules; the ratings are generally synchronized with the amount of available time, etc. Also, due to variances within a contest, ranges shown may overlap. A sample problem is provided with each entry, with a link to a solution. <br />
<br />
As you may have guessed with time many competitions got more challenging because many countries got more access to books targeted at olympiad preparation. But especially web site where one can discuss olympiad as our very own ML/AoPS! Thus when judging the difficulty level consider the last 10-15 years with more priority.<br />
<br />
If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. [http://www.mathlinks.ro/resources.php?c=182&cid=44 early AMC] problems and 10 is hardest level, e.g. [http://www.mathlinks.ro/resources.php?c=37&cid=47 China IMO Team Selection Test.] When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements.{{ref|1}}<br />
<br />
== Competitions ==<br />
<br />
<br />
=== [[AMC 8]] ===<br />
<br />
* Problem 1 - Problem 12: '''1''' <br />
*: ''What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?'' ([[1989 AJHSME Problems/Problem 10|Solution]])<br />
* Problem 13 - Problem 25: '''2'''<br />
*: ''A fifth number, <math>n</math>, is added to the set <math>\{ 3,6,9,10 \}</math> to make the mean of the set of five numbers equal to its median. What is the number of possible values of <math>n</math>? '' ([[1988 AJHSME Problems/Problem 21|Solution]])<br />
<br />
=== [[AMC 10]] ===<br />
<br />
* Problem 1 - 5: '''1'''<br />
*: ''The larger of two consecutive odd integers is three times the smaller. What is their sum?'' ([[2007 AMC 10A Problems/Problem 4|Solution]])<br />
* Problem 6 - 20: '''2'''<br />
*: ''How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?'' ([[2006 AMC 10A Problems/Problem 19|Solution]])<br />
* Problem 21 - 25: '''3'''<br />
*: ''Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?'' ([[2006 AMC 10B Problems/Problem 25|Solution]])<br />
<br />
=== [[AMC 12]] ===<br />
<br />
* Problem 1-10: '''2'''<br />
*: ''A solid box is <math>15</math> cm by <math>10</math> cm by <math>8</math> cm. A new solid is formed by removing a cube <math>3</math> cm on a side from each corner of this box. What percent of the original volume is removed?'' ([[2003 AMC 12A Problems/Problem 3|Solution]])<br />
* Problem 11-20: '''3'''<br />
*: ''An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?'' ([[2006 AMC 12B Problems/Problem 18|Solution]])<br />
* Problem 21-25: '''4'''<br />
*: ''Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>?'' ([[2009 AMC 12A Problems/Problem 23|Solution]])<br />
<br />
=== [[AIME]] ===<br />
<br />
* Problem 1 - 5: '''3'''<br />
*: ''If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is <math>\tan(x+y)</math>?'' ([[1986 AIME Problems/Problem 3|Solution]])<br />
* Problem 6 - 10: '''4''' <br />
*: ''Triangle <math>ABC</math> has <math>AB=21</math>, <math>AC=22</math> and <math>BC=20</math>. Points <math>D</math> and <math>E</math> are located on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>\overline{DE}</math> is [[parallel]] to <math>\overline{BC}</math> and contains the center of the inscribed circle of triangle <math>ABC</math>. Then <math>DE=m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.'' ([[2001 AIME I Problems/Problem 7|Solution]])<br />
* Problem 11 - 15: '''5.5'''<br />
*: ''A right cone|right circular cone has a base with radius <math>600</math> and height <math> 200\sqrt{7}. </math> A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is <math>125</math>, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is <math>375\sqrt{2}.</math> Find the least distance that the fly could have crawled.'' ([[2004 AIME II Problems/Problem 11|Solution]])<br />
<br />
=== [[APMO]] ===<br />
*Problem 1: '''6'''<br />
*Problem 2: '''7'''<br />
*Problem 3: '''7'''<br />
*Problem 4: '''7.5'''<br />
*Problem 5: '''8'''<br />
<br />
=== [[Austrian MO]] ===<br />
<br />
* Gebietswettbewerb Für Fortgeschrittene, Problems 1-4: '''2''' <br />
* Bundeswettbewerb Für Fortgeschrittene, Teil 1. Problems 1-4: '''3''' <br />
* Bundeswettbewerb Für Fortgeschrittene, Teil 2, Problems 1-6: '''4'''<br />
<br />
=== [[Canadian MO]] ===<br />
<br />
* Problem 1: '''5''' <br />
* Problem 2-3: '''6-6.5''' <br />
* Problems 4-5: '''7'''<br />
<br />
=== [[Indonesian MO]] ===<br />
* Problem 1/5: '''3.5'''<br />
*: '' In a drawer, there are at most <math>2009</math> balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is <math>\frac12</math>. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?'' <url>viewtopic.php?t=294065 (Solution)</url><br />
* Problem 2/6: '''4.5'''<br />
*: ''Find the lowest possible values from the function<br />
<math>f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009</math><br />
<br />
for any real numbers <math>x</math>.''<url>viewtopic.php?t=294067 (Solution)</url><br />
* Problem 3/7: '''5'''<br />
*: ''A pair of integers <math>(m,n)</math> is called ''good'' if<br />
<math>m\mid n^2 + n \ \text{and} \ n\mid m^2 + m</math><br />
<br />
Given 2 positive integers <math>a,b > 1</math> which are relatively prime, prove that there exists a ''good'' pair <math>(m,n)</math> with <math>a\mid m</math> and <math>b\mid n</math>, but <math>a\nmid n</math> and <math>b\nmid m</math>.'' <url>viewtopic.php?t=294068 (Solution)</url><br />
* Problem 4/8: '''6'''<br />
*: ''Given an acute triangle <math>ABC</math>. The incircle of triangle <math>ABC</math> touches <math>BC,CA,AB</math> respectively at <math>D,E,F</math>. The angle bisector of <math>\angle A</math> cuts <math>DE</math> and <math>DF</math> respectively at <math>K</math> and <math>L</math>. Suppose <math>AA_1</math> is one of the altitudes of triangle <math>ABC</math>, and <math>M</math> be the midpoint of <math>BC</math>.<br />
<br />
(a) Prove that <math>BK</math> and <math>CL</math> are perpendicular with the angle bisector of <math>\angle BAC</math>.<br />
<br />
(b) Show that <math>A_1KML</math> is a cyclic quadrilateral.'' <url>viewtopic.php?t=294069 (Solution)</url><br />
<br />
=== [[ARML]] ===<br />
<br />
* Individuals, Problem 1-5,7,9: '''3'''<br />
* Individuals, Problem 6,8: '''4'''<br />
* Individuals, Problem 10: '''6.5'''<br />
* Team/power, Problem 1-5: '''3.5'''<br />
* Team/power, Problem 6-10: '''5'''<br />
<br />
=== [[Balkan MO]] ===<br />
<br />
* Problem 1: '''6'''<br />
*: '' Solve the equation <math>3^x - 5^y = z^2</math> in positive integers. '' <br />
* Problem 2: '''6.5'''<br />
*: '' Let <math>MN</math> be a line parallel to the side <math>BC</math> of a triangle <math>ABC</math>, with <math>M</math> on the side <math>AB</math> and <math>N</math> on the side <math>AC</math>. The lines <math>BN</math> and <math>CM</math> meet at point <math>P</math>. The circumcircles of triangles <math>BMP</math> and <math>CNP</math> meet at two distinct points <math>P</math> and <math>Q</math>. Prove that <math>\angle BAQ = \angle CAP</math>. ''<br />
* Problem 3: '''7.5'''<br />
*: '' A <math>9 \times 12</math> rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres <math>C_1,C_2...,C_{96}</math> in such way that the following to conditions are both fulfilled<br />
<br />
<math>(i)</math> the distances <math>C_1C_2,...C_{95}C_{96}, C_{96}C_{1}</math> are all equal to <math>\sqrt {13}</math><br />
<br />
<math>(ii)</math> the closed broken line <math>C_1C_2...C_{96}C_1</math> has a centre of symmetry? ''<br />
* Problem 4: '''8'''<br />
*: '' Denote by <math>S</math> the set of all positive integers. Find all functions <math>f: S \rightarrow S</math> such that<br />
<br />
<math>f \bigg(f^2(m) + 2f^2(n)\bigg) = m^2 + 2 n^2</math> for all <math>m,n \in S</math>. ''<br />
<br />
=== [[CentroAmerican Olympiad]] ===<br />
* Problem 1: '''4'''<br />
*: ''Find all three-digit numbers <math>abc</math> (with <math>a \neq 0</math>) such that <math>a^{2} + b^{2} + c^{2}</math> is a divisor of 26.'' (<url>viewtopic.php?p=903856#903856 Solution</url>)<br />
* Problem 2,4,5: '''5-6'''<br />
*: ''Show that the equation <math>a^{2}b^{2} + b^{2}c^{2} + 3b^{2} - c^{2} - a^{2} = 2005</math> has no integer solutions.'' (<url>viewtopic.php?p=291301#291301 Solution</url>)<br />
* Problem 3/6: '''6.5''' <br />
*: ''Let <math>ABCD</math> be a convex quadrilateral. <math>I = AC\cap BD</math>, and <math>E</math>, <math>H</math>, <math>F</math> and <math>G</math> are points on <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively, such that <math>EF \cap GH = I</math>. If <math>M = EG \cap AC</math>, <math>N = HF \cap AC</math>, show that <math>\frac {AM}{IM}\cdot \frac {IN}{CN} = \frac {IA}{IC}.</math>'' (<url>viewtopic.php?p=828841#p828841 Solution</url><br />
<br />
=== [[China TST]] ===<br />
<br />
* Problem 1/4: '''7''' <br />
*: ''Given an integer <math>m,</math> prove that there exist odd integers <math>a,b</math> and a positive integer <math>k</math> such that <cmath>2m=a^{19}+b^{99}+k*2^{1000}.</cmath>''<br />
* Problem 2/5: '''8''' <br />
*: ''Given a positive integer <math>n>1</math> and real numbers <math>a_1 < a_2 < \ldots < a_n,</math> such that <math>\dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_n} \le 1,</math> prove that for any positive real number <math>x,</math> <cmath>\left(\dfrac{1}{a_1^2+x} + \dfrac{1}{a_2^2+x} + \ldots + \dfrac{1}{a_n^2+x}\right)^2 \ge \dfrac{1}{2a_1(a_1-1)+2x}.</cmath>''<br />
* Problem 3/6: '''9'''<br />
*: ''Let <math>f : \mathbb{R} \to \mathbb{R}</math> be a function so that for any real numbers <math>x, y,</math><br />
<br />
<cmath>f(x^3+y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2).</cmath><br />
<br />
Show that <math>f(1000)=1000f(1).</math><br />
''<br />
<br />
=== Germany Bundeswettbewerb Mathematik ===<br />
<br />
* Round 1, Problem 1: '''x'''<br />
*: '' Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter? '' <url>viewtopic.php?p=1194585#1194585 (Solution)</url><br />
* Round 1, Problem 2: '''x'''<br />
*: '' Represent the number <math>2008</math> as a sum of natural number such that the addition of the reciprocals of the summands yield 1. '' <url>viewtopic.php?p=1194595#1194595 (Solution)</url><br />
* Round 1, Problem 3: '''x'''<br />
*: '' Prove: In an acute triangle <math>ABC</math> angle bisector <math>w_{\alpha},</math> median <math>s_b</math> and the altitude <math>h_c</math> intersect in one point if <math>w_{\alpha},</math> side <math>BC</math> and the circle around foot of the altitude <math>h_c</math> have vertex <math>A</math> as a common point. '' <url>viewtopic.php?p=1194631#1194631 (Solution)</url><br />
* Round 1, Problem 4: '''x'''<br />
*: '' In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions <math>\{(0,0),(0,1),(1,0),(1,1)\}</math>. Given a finite number of moves can you yield the configuration <math>\{(0,0), (1,1), (3,0), (2, - 1)\}</math> ? '' <url>viewtopic.php?p=1194636#1194636 (Solution)</url><br />
* Round 2, Problem 1: '''x'''<br />
*: '' Determine all real <math>x</math> satisfying the equation<br />
<cmath>\sqrt [5]{x^3 + 2x} = \sqrt [3]{x^5 - 2x}.</cmath><br />
Odd roots for negative radicands shall be included in the discussion. '' <url>viewtopic.php?p=1249364#1249364 (Solution)</url><br />
* Round 2, Problem 2: '''x'''<br />
*: '' Let the positive integers <math>a,b,c</math> chosen such that the quotients <math>\frac {bc}{b + c},</math> <math>\frac {ca}{c + a}</math> and <math>\frac {ab}{a + b}</math> are integers. Prove that <math>a,b,c</math> have a common divisor greater than 1. '' <url>viewtopic.php?p=1249366#1249366 (Solution)</url><br />
* Round 2, Problem 3: '''x'''<br />
*: '' Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black. '' <url>viewtopic.php?p=1249370#1249370 (Solution)</url><br />
<br />
* Round 2, Problem 4: '''x'''<br />
*: '' On a bookcase there are <math>n \geq 3</math> books side by side by different authors. A librarian considers the first and second book from left and exchanges them iff they are not alphabetically sorted. Then he is doing the same operation with the second and third book from left etc. Using this procedure he iterates through the bookcase three times from left to right. Considering all possible initial book configurations how many of them will then be alphabetically sorted? '' <url>viewtopic.php?p=1249370#1249370 (Solution)</url><br />
<br />
=== [[HMMT]] ===<br />
<br />
* Individuals, Problem 1-5: '''4'''<br />
* Individuals, Problem 6-10: '''7'''<br />
<br />
=== [[IberoAmerican Olympiad]] ===<br />
<br />
* Problem 1/4: '''5.5'''<br />
* Problem 2/5: '''6.5'''<br />
* Problem 3/6: <br />
<br />
=== [[IMO]] ===<br />
<br />
* Problem 1/4: '''6.5'''<br />
*: ''Find all functions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that<br />
<center><math>\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}</math></center> for all positive real numbers <math>w,x,y,z,</math> satisfying <math>wx = yz.</math> ([[2008 IMO Problems/Problem 4|Solution]])<br />
''<br />
* Problem 2/5: '''7.5'''<br />
*: ''Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.'' ([[2006 IMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''9.5'''<br />
*: ''Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.'' (<url>viewtopic.php?p=572824#572824 Solution</url>)<br />
<br />
=== [[IMO Shortlist]] ===<br />
<br />
* Problem 1-2: '''5.5-7'''<br />
* Problem 3-4: '''7-8'''<br />
* Problem 5+: '''8-10'''<br />
<br />
=== [[Iran NMO]] ===<br />
<br />
=== [[Iran TST]] ===<br />
<br />
=== [[JBMO]] ===<br />
<br />
* Problem 1: '''4'''<br />
*: ''Find all real numbers <math>a,b,c,d</math> such that <br />
<cmath> \left\{\begin{array}{cc}a+b+c+d = 20,\\ ab+ac+ad+bc+bd+cd = 150.\end{array}\right. </cmath>''<br />
* Problem 2: '''5'''<br />
*: ''Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAC=\angle BDC=36^\circ</math>, <math>\angle CBD=18^\circ</math> and <math>\angle BAC=72^\circ</math>. The diagonals intersect at point <math>P</math>. Determine the measure of <math>\angle APD</math>.''<br />
* Problem 3: '''5'''<br />
*: ''Find all prime numbers <math>p,q,r</math>, such that <math>\frac pq-\frac4{r+1}=1</math>.''<br />
* Problem 4: '''6'''<br />
*: ''A <math>4\times4</math> table is divided into <math>16</math> white unit square cells. Two cells are called neighbors if they share a common side. A '''move''' consists in choosing a cell and changing the colors of neighbors from white to black or from black to white. After exactly <math>n</math> moves all the <math>16</math> cells were black. Find all possible values of <math>n</math>.''<br />
<br />
=== [[Mathcounts]] ===<br />
<br />
* Countdown: '''0.5''' (School, Chapter), '''1''' (State, National)<br />
* Sprint: '''1-1.5''' (school), '''1.5-2.5''' (Chapter, State), '''2-3''' (National)<br />
* Target: '''1.5'' (school), '''2''' (Chapter), '''2.5''' (State), '''2.75''' (National)<br />
<br />
=== [[Miklós Schweitzer]] ===<br />
<br />
* Problem 1-3: <br />
* Problem 4-6: <br />
* Problem 7-9: <br />
* Problem 10-12:<br />
<br />
=== [[MOEMS]] ===<br />
*Division E: '''1'''<br />
*: ''The whole number <math>N</math> is divisible by <math>7</math>. <math>N</math> leaves a remainder of <math>1</math> when divided by <math>2,3,4,</math> or <math>5</math>. What is the smallest value that <math>N</math> can be?'' ([http://www.moems.org/sample_files/SampleE.pdf Solution])<br />
*Division M: '''1'''<br />
*: ''The value of a two-digit number is <math>10</math> times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.'' ([http://www.moems.org/sample_files/SampleM.pdf Solution])<br />
<br />
=== [[Putnam]] ===<br />
<br />
* Problem A/B,1-2: '''6.5'''<br />
*: ''Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola <math>xy = 1</math> and both branches of the hyperbola <math>xy = - 1.</math> (A set <math>S</math> in the plane is called ''convex'' if for any two points in <math>S</math> the line segment connecting them is contained in <math>S.</math>)'' (<url>viewtopic.php?p=978383#p978383 Solution</url>)<br />
* Problem A/B,3-4: '''7.5'''<br />
*: ''Let <math>H</math> be an <math>n\times n</math> matrix all of whose entries are <math>\pm1</math> and whose rows are mutually orthogonal. Suppose <math>H</math> has an <math>a\times b</math> submatrix whose entries are all <math>1.</math> Show that <math>ab\le n</math>.'' (<url>viewtopic.php?p=383280#383280 Solution</url>)<br />
* Problem A/B,5-6: '''9'''<br />
*: ''For any <math>a > 0</math>, define the set <math>S(a) = \{[an]|n = 1,2,3,...\}</math>. Show that there are no three positive reals <math>a,b,c</math> such that <math>S(a)\cap S(b) = S(b)\cap S(c) = S(c)\cap S(a) = \emptyset,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' (<url>viewtopic.php?t=127810 Solution</url>)<br />
<br />
=== [[USAMO]] ===<br />
<br />
* Problem 1/4: '''7'''<br />
*: ''Let <math>\mathcal{P}</math> be a convex polygon with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n - 3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.'' ([[2008 USAMO Problems/Problem 4|Solution]]) <br />
* Problem 2/5: '''8'''<br />
*: ''Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1 + a_2r_2 + a_3r_3 = 0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x \le y</math>, then erase <math>y</math> and write <math>y - x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.'' ([[2008 USAMO Problems/Problem 5|Solution]])<br />
* Problem 3/6: '''8.5'''<br />
*: ''Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n </math> real roots.'' ([[2002 USAMO Problems/Problem 3|Solution]])<br />
<br />
=== [[USAJMO]] ===<br />
* Problem 1/4: '''6.5'''<br />
* Problem 2/5: '''7'''<br />
* Problem 3/6: '''8'''<br />
<br />
=== [[USAMTS]] ===<br />
<br />
* Problem 1-2: '''4'''<br />
*: ''Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle’s area is numerically equal to 6 times its perimeter.'' ([http://usamts.org/Solutions/Solution2_3_16.pdf Solution])<br />
* Problem 3-5: '''5'''<br />
*: ''Call a positive real number groovy if it can be written in the form <math>\sqrt{n} + \sqrt{n + 1}</math> for some positive integer <math>n</math>. Show that if <math>x</math> is groovy, then for any positive integer <math>r</math>, the number <math>x^r</math> is groovy as well.'' ([http://usamts.org/Solutions/Solutions_20_1.pdf Solution])<br />
<br />
=== [[USA TST]] ===<br />
<br />
(seems to vary more than other contests; estimates based on 08 and 09)<br />
<br />
* Problem 1/4/7: '''7.5'''<br />
* Problem 2/5/8: '''8'''<br />
* Problem 3/6/9: '''9'''<br />
<br />
== Scale ==<br />
{{ref|1}} All levels estimated and refer to ''averages''. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. Other contests can be interpolated against this. <br />
<br />
# Problems strictly for beginners, on the easiest elementary school or middle school levels. Examples would be MOEMS, easy Mathcounts questions, #1-20 on AMC 8s, very easy AMC 10/12 questions, and others that involve standard techniques introduced up to the middle school level<br />
# For motivated beginners, harder questions from the previous categories (hardest middle-school level questions, #5-20 on AMC 10, #5-10 on AMC 12, easiest AIME questions, etc).<br />
# For those not too familiar with standard techniques, #21-25 on AMC 10, #11-20ish on AMC 12, #1-5 on AIMEs, and analogous contests. <br />
# Intermediate-leveled problem solvers, the most difficult questions on AMC 12s (#22-25s), more difficult AIME-styled questions #6-10<br />
# Difficult AIME problems (#10-13), others, simple proof-based problems (JBMO etc)<br />
# High-leveled AIME-styled questions, not requiring proofs (#12-15). Introductory-leveled Olympiad-level questions (#1-4s).<br />
# Intermediate-leveled Olympiad-level questions, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc. <br />
# High-level difficult Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc. <br />
# Difficult Olympiad-level questions, eg #3,6s on difficult Olympiad contests. <br />
# Problems occasionally even unsuitable for normal grade school level competitions due to being exceedingly tedious/long/difficult (eg very few students are capable of solving, even on a worldwide basis), or involving techniques beyond high school level mathematics.<br />
<br />
== See also ==<br />
* <url>viewtopic.php?p=1565063#1565063 Forum discussion of wiki entry </url><br />
<br />
[[Category:Mathematics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_16&diff=695852003 AMC 10B Problems/Problem 162015-03-27T08:58:15Z<p>Djb86: /* Problem */ Fix spelling of dessert.</p>
<hr />
<div>==Problem==<br />
<br />
<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year <math>2003</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude><br />
<br />
<math>\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8</math><br />
<br />
==Solution==<br />
<br />
Let <math>m</math> be the number main courses the restaurant serves, so <math>2m</math> is the number of appetizers. Then the number of dinner combinations is <math>2m\times m\times3=6m^2</math>. Since the customer wants to eat a different dinner in all <math>365</math> days of <math>2003</math>, we must have<br />
<br />
<cmath>\begin{align*}<br />
6m^2 &\geq 365\\<br />
m^2 &\geq 60.83\ldots.\end{align*}</cmath><br />
<br />
The smallest integer value that satisfies this is <math>\boxed{\textbf{(E)}\ 8}</math>.<br />
<br />
==See Also==<br />
{{AMC10 box|year=2003|ab=B|num-b=15|num-a=17}}<br />
{{MAA Notice}}</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Number_theory/Advanced_topics&diff=61445Number theory/Advanced topics2014-04-14T19:24:19Z<p>Djb86: /* Elliptic Curves and Modular Forms */ Added a quick introduction. Should this be more of a list, or is this fine?</p>
<hr />
<div>== Algebraic Number Theory ==<br />
[[Algebraic number theory]] studies number theory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of <math>\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})</math>. Famous problems in algebraic number theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]].<br />
<br />
== Analytic Number Theory ==<br />
[[Analytic number theory]] studies number theory from the perspective of [[calculus]], and in particular [[real analysis]] and [[complex analysis]]. The techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].<br />
<br />
== Elliptic Curves and Modular Forms ==<br />
<br />
<br />
===Elliptic curves===<br />
<br />
An elliptic curve is the set of points <math>(x,y)</math> satisfying some two variable third degree equation. Using certain affine transformations it can be shown that it is sufficient to consider those equations which are in Weierstrass form:<br />
<cmath>y^2=x^3+g_2x+g^3.</cmath><br />
Technically, one should consider all pairs <math>(x,y)</math> of complex numbers satisfying such an equation, but often one can study the set of points where both coordinates lie in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the derivative <math>\frac{dy}{dx}</math> tends to infinity as well, and this should serve as some motivation to consider the point infinitely far vertically upward as the point at infinity. We denote it by <math>\mathcal{O}</math>.<br />
<br />
The most important aspect of studying elliptic curves is the fact that there is a natural abelian group structure on its points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commutativity and the existence of an identity and inverses. <br />
<br />
The addition can be described as follows. Take two points <math>P</math> and <math>Q</math> on the elliptic curve. The line through <math>P</math> and <math>Q</math> cuts the curve in a third point <math>R</math>. (One needs to take some care when <math>P=Q</math> or when this line is tangent to the curve, and hence cuts it in only two points.) We define <math>P+Q</math> to be the reflection of <math>R</math> in the <math>x</math>-axis. It takes some effort showing that this defines a group, but it can be done. The point at infinity <math>\mathcal{O}</math> is the identity for this group, and an inverse is obtained by reflecting a point in the <math>y</math>-axis. We may thus summarize the group law by saying <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line.<br />
<br />
===Modular forms===<br />
<br />
Denote by <math>\mathcal{H}</math> the upper half plane (those complex numbers with positive imaginary part). Then there are functions <math>G_4</math> and <math>G_6</math> defined by<br />
<cmath>G_k(z)=\sum_{(c,d)\in Z^2\backslash 0} (cz+d)^{-k}</cmath><br />
called Eisenstein series. If we set <math>g_2(z)=60G_4(z)</math> and <math>g_3(z)=140G_6(z)</math>, then there is a natural association with <math>z</math> of the elliptic curve defined by <math>y^2=x^3+g_2(z)x+g_3(z)</math>. Then every elliptic curve over the complex numbers is isomorphic to one given by some <math>z</math>, and two such curves, associated to <math>z</math> and <math>z'</math> are isomorphic if and only if there is a relation <br />
<cmath>z'=\frac{az+b}{cz+d},\quad a,b,c,d\in\mathbb{Z}, ad-bc=1.</cmath><br />
This encourages us to define an action of the matrix group <math>SL_2(\mathbb{Z})</math> on <math>\mathcal{H}</math> by setting<br />
<cmath>\left(\begin{array}{cc}a&b\\c&d\end{array}\right)z=\frac{az+b}{cz+d}.</cmath><br />
A modular form <math>f</math> is a function such that for all <math>z\in\mathcal{H}</math> and all <math>\gamma\in SL_2(\mathbb{Z})</math> we have<br />
<cmath>f(\gamma z)=(cz+d)^{-k}f(z)</cmath><br />
and such that <math>f</math> is holomorphic on <math>\mathcal{H}</math> and holomorphic at infinity. This last condition means that <math>f</math> can be written as an expansion in the parameter <math>q=e^{2\pi i z}</math> with no negative exponents:<br />
<cmath>f(z)=\sum_{n\ge 0}a_nq^n.</cmath><br />
As an example, the Eisenstein series <math>G_4</math> and <math>G_6</math> are modular forms of weight 4 and 6 respectively.<br />
<br />
===The connection between elliptic curves and modular forms===<br />
<br />
''It would be appreciated if you'd fill this section in.''<br />
<!-- I don't really feel like writing this right now. Any volunteers? --><br />
<br />
== See also ==<br />
* [[Number theory]]<br />
<br />
[[Category:Number theory]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Algebraic_geometry&diff=60851Algebraic geometry2014-03-12T21:50:18Z<p>Djb86: /* Affine Algebraic Varieties */</p>
<hr />
<div>'''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]], which is use of coordinates to solve geometrical problems.<br />
<br />
== Affine Algebraic Varieties ==<br />
<br />
One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^n</math> denote [[affine]] <math>n</math>-space, i.e. a [[vector space]] of [[dimension]] <math>n</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>n</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_n]</math> be the [[polynomial ring]] in <math>n</math> variables, and let <math>I</math> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''.<br />
<br />
== Projective Varieties ==<br />
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.<br />
<br />
== Quasiprojective Varieties ==<br />
<br />
The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.<br />
<br />
== General Algebraic Varieties ==<br />
<br />
Defined in terms of sheafs and patchings. <br />
<br />
== Schemes ==<br />
Let <math>A</math> be a ring and <math>X=\operatorname{Spec}A</math>. An affine scheme is a ringed topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>.<br />
A scheme is a ringed topological space <math>(X,\mathcal{O}_X)</math> admitting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>.<br />
<br />
{{stub}}<br />
<br />
[[Category:Algebra]]<br />
[[Category:Geometry]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Algebraic_geometry&diff=60850Algebraic geometry2014-03-12T21:49:45Z<p>Djb86: /* Affine Algebraic Varieties */</p>
<hr />
<div>'''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]], which is use of coordinates to solve geometrical problems.<br />
<br />
== Affine Algebraic Varieties ==<br />
<br />
One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^n</math> denote [[affine]] <math>n</math>-space, i.e. a [[vector space]] of [[dimension]] <math>k</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>n</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_n]</math> be the [[polynomial ring]] in <math>n</math> variables, and let <math>I</math> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''.<br />
<br />
== Projective Varieties ==<br />
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.<br />
<br />
== Quasiprojective Varieties ==<br />
<br />
The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.<br />
<br />
== General Algebraic Varieties ==<br />
<br />
Defined in terms of sheafs and patchings. <br />
<br />
== Schemes ==<br />
Let <math>A</math> be a ring and <math>X=\operatorname{Spec}A</math>. An affine scheme is a ringed topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>.<br />
A scheme is a ringed topological space <math>(X,\mathcal{O}_X)</math> admitting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>.<br />
<br />
{{stub}}<br />
<br />
[[Category:Algebra]]<br />
[[Category:Geometry]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Category_talk:Stubs&diff=60849Category talk:Stubs2014-03-12T21:35:29Z<p>Djb86: Created page with "Looking at the entries that are labeled as stubs, I see that many of them fall in two categories: (1) entries that are quite advanced for AoPS and (2) entries that don't have a l..."</p>
<hr />
<div>Looking at the entries that are labeled as stubs, I see that many of them fall in two categories: (1) entries that are quite advanced for AoPS and (2) entries that don't have a lot to say about them. While I think entries of type (1) serve some goal and can be expanded, there are many better places to read about them, e.g. on Wikipedia. My bigger concern lies in entries of type (2), for example [[coefficient]]. How much can one really say about coefficients? I understand that not everyone will know what a coefficient is, and the entry could serve simply as a definition. However, that is all that is necessary. So, should this still be a stub? <br />
<br />
One approach may be to keep these entries, but to unlabel them as stubs. Another might be to include this as a subsection in [[polynomial]] and for the coefficient page to redirect to this subsection. I am hoping to start some kind of discussion to see what people prefer, because at the moment there seem (to me) to be too many stubs which will not benefit much from improvement.</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Algebraically_closed&diff=60848Algebraically closed2014-03-12T21:17:37Z<p>Djb86: </p>
<hr />
<div>In [[abstract algebra]], a [[field]] is said to be '''algebraically closed''' if any nonconstant [[polynomial]] with [[coefficient]]s in the field also has a [[root]] in the field. The field of [[complex number]]s, denoted <math>\mathbb{C}</math>, is a well-known example of an algebraically closed field. <br />
<br />
{{stub}}<br />
<br />
[[Category:Field theory]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Number_theory/Advanced_topics&diff=51995Number theory/Advanced topics2013-03-30T21:18:31Z<p>Djb86: /* Algebraic Number Theory */</p>
<hr />
<div>== Algebraic Number Theory ==<br />
[[Algebraic number theory]] studies number theory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of <math>\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})</math>. Famous problems in algebraic number theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]].<br />
<br />
== Analytic Number Theory ==<br />
[[Analytic number theory]] studies number theory from the perspective of [[calculus]], and in particular [[real analysis]] and [[complex analysis]]. The techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].<br />
<br />
== Elliptic Curves and Modular Forms ==<br />
''It would be appreciated if you'd fill this section in.''<br />
<!-- I don't really feel like writing this right now. Any volunteers? --><br />
<br />
== See also ==<br />
* [[Number theory]]<br />
<br />
[[Category:Number theory]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Fundamental_Theorem_of_Algebra&diff=51994Fundamental Theorem of Algebra2013-03-30T19:48:05Z<p>Djb86: Fix brackets</p>
<hr />
<div>The '''fundamental theorem of algebra''' states that every [[nonconstant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some [[analysis]], since the result depends on certain properties of the complex numbers that are most naturally described in [[topology | topological]] terms.<br />
<br />
It follows from the [[division algorithm]] that every complex polynomial of degree <math>n</math> has <math>n</math> complex roots, counting multiplicities. In other words, every polynomial over <math>\mathbb{C}</math> splits over <math>\mathbb{C}</math>, or decomposes into linear factors.<br />
<br />
== Proofs ==<br />
<br />
=== Proof by Liouville's Theorem ===<br />
<br />
We use [[Liouville's Boundedness Theorem]] of [[complex analysis]], which says that every [[bounded]] [[entire function]] is [[constant]].<br />
<br />
Suppose that <math>P(z)</math> is a complex polynomial of degree <math>n</math> with no complex roots; without loss of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0</math>; we now consider the case when <math>n>0</math>.<br />
<br />
Let <math>R</math> be the sum of absolute values of the coefficients of <math>P</math>, so that <math>R \ge 1</math>. Then for <math>\lvert z \rvert \ge S</math>,<br />
<cmath> \lvert P(z) \rvert \ge \lvert z^n \rvert - (R-1) \lvert z^{n-1} \rvert<br />
= \lvert z^{n-1} \rvert \cdot \bigl[ \lvert z \rvert - (R-1) \bigr]<br />
\ge R^{n-1} . </cmath><br />
It follows that <math>1/P(z)</math> is a bounded entire function for <math>\lvert z \rvert > R</math>. On the other hand, by the [[Heine-Borel Theorem]], the set of <math>z</math> for which <math>\lvert z \rvert \le R</math> is a [[compact set]] so its image under <math>1/P</math> is also compact; in particular, it is bounded. Therefore the function <math>1/P(z)</math> is bounded on the entire complex plane when <math>n>0</math>.<br />
<br />
Now we apply Liouville's theorem and see that <math>1/P(z)</math> is constant, so <math>P(z)</math> is a constant polynomial. The theorem then follows. <math>\blacksquare</math><br />
<br />
=== Algebraic Proof ===<br />
<br />
Let <math>P(x)</math> be a polynomial with complex coefficients. Since <math>F(x) = P(x) \overline{P(x)}</math> is a polynomial with real coefficients such that the roots of <math>P</math> are also roots of <math>F</math>, it suffices to show that every polynomial with ''real'' coefficients has a complex root. To this end, let the degree of <math>F</math> be <math>d = 2^n q</math>, where <math>q</math> is odd. We induct on the quantity <math>n</math>.<br />
<br />
For <math>n=0</math>, we note that for sufficiently large negative real numbers <math>x</math>, <math>F(x) < 0</math>; for sufficiently large positive real numbers <math>x</math>, <math>F(x) > 0</math>. It follows from the [[Intermediate Value Theorem]] that <math>F(x)</math> has a real root.<br />
<br />
Now suppose that <math>n > 0</math>. Let <math>C</math> be a [[splitting field]] of <math>F</math> over <math>\mathbb{C}</math>, and let <math>x_1, \dotsc, x_d</math> be the roots of <math>F</math> in <math>C</math>.<br />
<br />
Let <math>c</math> be an arbitrary real number, and let <math>y_{c,i,j} = x_i + x_j + cx_ix_j</math> for <math>1 \le i \le j \le d</math>. Let<br />
<cmath> G_c(x) = \prod_{1 \le i \le j \le d} (x-y_{c,i,j}) . </cmath><br />
The coefficients of <math>G</math> are symmetric in <math>x_1, \dotsc, x_d</math>. Therefore they can be expressed as linear combinations of real numbers times the [[elementary symmetric polynomial]]s in <math>x_1, \dotsc, x_n</math>; thus they are real numbers. Since the degree of <math>G_c</math> is <math>\binom{d+1}{2} = 2^{n-1}q(d+1)</math>, it follows by inductive hypothesis that <math>G_c</math> has a complex root; that is, <math>y_{c,i(c),j(c)} \in \mathbb{C}</math> for some <math>1 \le i(c) \le j(c) \le d</math>.<br />
<br />
Now, since there are infinitely many real numbers but only finitely many integer pairs <math>(i,j)</math> with <math>1 \le i \le j \le d</math>, it follows that for two distinct numbers <math>c,c'</math>, <math>(i(c),j(c)) = (i(c'),j(c')) = (i,j)</math>. It follows that <math>x_i + x_j</math> and <math>x_ix_j</math> are both complex numbers, so <math>x_i</math> and <math>x_j</math> satisfy a quadratic equation with complex coefficients. Hence they are complex numbers. Therefore <math>F</math> has a complex root, as desired. <math>\blacksquare</math><br />
<br />
== References ==<br />
<br />
* Samuel, Pierre (trans. A. Silberger), ''Algebraic Theory of Numbers'', Dover 1970, ISBN 978-0-486-46666-8 .<br />
* [http://www.cut-the-knot.org/fta/analytic.shtml Proofs of the Fundamental Theorem of Algebra on Cut the Knot]<br />
<br />
== See also ==<br />
* [[Algebra]]<br />
<br />
<br />
[[Category:Complex numbers]]<br />
[[Category:Algebra]]<br />
[[Category:Complex Analysis]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Matrix&diff=51993Matrix2013-03-30T19:29:16Z<p>Djb86: /* Matrix Product */</p>
<hr />
<div>A '''matrix''' over a field <math>F</math> is a [[function]] from <math>A\times B</math> to <math>F</math>, where <math>A</math> and <math>B</math> are the sets <math>A=\{1,2,\ldots,m\}</math> and <math>B=\{1,2,\ldots,n\}</math>.<br />
A matrix is usually represented as a rectangular array of scalars from the [[field]], such that each column belongs to the [[vector space]] <math>F^m</math>, where <math>m</math> is the number of rows. If a matrix <math>A</math> has <math>m</math> rows and <math>n</math> columns, its order is said to be <math>m \times n</math>, and it is written as <math>A_{m \times n}</math>.<br />
<br />
The element in the <math>i^{th}</math> row and <math>j^{th}</math> column of <math>A</math> is written as <math>(A)_{ij}</math>. It is more often written as <math>a_{ij}</math>, in which case <math>A</math> can be written as <math>[a_{ij}]</math>.<br />
<br />
==Determinant==<br />
If <math>A_{m\times n}</math> is a matrix over <math>F</math> with <math>m=n</math>, a '''Determinant''' assigns <math>A_{m\times n}</math> to a member of <math>F</math> and is denoted by <math>|A|</math> or <math>\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots<br />
& \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}\end{vmatrix}</math> <br />
<br />
It is defined recursively. <br />
<br />
<center><math>\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\dot{=}a_{11} a_{22} - a_{21} a_{12}</math><br />
<br />
<math>\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots<br />
& \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}\end{vmatrix}\dot{=}\sum_{k=1}^n (-1)^{k+1} a_{1k} |A'_{1k}|</math></center> where <math>A'_{cd}</math> is the matrix <math>A</math> with the <math>c^{th}</math> row and <math>d^{th}</math> column removed. <br />
== Transposes ==<br />
<br />
Let <math>A</math> be <math>[a_{ij}]</math>. Then <math>[a_{ji}]</math> is said to be the transpose of <math>A</math>, written as <math>A^T</math> or simply <math>A'</math>. If A is over the complex field, replacing each element of <math>A^T</math> by its complex conjugate gives us the conjugate transpose <math>A^*</math> of <math>A</math>. In other words, <math>A^*=[\bar {a_{ji}}]</math><br />
<br />
<math>A</math> is said to be symmetric if and only if <math>A=A^T</math>. <math>A</math> is said to be hermitian if and only if <math>A=A^*</math>. <math>A</math> is said to be skew symmetric if and only if <math>A=-A^T</math>. <math>A</math> is said to be skew hermitian if and only if <math>A=-A^*</math>.<br />
<br />
== Matrix Product ==<br />
<br />
Let <math>A</math> be a matrix of order <math>m_1 \times m_2</math> and <math>B</math> a matrix of order <math>n_1 \times n_2</math>. Then the product <math>AB</math> exists if and only if <math>m_2=n_1</math> and in that case we define the product <math>C=AB</math> as the matrix of order <math>m_1 \times n_2</math> for which <br />
<cmath>(C)_{ij}=\sum ^{n_1} _{k=1} (A)_{ik} (B)_{kj}</cmath><br />
for all <math>i</math> and <math>j</math> such that <math>1\le i\le m_1</math> and <math>1\le j\le n_2</math>.<br />
<br />
== Vector spaces associated with a matrix ==<br />
<br />
As already stated before, the columns of <math>A</math> form a subset of <math>F^m</math>. The subspace of <math>F^m</math> generated by these columns is said to be the column space of <math>A</math>, written as <math>C(A)</math>. Similarly, the transposes of the rows form a subset of the vector space <math>F^n</math>. The subspace of <math>F^n</math> generated by these is known as the row space of <math>A</math>, written as <math>R(A)</math>.<br />
<br />
<math>y \in C(A) </math>implies <math>\exists x </math> such that <math> y_{m \times 1} = A_{m \times n} x_{n \times 1}</math><br />
<br />
Similarly, <math>y \in C(A) </math>implies <math>\exists x </math> such that <math> y_{n \times 1} = A^T_{n \times m} x_{m \times 1}</math><br />
<br />
The set <math>\{x:A_{m \times n}x_{n \times 1} = \phi\}</math> forms a subspace of <math>F^n</math>, known as the null space <math>N(A)</math> of <math>A</math>.<br />
<br />
== Rank and nullity ==<br />
<br />
The dimension of <math>C(A)</math> is known as the column rank of <math>A</math>. The dimension of <math>R(A)</math> is known as the row rank of <math>A</math>. These two ranks are found to be equal, and the common value is known as the rank <math>r(A)</math> of <math>A</math>.<br />
<br />
The dimension of <math>N(A)</math> is known as the nullity <math>\eta (A)</math> of A.<br />
<br />
If <math>A</math> is a square matrix of order <math>n \times n</math>, then <math>r(A) + \eta (A) = n</math>.<br />
<br />
[[Category:Linear algebra]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Polynomial&diff=51992Polynomial2013-03-30T18:39:14Z<p>Djb86: Added non-examples</p>
<hr />
<div>A '''polynomial''' is a [[function]] in one or more [[variable]]s that consists of a sum of variables raised to [[nonnegative]], [[integer|integral]] powers and multiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[real]] or [[complex]] numbers; but in [[abstract algebra]] often an arbitrary [[field]]).<br />
<br />
For example, these are polynomials:<br />
* <math>4x^2 + 6x - 9</math>, in the variable <math>x</math><br />
* <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math><br />
* <math>5x^4 - 2x^2 + 9</math>, in the variable <math>x</math><br />
* <math>\sin^2{x} + 5</math>, in the variable <math>\sin x</math><br />
<br />
However, <br />
* <math>\sin^2{x} + 5</math><br />
* <math>\frac{4x+3}{2x-9}</math><br />
*<math>x^{-1}+2+3x+x^2</math><br />
*<math>x^{1/3}=\sqrt[3]{x}</math><br />
are functions, but ''not'' polynomials, in the variable <math>x</math><br />
<br />
==Introductory Topics==<br />
===A More Precise Definition===<br />
<br />
A polynomial in one variable is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. Here, <math>a_i</math> is the <math>i</math>th coefficient and <math>a_n \neq 0</math>. Often, the leading coefficient of a polynomial will be equal to 1. In this case, we say we have a ''monic'' polynomial.<br />
<br />
=== The Degree of a Polynomial ===<br />
<br />
The simplest piece of information that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as the ''degree'' of the polynomial and is written <math>\deg(P)</math>. For instance, <math>\deg(x^2 + 3x + 4) = 2</math> and <math>\deg(x^5 - 1) = 5</math>. When a polynomial is written in the form <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math> with <math>a_n \neq 0</math>, the integer <math>n</math> is the degree of the polynomial. <br />
<br />
The degree, together with the coefficient of the largest term, provides a surprisingly large amount of information about the polynomial: how it behaves in the [[limit]] as the variable grows very large (either in the positive or negative direction) and how many roots it has.<br />
<br />
===Finding Roots of Polynomials===<br />
<br />
====What is a root?====<br />
<br />
A [[root]] is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>. For some polynomials, you can easily set the polynomial equal to zero and solve or otherwise find roots, but in some cases it is much more complicated.<br />
<br />
====The Fundamental Theorem of Algebra====<br />
<br />
The [[Fundamental Theorem of Algebra]] states that any polynomial with [[complex number|complex]] coefficients can be written as<br />
<br />
<math>P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)</math> where <math>k</math> is a constant, the <math>x_i</math> are (not necessarily distinct) complex numbers and <math>n</math> is the degree of the polynomial in exactly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots will be <math>x_1,x_2,...,x_n</math>. This also tells us that the degree of a given polynomial is at least as large as the number of distinct roots of that polynomial. In quadratics roots are more complex and can simply be the sqrt of a prime number.<br />
<br />
====Factoring====<br />
<br />
Different methods of [[factoring]] can help find roots of polynomials. Consider this polynomial:<br />
<br />
<math>x^3 + 3x^2 - 4x - 12 = 0</math><br />
<br />
This polynomial easily factors to:<br />
<br />
<math>(x+3)(x^2-4) = 0</math><br />
<br />
<math>(x+3)(x-2)(x+2) = 0</math><br />
<br />
Now, the roots of the polynomial are clearly -3, -2, and 2.<br />
<br />
====The Rational Root Theorem====<br />
We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. The [[Rational Root Theorem]] states that if <math>P(x)</math> has a rational root <math>\pm\frac{p}{q}</math> and this [[fraction]] is fully reduced, then <math>p</math> is a [[divisor]] of <math>a_0</math> and <math>q</math> is a divisor of <math>a_n</math>. This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It is also especially convenient when dealing with monic polynomials.<br />
<br />
====Descartes' Law of Signs====<br />
By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decartes' Rule of Signs says that for a polynomial <math>P(x)</math>, the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of <math>P(-x)</math>, or is less than that by a multiple of 2.<br />
<br />
===Binomial Theorem===<br />
The [[Binomial Theorem]] can be very useful for factoring and expanding polynomials.<br />
<br />
<br />
===Special Values===<br />
Given the coefficients of a polynomial, it is very easy to figure out the value of the polynomial on different inputs. In some cases, the reverse is also true. The most obvious example is also the simplest: for any polynomial <math>P(x) = a_nx^n + \ldots + a_1 x + a_0</math>, <math>P(0) = a_0</math> so the value of a polynomial at 0 is also the constant coefficient.<br />
<br />
Similarly, <math>P(1) = a_n + a_{n - 1} + \ldots + a_1 + a_0</math>, so the value at 1 is equal to the sum of the coefficients.<br />
<br />
In fact, the value at any point gives us a linear equation in the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given the value of a polynomial <math>P</math> and <math>\deg(P) + 1</math> different points, we can always find the coefficients of the polynomial.<br />
<br />
==Intermediate Topics==<br />
*[[Complex numbers]]<br />
*[[Fundamental Theorem of Algebra]]<br />
*[[Roots of unity]]<br />
<br />
==Olympiad Topics==<br />
<br />
* [[Vieta's formulas]]<br />
* [[Newton's identities]]<br />
* [[Newton sums]]<br />
<br />
==Other Resources==<br />
An extensive coverage of this topic is given in [http://www.artofproblemsolving.com/Resources/Papers/PolynomialsAK.pdf A Few Elementary Properties of Polynomials] by Adeel Khan.<br />
<br />
<br />
== See also ==<br />
* [[Algebra]]<br />
<br />
[[Category:Definition]]<br />
[[Category:Elementary algebra]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Polynomial&diff=51991Polynomial2013-03-30T18:35:57Z<p>Djb86: Slightly modified wording in introduction</p>
<hr />
<div>A '''polynomial''' is a [[function]] in one or more [[variable]]s that consists of a sum of variables raised to [[nonnegative]], [[integer|integral]] powers and multiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[real]] or [[complex]] numbers; but in [[abstract algebra]] often an arbitrary [[field]]).<br />
<br />
For example, these are polynomials:<br />
* <math>4x^2 + 6x - 9</math>, in the variable <math>x</math><br />
* <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math><br />
* <math>5x^4 - 2x^2 + 9</math>, in the variable <math>x</math><br />
* <math>\sin^2{x} + 5</math>, in the variable <math>\sin x</math><br />
<br />
However, <br />
* <math>\sin^2{x} + 5</math><br />
* <math>\frac{4x+3}{2x-9}</math><br />
are functions, but ''not'' polynomials, in the variable <math>x</math><br />
<br />
==Introductory Topics==<br />
===A More Precise Definition===<br />
<br />
A polynomial in one variable is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. Here, <math>a_i</math> is the <math>i</math>th coefficient and <math>a_n \neq 0</math>. Often, the leading coefficient of a polynomial will be equal to 1. In this case, we say we have a ''monic'' polynomial.<br />
<br />
=== The Degree of a Polynomial ===<br />
<br />
The simplest piece of information that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as the ''degree'' of the polynomial and is written <math>\deg(P)</math>. For instance, <math>\deg(x^2 + 3x + 4) = 2</math> and <math>\deg(x^5 - 1) = 5</math>. When a polynomial is written in the form <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math> with <math>a_n \neq 0</math>, the integer <math>n</math> is the degree of the polynomial. <br />
<br />
The degree, together with the coefficient of the largest term, provides a surprisingly large amount of information about the polynomial: how it behaves in the [[limit]] as the variable grows very large (either in the positive or negative direction) and how many roots it has.<br />
<br />
===Finding Roots of Polynomials===<br />
<br />
====What is a root?====<br />
<br />
A [[root]] is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>. For some polynomials, you can easily set the polynomial equal to zero and solve or otherwise find roots, but in some cases it is much more complicated.<br />
<br />
====The Fundamental Theorem of Algebra====<br />
<br />
The [[Fundamental Theorem of Algebra]] states that any polynomial with [[complex number|complex]] coefficients can be written as<br />
<br />
<math>P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)</math> where <math>k</math> is a constant, the <math>x_i</math> are (not necessarily distinct) complex numbers and <math>n</math> is the degree of the polynomial in exactly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots will be <math>x_1,x_2,...,x_n</math>. This also tells us that the degree of a given polynomial is at least as large as the number of distinct roots of that polynomial. In quadratics roots are more complex and can simply be the sqrt of a prime number.<br />
<br />
====Factoring====<br />
<br />
Different methods of [[factoring]] can help find roots of polynomials. Consider this polynomial:<br />
<br />
<math>x^3 + 3x^2 - 4x - 12 = 0</math><br />
<br />
This polynomial easily factors to:<br />
<br />
<math>(x+3)(x^2-4) = 0</math><br />
<br />
<math>(x+3)(x-2)(x+2) = 0</math><br />
<br />
Now, the roots of the polynomial are clearly -3, -2, and 2.<br />
<br />
====The Rational Root Theorem====<br />
We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. The [[Rational Root Theorem]] states that if <math>P(x)</math> has a rational root <math>\pm\frac{p}{q}</math> and this [[fraction]] is fully reduced, then <math>p</math> is a [[divisor]] of <math>a_0</math> and <math>q</math> is a divisor of <math>a_n</math>. This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It is also especially convenient when dealing with monic polynomials.<br />
<br />
====Descartes' Law of Signs====<br />
By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decartes' Rule of Signs says that for a polynomial <math>P(x)</math>, the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of <math>P(-x)</math>, or is less than that by a multiple of 2.<br />
<br />
===Binomial Theorem===<br />
The [[Binomial Theorem]] can be very useful for factoring and expanding polynomials.<br />
<br />
<br />
===Special Values===<br />
Given the coefficients of a polynomial, it is very easy to figure out the value of the polynomial on different inputs. In some cases, the reverse is also true. The most obvious example is also the simplest: for any polynomial <math>P(x) = a_nx^n + \ldots + a_1 x + a_0</math>, <math>P(0) = a_0</math> so the value of a polynomial at 0 is also the constant coefficient.<br />
<br />
Similarly, <math>P(1) = a_n + a_{n - 1} + \ldots + a_1 + a_0</math>, so the value at 1 is equal to the sum of the coefficients.<br />
<br />
In fact, the value at any point gives us a linear equation in the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given the value of a polynomial <math>P</math> and <math>\deg(P) + 1</math> different points, we can always find the coefficients of the polynomial.<br />
<br />
==Intermediate Topics==<br />
*[[Complex numbers]]<br />
*[[Fundamental Theorem of Algebra]]<br />
*[[Roots of unity]]<br />
<br />
==Olympiad Topics==<br />
<br />
* [[Vieta's formulas]]<br />
* [[Newton's identities]]<br />
* [[Newton sums]]<br />
<br />
==Other Resources==<br />
An extensive coverage of this topic is given in [http://www.artofproblemsolving.com/Resources/Papers/PolynomialsAK.pdf A Few Elementary Properties of Polynomials] by Adeel Khan.<br />
<br />
<br />
== See also ==<br />
* [[Algebra]]<br />
<br />
[[Category:Definition]]<br />
[[Category:Elementary algebra]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Pigeonhole_Principle/Solutions&diff=51985Pigeonhole Principle/Solutions2013-03-30T16:58:00Z<p>Djb86: Add solution and remove dead link</p>
<hr />
<div>These are the solutions to the problems related to the '''[[Pigeonhole Principle]]'''. <br />
<br />
==Introductory==<br />
===I1===<br />
The Martian must pull 5 socks out of the drawer to guarantee he has a pair.<br />
In this case the pigeons are the socks he pulls out and the holes are the colors.<br />
Thus, if he pulls out 5 socks, the Pigeonhole Principle states that some two of them have the same color.<br />
Also, note that it is possible to pull out 4 socks without obtaining a pair.<br />
<br />
===I2===<br />
Consider the residues of the elements of <math>S</math>, modulo <math>n</math>. By the Pigeonhole Principle, there exist distinct <math>a, b \in S</math> such that <math>a \equiv b \pmod n</math>, as desired.<br />
<br />
==Intermediate==<br />
===M1===<br />
The maximum number of friends one person in the group can have is n-1, and the minimum is 0. If all of the members have at least one friend, then each individual can have somewhere between <math>1</math> to <math>4</math> friends; as there are <math>n</math> individuals, by pigeonhole there must be at least two with the same number of friends. If one individual has no friends, then the remaining friends must have from <math>1</math> to <math>n-2</math> friends for the remaining friends not to also have no friends. By pigeonhole again, this leaves at least <math>1</math> other person with <math>0</math> friends.<br />
<br />
===M2===<br />
For the difference to be a multiple of 5, the two integers must have the same remainder when divided by 5. Since there are 5 possible remainders (0-4), by the pigeonhole principle, at least two of the integers must share the same remainder. Thus, the answer is 1 (E).<br />
<br />
===M3===<br />
Multiplying both sides by <math>q</math>, we have<br />
<cmath>|xq - p| < \frac{1}{n}.</cmath><br />
Now, we wish to find a <math>q</math> between 1 and <math>n</math> such that <math>xq</math> is within <math>\frac{1}{n}</math> of some integer.<br />
Let <math>\{a\}</math> denote the fractional part of <math>a</math>.<br />
Now, we sort the pigeons <math>\{x\}, \{2x\}, \hdots, \{nx\}</math> into the holes <math>(0, 1/n}), (1/n, 2/n), \hdots, ((n - 1)/n, 1).</math><br />
If any pigeon falls into the first hole, we are done.<br />
Therefore assume otherwise; then some two pigeons <math>\{ix\}, \{jx\} \in (k/n, (k + 1)/n)</math> for <math>1 \le k < n</math>.<br />
Assume, without loss of generality, that <math>j - i > 0</math>.<br />
Then we have that <math>\{(j - i)x\}</math> must fall into the first or last hole, contradiction.<br />
<br />
==Olympiad==<br />
===O1===<br />
By the [[Triangle Inequality]] Theorem, a side of a triangle can be less than the sum of the other two sides. Assume that the first two segments are as short as possible (1&nbsp;inch). For a triangle to not exist, the next term must be 2, then 3, 5, 8, and 13 (this is the [[Fibonacci number|Fibonacci sequence]]). The 7th term is 13, which is greater than 10 inches.<!--Where does pigeonhole come in--><br />
===O2===<br />
For the difference to be a multiple of 7, the integers must have equal [[Modular arithmetic/Introduction |modulo]] 7 residues. To avoid having 15 with the same residue, 14 numbers with different modulo 7 residues can be picked (<math>14 * 7 = 98</math>). Thus, two numbers are left over and have to share a modulo 7 residue with the other numbers under the pigeonhole principle.<br />
===O3===<br />
Label the numbers in the set <math>x_1,\dots,x_{100}</math>, consider the 100 subsets <math>\{x_1\}, \{x_1,x_2\},\dots,\{x_1,\dots,x_{100}\}</math> and for each of these subsets, compute its sum. If none of these sums is divisible by <math>100</math>, then there are <math>100</math> sums and <math>99</math> residue classes mod <math>100</math> (excluding <math>0</math>). Therefore two of these sums are the same mod <math>100</math>, say <math>x_1+\cdots+x_i\equiv x_1+\cdots+x_j\pmod{100}</math> (with <math>i<j</math>). Then <math>x_{i+1}+\cdots+x_j\equiv 0\pmod{100}</math>, and the subset <math>\{x_{i+1},\dots,x_j\}</math> suffices.<br />
<br />
===O4===<br />
Inscribe a regular <math>9</math>-gon, it will divide the circle into <math>9</math> equal arcs. The length of the side of this <math>9</math>-gon is <math>\simeq 1.71</math>, and this is an upper bound on the distance of any two points on the arc. From the pigeonhole principle one of the arcs contains at least two of the points.<br />
<br />
===O5===<br />
The pigeonhole principle is used in [http://usamts.org/Solutions/Solution4_1_18.pdf these solutions (PDF)].<br />
===O6===<br />
{{solution}}<br />
<!--todo--></div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51964Resources for mathematics competitions2013-03-29T16:07:59Z<p>Djb86: Fix a bracket</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
* [https://brilliant.org/ Brilliant] is a website where one can solve problems to gain points and go to higher levels.<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline - but a few mirrors are available, e.g [https://webspace.utexas.edu/ag6823/www/www.kalva.demon.co.uk/index.html here].)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions. [Site no longer exists.]<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions. [Site no longer exists.]<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recommendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal recommendations.]<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
<br />
'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
<br />
== See also ==<br />
<br />
* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51963Resources for mathematics competitions2013-03-29T16:07:25Z<p>Djb86: Add a link to a problem solving site</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
* [[https://brilliant.org/ Brilliant] is a website where one can solve problems to gain points and go to higher levels.<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline - but a few mirrors are available, e.g [https://webspace.utexas.edu/ag6823/www/www.kalva.demon.co.uk/index.html here].)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions. [Site no longer exists.]<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions. [Site no longer exists.]<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recommendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal recommendations.]<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
<br />
'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
<br />
== See also ==<br />
<br />
* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=List_of_mathematics_competitions&diff=51962List of mathematics competitions2013-03-29T15:55:31Z<p>Djb86: Fix link of Competition Ratings</p>
<hr />
<div>This list is intended to be global. If other international or contests from other nations or regions are documented elsewhere, they should be added here as well.<br />
<br />
This is a directory of internal links to more helpful pages about mathematics competitions. This is not the place to list individual competitions.<br />
<br />
<br />
<br />
== International mathematics competitions ==<br />
<br />
[[List of international mathematics competitions]].<br />
<br />
== Regional mathematics competitions ==<br />
<br />
[[List of regional mathematics competitions]].<br />
<br />
== National mathematics competitions ==<br />
=== United States ===<br />
Mathematics competitions in the United States are so numerous that we categorize them according to the level of schooling of competing students.<br />
<br />
*[[List of United States elementary school mathematics competitions]].<br />
*[[List of United States middle school mathematics competitions]].<br />
*[[List of United States high school mathematics competitions]].<br />
*[[List of United States college mathematics competitions]].<br />
<br />
<br />
=== Argentina ===<br />
<br />
[[Argentina mathematics competitions]].<br />
<br />
=== Australia ===<br />
<br />
[[Australia mathematics competitions]].<br />
<br />
=== Austria ===<br />
<br />
[[Austria mathematics competitions]].<br />
<br />
=== Belgium ===<br />
<br />
[[Belgium mathematics competitions]].<br />
<br />
=== Brazil ===<br />
<br />
[[Brazil mathematics competitions]].<br />
<br />
=== Bulgaria ===<br />
<br />
[[Bulgaria mathematics competitions]].<br />
<br />
=== Canada ===<br />
<br />
[[List of Canada mathematics competitions]].<br />
<br />
=== China ===<br />
<br />
[[List of China mathematics competitions]].<br />
<br />
=== Cyprus ===<br />
<br />
[[Cyprus mathematics competitions]].<br />
<br />
=== Denmark ===<br />
<br />
[[Denmark mathematics competitions]].<br />
<br />
=== Germany ===<br />
<br />
[[List of Germany mathematics competitions]].<br />
<br />
=== Hungary ===<br />
<br />
[[Hungary mathematics competitions]].<br />
<br />
=== Greece ===<br />
<br />
[[Greece mathematics competitions]].<br />
<br />
===India===<br />
<br />
[[India mathematics competitions]].<br />
<br />
===Ireland===<br />
<br />
[[Ireland mathematics competitions]].<br />
<br />
===Israel===<br />
<br />
[[Israeli mathematics competitions]].<br />
<br />
=== Mexico ===<br />
<br />
[[Mexico mathematics competitions]].<br />
<br />
=== Netherlands ===<br />
<br />
[[Netherlands mathematics competitions]].<br />
<br />
=== Norway ===<br />
<br />
[[Norway mathematics competitions]].<br />
<br />
=== Philippines ===<br />
<br />
[[Philippines mathematics competitions]].<br />
<br />
=== Poland ===<br />
<br />
[[Poland mathematics competitions]].<br />
<br />
=== Portugal ===<br />
<br />
[[Portugal mathematics competitions]].<br />
<br />
=== Romania ===<br />
<br />
[[Romania mathematics competitions]].<br />
<br />
=== Singapore ===<br />
<br />
[[Singapore mathematics competitions]].<br />
<br />
=== South Korea ===<br />
<br />
[[Korean mathematics competitions]].<br />
<br />
=== Slovakia ===<br />
<br />
[[Slovaki mathematics competitions]].<br />
<br />
=== South Africa ===<br />
<br />
[[South Africa mathematics competitions]].<br />
<br />
=== Sweden ===<br />
<br />
[[Sweden mathematics competitions]].<br />
<br />
=== Thailand ===<br />
<br />
[[Thailand mathematics competitions]].<br />
<br />
=== United Kingdom ===<br />
<br />
[[United Kingdom mathematics competitions]].<br />
<br />
<br />
<br />
=== Uruguay ===<br />
<br />
[[Uruguay mathematics competitions]].<br />
<br />
== See also == <br />
* [http://en.wikipedia.org/wiki/List_of_mathematics_competitions List of mathematics competitions on Wikipedia]<br />
* [[Mathematics competition resources]]<br />
* [[Mathematics scholarships]]<br />
* [[Mathematical olympiads]]<br />
* [[Mathematical problem solving]]<br />
* [[World Federation of National Mathematics Competitions]]<br />
* [[Academic competitions]]<br />
* [[AoPSWiki:Competition ratings]]<br />
<br />
[[Category:Mathematics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51961Resources for mathematics competitions2013-03-29T15:42:12Z<p>Djb86: Correcting spelling</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline - but a few mirrors are available, e.g [https://webspace.utexas.edu/ag6823/www/www.kalva.demon.co.uk/index.html here].)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions. [Site no longer exists.]<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions. [Site no longer exists.]<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recommendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal recommendations.]<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
<br />
'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
<br />
== See also ==<br />
<br />
* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51959Resources for mathematics competitions2013-03-29T15:38:47Z<p>Djb86: Mentioned in 2 links that site no longer exists - should they be removed from the list?</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline - but a few mirrors are available, e.g [https://webspace.utexas.edu/ag6823/www/www.kalva.demon.co.uk/index.html here].)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions. [Site no longer exists.]<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions. [Site no longer exists.]<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recomendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal reccomendations]<br />
<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
<br />
'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
<br />
== See also ==<br />
<br />
* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51957Resources for mathematics competitions2013-03-29T15:29:44Z<p>Djb86: Add a link to a mirror site of Kalva</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline - but a few mirrors are available, e.g [https://webspace.utexas.edu/ag6823/www/www.kalva.demon.co.uk/index.html here].)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions.<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions.<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recomendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal reccomendations]<br />
<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
<br />
'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
<br />
== See also ==<br />
<br />
* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86https://artofproblemsolving.com/wiki/index.php?title=Resources_for_mathematics_competitions&diff=51956Resources for mathematics competitions2013-03-29T15:10:24Z<p>Djb86: Update link of South African MO website</p>
<hr />
<div>The [[Art of Problem Solving]] hosts this [[AoPSWiki]] as well as many other online resources for students interested in [[mathematics competitions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources relevant to the topic.<br />
<br />
* [[Math books]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics websites]]<br />
<br />
<br />
== Math competition classes ==<br />
* [[Art of Problem Solving]] hosts classes that are popular among many of the highest performing students in the United States. [http://www.artofproblemsolving.com/Classes/AoPS_C_PSeries.php AoPS Problem Series].<br />
* [[EPGY]] hosts classes for [[AMC]] students.<br />
<br />
<br />
== Math competition problems ==<br />
=== Problem books ===<br />
Many mathematics competitions sell books of past competitions and solutions. These books can be great supplementary material for avid students of mathematics.<br />
* [[ARML]] has four problem books covering most ARML as well as some [[NYSML]] competitions. However, they are generally difficult to find. Some can be ordered [http://www.arml.com/books.htm here].<br />
* [[MOEMS]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php here] at [[AoPS]].<br />
* [[MathCounts]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php here] at [[AoPS]].<br />
* [[AMC]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php here] at [[AoPS]].<br />
* [[Mandelbrot Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php here] at [[AoPS]].<br />
* [[William Lowell Putnam Mathematical Competition | Putnam Competition]] books are available [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php here] at [[AoPS]].<br />
<br />
<br />
=== Problems online ===<br />
[[Art of Problem Solving]] maintains a very large database of [http://www.artofproblemsolving.com/Forum/resources.php math contest problems]. Many math contest websites include archives of past problems. The [[List of mathematics competitions]] leads to links for many of these competition homepages. Here are a few exmaples:<br />
==== Introductory Problem Solvers ====<br />
* [[Mu Alpha Theta]].org hosts past [http://www.mualphatheta.org/National_Convention/PastTests.aspx contest problems].<br />
* Noetic Learning [http://www.noetic-learning.com/gifted/index.jsp Challenge Math] - Problem Solving for the Gifted Elementary Students .<br />
* Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].<br />
* [[Alabama Statewide High School Mathematics Contest]] [http://mcis.jsu.edu/mathcontest/ homepage].<br />
* The [[South African Mathematics Olympiad]] [http://www.samf.ac.za/QuestionPapers.aspx here] includes many years of past problems with solutions.<br />
* [http://www.beestar.org/index.jsp?adid=106 Beestar.org] - Beestar weekly problem solving tests for grade 1 - 8<br />
<br />
==== Intermediate Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet.<br />
* Past [[Colorado Mathematical Olympiad]] (CMO) problems can be found at the [http://www.uccs.edu/%7Easoifer/olympiad.html CMO homepage].<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.mathprob.com/ MathProb] is a website that presents problems and solutions based by categories and is run by [[User:Anirudh | Anirudh]].<br />
<br />
==== Olympiad Problem Solvers ====<br />
* [[AoPS]] [http://www.artofproblemsolving.com/Forum/resources.php math contest problems and solutions]<br />
* Past [[United States of America Mathematical Talent Search | USAMTS]] problems can be found at the [http://usamts.org USAMTS homepage].<br />
* The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offline.)<br />
* [http://www.qbyte.org/puzzles/ Nick's Mathematical Puzzles] -- Challenging problems with hints and solutions.<br />
* [[Canadian Mathematical Olympiad]] are hosted [http://www.cms.math.ca/Competitions/CMO/ here by the Canadian Mathematical Society].<br />
* [http://members.tripod.com/%7EPertselV/RusMath.html Problems of the All-Soviet-Union math competitions 1961-1986] - Many problems, no solutions.<br />
* Past [[International Mathematical Talent Search]] (IMTS) problems can be found [http://www.cms.math.ca/Competitions/IMTS/ here]<br />
* [http://www.geocities.com/CapeCanaveral/Lab/4661/ Olympiad Math Madness] - Stacks of challenging problems, no solutions.<br />
<br />
== Articles ==<br />
<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Contests.php Pros and Cons and Math Competitions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_CultureofExpectation.php Establishing a Positive Culture of Expectation in Math Education] by [[Sister Scholastica Award]] winner Darryl Hill.<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php Stop Making Stupid Mistakes] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Questions.php Stupid Questions] by [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_Teaching.php Learning Through Teaching]<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[user:MCrawford | Mathew Crawford]].<br />
<br />
== A Huge List of Links ==<br />
=== AoPS Course Recommendations ===<br />
[http://www.artofproblemsolving.com/School/recommendations.php Art of Problem Solving Course Recomendations]<br />
[http://www.artofproblemsolving.com/Store/personalrec.php Still have trouble deciding which course? Ask for personal reccomendations]<br />
<br />
<br />
===AMC 8 Preparation===<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=42 AMC 8 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_8_Problems_and_Solutions AMC 8 Problems in the AoPS wiki]<br />
<br />
===AMC 10/12 Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=388108&hilit=preparation How preparing for the AIME will help AMC 10/12 Score] <br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396741&hilit=preparation What class to take?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=387918&hilit=preparation AMC 10 for AMC 12 practice]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385418&hilit=preparation AMC prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384828&hilit=preparation AMC 10/12 Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=384747&hilit=preparation AIME/AMC 10 Overlap and Preparation]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=378851&hilit=preparation How to prepare for amc10 and aime?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=369849&hilit=preparation Preparation for AMC 10?]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43 AMC 10 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_10_Problems_and_Solutions AMC 10 Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44 AMC 12 Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AHSME_Problems_and_Solutions AHSME (Old AMC 12) Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AMC_12_Problems_and_Solutions AMC 12 Problems in the AoPS Wiki]<br />
<br />
===AIME Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397954&hilit=preparation Studying to qualify for USAMO]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=400442&hilit=preparation How to prepare for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399160&hilit=preparation Preparation for the AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=357602&hilit=preparation Using non-AIME questions to prepare for AIME]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=355918&hilit=preparation Best books to prepare for AIME?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344816&hilit=preparation How to improve AIME score to make JMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=341827&hilit=preparation Preparation for AIME and USAMO]<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45 AIME Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/AIME_Problems_and_Solutions AIME Problems in the AoPS Wiki]<br />
<br />
===Beginning Olympiad Preparation===<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=480253 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=481746&p=2698978 General]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401061&hilit=preparation How to Prepare for USAJMO?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=399023&hilit=preparation USAMO preparation/doing problems]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=396736&hilit=preparation Easier Olympiads for USAJMO practice?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=366383&hilit=preparation For the USAMO: ACoPS or Engel?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=360619&hilit=preparation Olympiad problems- how to prepare]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=354103&hilit=preparation USAMO/Olympiads Preparation: Where to start?]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=preparation USAJMO prep]<br />
====Bunch of General links====<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=31888&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=71008&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=79077&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=81296&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=143168&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=273572&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=294132&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=344929&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385092&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=397424&hilit=olympiad+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401201&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=401640&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=406402&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=411476&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=419800&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=447454]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=453638&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=474960&hilit=USAMO+prep]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=385654]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=420845]<br />
'''[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2379622#p2379622]'''<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176 USAJMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAJMO_Problems_and_Solutions USAJMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
<br />
===Middle/Advanced Olympiad Preparation===<br />
<br />
====Problems====<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38803 Practice Olympiad 1]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38804 Practice Olympiad 2]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38805 Practice Olympiad 3]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38806 Practice Olympiad Solutions]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27 USAMO Problems in the Resources Section] <br />
[http://www.artofproblemsolving.com/Wiki/index.php/USAMO_Problems_and_Solutions USAMO Problems in the AoPS Wiki]<br />
[http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO Problems in the Resources Section]<br />
[http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions IMO Problems in the AoPS Wiki]<br />
<br />
<br />
===Book Links:===<br />
====Olympiad Level====<br />
=====Free=====<br />
[http://analgeomatica.files.wordpress.com/2008/11/geometryrevisited_coxetergreitzer_0883856190.pdf Geometry Revisited]<br />
[http://www-math.mit.edu/~kedlaya/geometryunbound/gu-060118.pdf Geometry Unbound]<br />
[http://students.imsa.edu/~tliu/Math/planegeo.pdf Plane Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38802 Hidden Discoveries -- How To]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38801 Infinity]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38816 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38817 Diophantine Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38818 More Diophantine Number Theory]<br />
<br />
=====Not Free=====<br />
[http://www.amazon.com/Plane-Euclidean-Geometry-Theory-Problems/dp/0953682366/ref=sr_1_1?s=books&ie=UTF8&qid=1338742080&sr=1-1 Plane Euclidean Geometry: Theory and Problems]<br />
[http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100/ref=sr_1_1?s=books&ie=UTF8&qid=1338742131&sr=1-1 Complex Numbers and Geometry]<br />
[http://www.amazon.com/Geometry-Complex-Numbers-Dover-Mathematics/dp/0486638308/ref=sr_1_1?s=books&ie=UTF8&qid=1338742156&sr=1-1 Geometry of Complex Numbers]<br />
[http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?s=books&ie=UTF8&qid=1338741912&sr=1-1 Complex Numbers from A to …Z]<br />
[http://www.amazon.com/103-Trigonometry-Problems-Training-Team/dp/0817643346/ref=sr_1_1?s=books&ie=UTF8&qid=1338742048&sr=1-1 103 Trigonometry Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/An-Introduction-Diophantine-Equations-Problem-Based/dp/0817645489/ref=sr_1_1?ie=UTF8&qid=1338741533&sr=8-1 An Introduction to Diophantine Equations: A Problem-Based Approach]<br />
[http://www.amazon.com/Introductions-Number-Theory-Inequalities-Bradley/dp/0953682382/ref=sr_1_1?s=books&ie=UTF8&qid=1338741653&sr=1-1 Introductions to Number Theory and Inequalities]<br />
[http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=sr_1_1?s=books&ie=UTF8&qid=1338741697&sr=1-1 104 Number Theory Problems: From the Training of the USA IMO Team]<br />
[http://www.amazon.com/102-Combinatorial-Problems-Titu-Andreescu/dp/0817643176/ref=sr_1_1?s=books&ie=UTF8&qid=1338741741&sr=1-1 102 Combinatorial Problems]<br />
[http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/8181283368/ref=sr_1_2?s=books&ie=UTF8&qid=1338741874&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies]<br />
[http://www.amazon.com/Mathematical-Olympiads-1972-1986-Problems-Solutions/dp/0883856344/ref=sr_1_fkmr1_1?s=books&ie=UTF8&qid=1338742228&sr=1-1 -fkmr1 USA Mathematical Olympiads 1972-1986 Problems and Solutions]<br />
[http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=art+and+craft+of+problem+solving Art and Craft of Problem Solving]<br />
[http://www.amazon.com/Problem-Solving-Strategies-Problem-Books-Mathematics/dp/0387982191/ref=sr_1_1?ie=UTF8&qid=1338865322&sr=8-1 Problem Solving Strategies]<br />
<br />
===Problem Sets===<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=1068 31 Olympiad problems about Probabilistic Method]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30721 567 Nice and Hard Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32201 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32093 100 Polynomial Problems]<br />
[http://http://www.artofproblemsolving.com/Forum/download/file.php?id=32270 161 Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=31329 Trigonometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32212 General all levels]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32310 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32228 Olympiad Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33993 33 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33874 Induction Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33873 Induction Solutions]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32128 260 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30649 150 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35398 50 Diophantine Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35716 60 Geometry Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33026 116 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32361 Algebraic Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33543 100 Combinatorics Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=30597 100 Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=32007 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37234 Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37233 General]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33551 100 Number Theory Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=33486 100 Functional Equation Problems]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37457 Beginning/Intermediate Counting and Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=37628 40 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35164 100 Geometric Inequalities]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38538 10 Fun Unconventional Problems :)]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=35831 169 Functional Equations]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38916 Triangle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38915 Probability]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38914 Algebra]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38920 Number Theory]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38918 Circle Geometry]<br />
[http://www.artofproblemsolving.com/Forum/download/file.php?id=38919 Other Geometry]<br />
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'''[http://www.artofproblemsolving.com/Wiki/index.php/AoPSWiki:Competition_ratings Ranking of all Olympiads (Difficulty Level)]'''<br />
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== See also ==<br />
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* [[List of mathematics competitions]]<br />
* [[Mathematics scholarships]]<br />
* [[Science competitions]]<br />
* [[Informatics competitions]]</div>Djb86