# Difference between revisions of "AoPS Wiki:Competition ratings"

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== Competitions == | == Competitions == | ||

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=== [[AMC 8]] === | === [[AMC 8]] === | ||

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* Problem 14 - 15: '''6''' | * Problem 14 - 15: '''6''' | ||

*: ''Find the largest integer <math>n</math> satisfying the following conditions: (i) <math>n^2</math> can be expressed as the difference of two consecutive cubes; (ii) <math>2n + 79</math> is a perfect square.'' ([[2008 AIME II Problems/Problem 15|Solution]]) | *: ''Find the largest integer <math>n</math> satisfying the following conditions: (i) <math>n^2</math> can be expressed as the difference of two consecutive cubes; (ii) <math>2n + 79</math> is a perfect square.'' ([[2008 AIME II Problems/Problem 15|Solution]]) | ||

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+ | === [[APMO]] === | ||

+ | Problem 1: '''6.5''' | ||

+ | Problem 2: '''7''' | ||

+ | |||

+ | === [[Austrian MO]] === | ||

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+ | * Gebietswettbewerb Für Fortgeschrittene, Problems 1-4: '''2''' | ||

+ | * Bundeswettbewerb Für Fortgeschrittene, Teil 1. Problems 1-4: '''3''' | ||

+ | * Bundeswettbewerb Für Fortgeschrittene, Teil 2, Problems 1-6: '''4''' | ||

=== [[ARML]] === | === [[ARML]] === | ||

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* Team/power, Problem 1-5: '''3.5''' | * Team/power, Problem 1-5: '''3.5''' | ||

* Team/power, Problem 6-10: '''5''' | * Team/power, Problem 6-10: '''5''' | ||

+ | |||

+ | === [[China TST]] === | ||

+ | |||

+ | * Problem 1/4: | ||

+ | * Problem 2/5: | ||

+ | * Problem 3/6: '''8.5''' | ||

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+ | === Germany Bundeswettbewerb Mathematik === | ||

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+ | * Round 1, Problem 1: | ||

+ | * Round 1, Problem 2: | ||

+ | * Round 1, Problem 3: | ||

+ | * Round 1, Problem 4: | ||

+ | * Round 2, Problem 1: | ||

+ | * Round 2, Problem 2: | ||

+ | * Round 2, Problem 3: | ||

+ | * Round 2, Problem 4: | ||

=== [[HMMT]] === | === [[HMMT]] === | ||

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* Individuals, Problem 1-5: '''4''' | * Individuals, Problem 1-5: '''4''' | ||

* Individuals, Problem 6-10: '''5''' | * Individuals, Problem 6-10: '''5''' | ||

+ | |||

+ | === [[IMO]] === | ||

+ | |||

+ | * Problem 1/4: '''6.5''' | ||

+ | *: ''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 | ||

+ | <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]]) | ||

+ | '' | ||

+ | * Problem 2/5: '''7.5''' | ||

+ | *: ''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]]) | ||

+ | * Problem 3/6: '''9''' | ||

+ | *: ''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>) | ||

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+ | === [[IMO Shortlist]] === | ||

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+ | * Problem 1-2: '''5''' | ||

+ | * Problem 3-4: '''7''' | ||

+ | * Problem 5+: '''8.5+''' | ||

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+ | === [[Iran NMO]] === | ||

+ | |||

+ | === [[Iran TST]] === | ||

=== [[JBMO]] === | === [[JBMO]] === | ||

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*: ''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>.'' | *: ''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>.'' | ||

− | === [[ | + | === [[Mathcounts]] === |

− | + | === [[Miklós Schweitzer]] === | |

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− | + | * Problem 1-3: | |

+ | * Problem 4-6: | ||

+ | * Problem 7-9: | ||

+ | * Problem 10-12: | ||

− | * | + | === [[MOEMS]] === |

− | * | + | *Division E: '''1''' |

− | + | *: ''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]) | |

− | + | *Division M: '''1''' | |

− | + | *: ''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]) | |

− | * | ||

− | * | ||

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− | === [[ | + | === [[Putnam]] === |

− | * Problem 1-2: '''5''' | + | * Problem A/B,1-2: '''5.5''' |

− | * Problem 3-4: '''7''' | + | *: ''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>) |

− | * Problem 5 | + | * Problem A/B,3-4: '''7''' |

+ | *: ''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>) | ||

+ | * Problem A/B,5-6: '''9''' | ||

+ | *: ''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) = \phi,S(a)\cup S(b)\cup S(c) = \{1,2,3,...\}</math>.'' (<url>viewtopic.php?t=127810 Solution</url>) | ||

=== [[USAMO]] === | === [[USAMO]] === | ||

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*: ''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]]) | *: ''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]]) | ||

− | === [[ | + | === [[USAMTS]] === |

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− | + | * Problem 1-2: '''4''' | |

+ | *: ''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]) | ||

+ | * Problem 3-5: '''5''' | ||

+ | *: ''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]) | ||

=== [[USA TST]] === | === [[USA TST]] === | ||

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* Problem 3/6/9: '''8.5''' | * Problem 3/6/9: '''8.5''' | ||

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== Scale == | == Scale == |

## Revision as of 17:19, 27 July 2009

This page contains an approximate estimation of the difficulty level of various 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.

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.

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.

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, i.e. early AMC problems and 10 is hardest level, i.e. China IMO Team Selection Test. When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements.^{[1]}

## Contents

- 1 Competitions
- 1.1 AMC 8
- 1.2 AMC 10
- 1.3 AMC 12
- 1.4 AIME
- 1.5 APMO
- 1.6 Austrian MO
- 1.7 ARML
- 1.8 China TST
- 1.9 Germany Bundeswettbewerb Mathematik
- 1.10 HMMT
- 1.11 IMO
- 1.12 IMO Shortlist
- 1.13 Iran NMO
- 1.14 Iran TST
- 1.15 JBMO
- 1.16 Mathcounts
- 1.17 Miklós Schweitzer
- 1.18 MOEMS
- 1.19 Putnam
- 1.20 USAMO
- 1.21 USAMTS
- 1.22 USA TST

- 2 Scale
- 3 See also

## Competitions

### AMC 8

- Problem 1 - Problem 12:
**1***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?*(Solution)

- Problem 13 - Problem 25:
**2***A fifth number, , is added to the set to make the mean of the set of five numbers equal to its median. What is the number of possible values of ?*(Solution)

### AMC 10

- Problem 1 - 5:
**1***The larger of two consecutive odd integers is three times the smaller. What is their sum?*(Solution)

- Problem 6 - 22:
**2***How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?*(Solution)

- Problem 18 - 25:
**3***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?*(Solution)

### AMC 12

- Problem 1-5:
**2***A solid box is cm by cm by cm. A new solid is formed by removing a cube cm on a side from each corner of this box. What percent of the original volume is removed?*(Solution)

- Problem 3-18:
**3***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?*(Solution)

- Problem 22-25:
**4***Functions and are quadratic, , and the graph of contains the vertex of the graph of . The four -intercepts on the two graphs have -coordinates , , , and , in increasing order, and . The value of is , where , , and are positive integers, and is not divisible by the square of any prime. What is ?*(Solution)

### AIME

- Problem 1 - 5:
**3***If and , what is ?*(Solution)

- Problem 5 - 10:
**4** - Problem 11 - 15:
**5***A right cone|right circular cone has a base with radius and height A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is , 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 Find the least distance that the fly could have crawled.*(Solution)

- Problem 14 - 15:
**6***Find the largest integer satisfying the following conditions: (i) can be expressed as the difference of two consecutive cubes; (ii) is a perfect square.*(Solution)

### APMO

Problem 1: **6.5**
Problem 2: **7**

### Austrian MO

- Gebietswettbewerb Für Fortgeschrittene, Problems 1-4:
**2** - Bundeswettbewerb Für Fortgeschrittene, Teil 1. Problems 1-4:
**3** - Bundeswettbewerb Für Fortgeschrittene, Teil 2, Problems 1-6:
**4**

### ARML

- Individuals, Problem 1-5,7,9:
**3** - Individuals, Problem 6,8:
**4** - Individuals, Problem 10:
**6.5** - Team/power, Problem 1-5:
**3.5** - Team/power, Problem 6-10:
**5**

### China TST

- Problem 1/4:
- Problem 2/5:
- Problem 3/6:
**8.5**

### Germany Bundeswettbewerb Mathematik

- Round 1, Problem 1:
- Round 1, Problem 2:
- Round 1, Problem 3:
- Round 1, Problem 4:
- Round 2, Problem 1:
- Round 2, Problem 2:
- Round 2, Problem 3:
- Round 2, Problem 4:

### HMMT

- Individuals, Problem 1-5:
**4** - Individuals, Problem 6-10:
**5**

### IMO

- Problem 1/4:
**6.5***Find all functions (so is a function from the positive real numbers) such that*

for all positive real numbers satisfying (Solution)

- Problem 2/5:
**7.5***Let be a polynomial of degree with integer coefficients, and let be a positive integer. Consider the polynomial , where occurs times. Prove that there are at most integers such that .*(Solution)

- Problem 3/6:
**9***Assign to each side of a convex polygon the maximum area of a triangle that has as a side and is contained in . Show that the sum of the areas assigned to the sides of is at least twice the area of .*(<url>viewtopic.php?p=572824#572824 Solution</url>)

### IMO Shortlist

- Problem 1-2:
**5** - Problem 3-4:
**7** - Problem 5+:
**8.5+**

### Iran NMO

### Iran TST

### JBMO

- Problem 1:
**4***Find all real numbers such that*

- Problem 2:
**5***Let be a convex quadrilateral with , and . The diagonals intersect at point . Determine the measure of .*

- Problem 3:
**5***Find all prime numbers , such that .*

- Problem 4:
**6***A table is divided into 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 moves all the cells were black. Find all possible values of .

### Mathcounts

### Miklós Schweitzer

- Problem 1-3:
- Problem 4-6:
- Problem 7-9:
- Problem 10-12:

### MOEMS

- Division E:
**1***The whole number is divisible by . leaves a remainder of when divided by or . What is the smallest value that can be?*(Solution)

- Division M:
**1***The value of a two-digit number is times more than the sum of its digits. The units digit is 1 more than twice the tens digit. Find the two-digit number.*(Solution)

### Putnam

- Problem A/B,1-2:
**5.5***Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola and both branches of the hyperbola (A set in the plane is called*convex*if for any two points in the line segment connecting them is contained in )*(<url>viewtopic.php?p=978383#p978383 Solution</url>)

- Problem A/B,3-4:
**7***Let be an matrix all of whose entries are and whose rows are mutually orthogonal. Suppose has an submatrix whose entries are all Show that .*(<url>viewtopic.php?p=383280#383280 Solution</url>)

- Problem A/B,5-6:
**9***For any , define the set . Show that there are no three positive reals such that .*(<url>viewtopic.php?t=127810 Solution</url>)

### USAMO

- Problem 1/4:
**5.5***Let be a convex polygon with sides, . Any set of diagonals of that do not intersect in the interior of the polygon determine a*triangulation*of into triangles. If is regular and there is a triangulation of consisting of only isosceles triangles, find all the possible values of .*(Solution)

- Problem 2/5:
**7***Three nonnegative real numbers , , are written on a blackboard. These numbers have the property that there exist integers , , , not all zero, satisfying . We are permitted to perform the following operation: find two numbers , on the blackboard with , then erase and write in its place. Prove that after a finite number of such operations, we can end up with at least one on the blackboard.*(Solution)

- Problem 3/6:
**8.5***Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree with real coefficients is the average of two monic polynomials of degree with real roots.*(Solution)

### USAMTS

- Problem 1-2:
**4***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.*(Solution)

- Problem 3-5:
**5***Call a positive real number groovy if it can be written in the form for some positive integer . Show that if is groovy, then for any positive integer , the number is groovy as well.*(Solution)

### USA TST

(seems to vary more than other contests; estimates based on 08 and 09)

- Problem 1/4/7:
**7.5** - Problem 2/5/8:
**8** - Problem 3/6/9:
**8.5**

## Scale

^{[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.

- 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
- 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).
- 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.
- Intermediate-leveled problem solvers, the most difficult questions on AMC 12s (#22-25s), more difficult AIME-styled questions #6-10
- Difficult AIME problems (#10-13), others, simple proof-based problems (JBMO etc)
- High-leveled AIME-styled questions, not requiring proofs (#12-15). Introductory-leveled Olympiad-level questions (#1-4s).
- Intermediate-leveled Olympiad-level questions, #1,3s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,4s, etc.
- High-level difficult Olympiad-level questions, eg #2,4s on difficult Olympiad contest and easier #3,6s, etc.
- Difficult Olympiad-level questions, eg #3,6s on difficult Olympiad contests.
- 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.

## See also

- <url>viewtopic.php?p=1565063#1565063 Forum discussion of wiki entry </url>