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

(→USAJMO) |
m (→Scale) |
||

Line 237: | Line 237: | ||

# High-level difficult Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc. | # High-level difficult Olympiad-level questions, eg #2,5s on difficult Olympiad contest and easier #3,6s, etc. | ||

# Difficult Olympiad-level questions, eg #3,6s on difficult Olympiad contests. | # Difficult Olympiad-level questions, eg #3,6s on difficult Olympiad contests. | ||

− | # Problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult ( | + | # Problems occasionally even unsuitable for very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis), or involving techniques beyond high school level mathematics. |

== See also == | == See also == | ||

[[Category:Mathematics competitions]] | [[Category:Mathematics competitions]] |

## Revision as of 13:28, 31 May 2017

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 Olympiads such as our very own AoPS!

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. early AMC problems and 10 is hardest level, e.g. 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 MOEMS
- 1.2 AMC 8
- 1.3 Mathcounts
- 1.4 AMC 10
- 1.5 Austrian MO
- 1.6 AMC 12
- 1.7 AIME
- 1.8 Indonesian MO
- 1.9 ARML
- 1.10 Central American Olympiad
- 1.11 Canadian MO
- 1.12 APMO
- 1.13 Balkan MO
- 1.14 HMMT
- 1.15 Ibero American Olympiad
- 1.16 IMO
- 1.17 IMO Shortlist
- 1.18 JBMO
- 1.19 Putnam
- 1.20 USAJMO
- 1.21 USAMO
- 1.22 USAMTS
- 1.23 USA TST
- 1.24 China TST

- 2 Scale
- 3 See also

## Competitions

### 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)

### 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:
**1.5***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)

### Mathcounts

- Countdown:
**0.5**(School, Chapter),**1**(State, National) - Sprint:
**1-1.5**(school),**1.5**(Chapter),**1.5-2**(State),**2**(National) - Target:
**1.5**(school),**2**(Chapter),**2**(State),**2.5**(National)

### 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 - 20:
**2***How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?*(Solution)

- Problem 21 - 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)

### Austrian MO

- Regional Competition for Advanced Students, Problems 1-4:
**2** - Federal Competition for Advanced Students, Part 1. Problems 1-4:
**3** - Federal Competition for Advanced Students, Part 2, Problems 1-6:
**4**

### AMC 12

- Problem 1-10:
**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 11-20:
**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 21-25:
**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**- Starting at an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let be the probability that the object reaches in six or fewer steps. Given that can be written in the form where and are relatively prime positive integers, find (Solution)

- Problem 6 - 10:
**4** - Problem 10 - 12:
**5**- Let be a complex number with . Let be the polygon in the complex plane whose vertices are and every such that . Then the area enclosed by can be written in the form , where is an integer. Find the remainder when is divided by . (Solution)

- Problem 12 - 15:
**6***Let*

Let be the distinct zeros of and let for where and and are real numbers. Let

where and are integers and is not divisible by the square of any prime. Find .* (Solution)*

### Indonesian MO

- Problem 1/5:
**3.5***In a drawer, there are at most 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 . Determine the maximum amount of white balls in the drawer, such that the probability statement is true?*<url>viewtopic.php?t=294065 (Solution)</url>

- Problem 2/6:
**4.5***Find the lowest possible values from the function*

for any real numbers .*<url>viewtopic.php?t=294067 (Solution)</url>*

- Problem 3/7:
**5***A pair of integers is called*good*if*

Given 2 positive integers which are relatively prime, prove that there exists a *good* pair with and , but and .* <url>viewtopic.php?t=294068 (Solution)</url>*

- Problem 4/8:
**6***Given an acute triangle . The incircle of triangle touches respectively at . The angle bisector of cuts and respectively at and . Suppose is one of the altitudes of triangle , and be the midpoint of .*

(a) Prove that and are perpendicular with the angle bisector of .

(b) Show that is a cyclic quadrilateral.* <url>viewtopic.php?t=294069 (Solution)</url>*

### ARML

- Individuals, Problem 6 and 8:
**4**

- Individuals, Problem 10:
**6**

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

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

### Central American Olympiad

- Problem 1:
**4***Find all three-digit numbers (with ) such that is a divisor of 26.*(<url>viewtopic.php?p=903856#903856 Solution</url>)

- Problem 2,4,5:
**5-6***Show that the equation has no integer solutions.*(<url>viewtopic.php?p=291301#291301 Solution</url>)

- Problem 3/6:
**6.5***Let be a convex quadrilateral. , and , , and are points on , , and respectively, such that . If , , show that*(<url>viewtopic.php?p=828841#p828841 Solution</url>

### Canadian MO

- Problem 1:
**5** - Problem 2:
**6** - Problem 3:
**6.5** - Problem 4:
**7-7.5** - Problem 5:
**7.5-8**

### APMO

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

### Balkan MO

- Problem 1:
**6***Solve the equation in positive integers.*

- Problem 2:
**6.5***Let be a line parallel to the side of a triangle , with on the side and on the side . The lines and meet at point . The circumcircles of triangles and meet at two distinct points and . Prove that .*

- Problem 3:
**7.5***A 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 in such way that the following to conditions are both fulfilled*

the distances are all equal to

the closed broken line has a centre of symmetry?

- Problem 4:
**8***Denote by the set of all positive integers. Find all functions such that*

for all . '

### HMMT

**February Contest**

- Individual Round, Problem 1-5:
**5** - Individual Round, Problem 6-10:
**6** - Team Round:
**7.5** - HMIC:
**8**

### Ibero American Olympiad

- Problem 1/4:
**5.5** - Problem 2/5:
**6.5** - Problem 3/6:
**7.5**

### IMO

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

for all positive real numbers satisfying (Solution)

- Problem 2/5:
**7.5-8***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:
**9.5***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:
**5.5-7** - Problem 3-4:
**7-8** - Problem 5+:
**8-10**

### 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 .

### Putnam

- Problem A/B,1-2:
**7***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:
**8***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>)

### USAJMO

- Problem 1/4:
**6** - Problem 2/5:
**6.5** - Problem 3/6:
**7**

### USAMO

- Problem 1/4:
**7***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:
**8***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:
**9***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

USAMTS generally has a different feel to it than olympiads, and is mainly for proofwriting practice instead of olympiad practice depending on how one takes the test. USAMTS allows an entire month to solve problems, with internet resources and books being allowed. However, the ultimate gap is that it permits computer programs to be used, and that Problem 1 is not a proof problem. However, it can still be roughly put to this rating scale:

- Problem 1-2:
**3-4***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:
**5-6***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** - Problem 2/5/8:
**8** - Problem 3/6/9:
**9.5**

### China TST

- Problem 1/4:
**7***Given an integer prove that there exist odd integers and a positive integer such that*

- Problem 2/5:
**8.5***Given a positive integer and real numbers such that prove that for any positive real number*

- Problem 3/6:
**10***Let be an integer and let be non-negative real numbers. Define for . Prove that*

## 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/USAJMO, 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, #1-5 AMC 10s, and others that involve standard techniques introduced up to the middle school level
- For motivated beginners, harder questions from the previous categories (#21-25 on AMC 8, Challenging Mathcounts questions, #5-20 on AMC 10, #5-10 on AMC 12, the easiest AIME questions, etc).
- For those not too familiar with standard techniques, MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, #1-5 on AIMEs, etc.
- Intermediate-leveled problem solvers, the most difficult questions on AMC 12s (#21-25s), 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, #1,4s that require more technical knowledge than new students to Olympiad-type questions have, easier #2,5s, etc.
- High-level difficult Olympiad-level questions, eg #2,5s 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 very hard competitions (like the IMO) due to being exceedingly tedious/long/difficult (e.g. very few students are capable of solving, even on a worldwide basis), or involving techniques beyond high school level mathematics.