AoPS Wiki:Competition ratings
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.
Contents
Scale
All levels are 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 beginner, on the easiest elementary school or middle school levels (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), most traditional middle/high school word problems
- 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), traditional middle/high school word problems with extremely complex problem solving
- Beginner/novice problems that require more creative thinking (MathCounts National, #21-25 on AMC 10, #11-20ish on AMC 12, #1-5 on AIMEs, etc.)
- Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions #6-10
- Difficult AIME problems (#10-13), simple proof-based problems (JBMO), etc
- High-leveled AIME-styled questions (#12-15). Introductory-leveled Olympiad-level questions (#1,4s).
- Tougher 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.
- Expert Olympiad-level questions, eg #3,6s on difficult Olympiad contests.
- Super Expert problems, 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).
Competitions
Introductory Competitions
Most middle school and first-stage high school competitions would fall under this category. Problems in these competitions are usually ranked from 1 to 3. A full list is available here.
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)
- The whole number
- 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)
- The value of a two-digit number is
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)
- A fifth number,
Mathcounts
- Countdown: 0.5 (School, Chapter), 1 (State, National)
- Sprint: 1-1.5 (school), 1.5 (Chapter),2 (State), 2-2.5 (National)
- Target: 1.5 (school), 2 (Chapter), 2-2.5 (State), 2.5 (National)
AMC 10
- Problem 1 - 5: 1
- A rectangular box has integer side lengths in the ratio
. Which of the following could be the volume of the box? (Solution)
- A rectangular box has integer side lengths in the ratio
- Problem 6 - 20: 2
- Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run? (Solution)
- Problem 21 - 25: 3
- The vertices of an equilateral triangle lie on the hyperbola
, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? (Solution)
- The vertices of an equilateral triangle lie on the hyperbola
CEMC Multiple Choice Tests
This covers the CEMC Gauss, Pascal, Cayley, and Fermat tests.
- Part A: 0.5-1.5
- How many different 3-digit whole numbers can be formed using the digits 4, 7, and 9, assuming that no digit can be repeated in a number? (2015 Gauss 7 Problem 10)
- Part B: 1-2
- Two lines with slopes
and
intersect at
. What is the area of the triangle formed by these two lines and the vertical line
? (2017 Cayley Problem 19)
- Two lines with slopes
- Part C (Gauss/Pascal): 2-2.5
- Suppose that
, where
,
, and
are positive integers with
in lowest terms. What is the sum of the digits of the smallest positive integer
for which
is a multiple of 1004? (2014 Pascal Problem 25)
- Suppose that
- Part C (Cayley/Fermat): 2.5-3
- Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puck among the yellow buckets. Once he is finished, what is the probability that a green bucket contains more pucks than each of the other 11 buckets? (2018 Fermat Problem 24)
CEMC Fryer/Galois/Hypatia
- Problem 1-2: 1-2
- Problem 3-4 (early parts): 2-3
- Problem 3-4 (later parts): 3-5
Intermediate Competitions
This category consists of all the non-proof math competitions for the middle stages of high school. The difficulty range would normally be from 3 to 6. A full list is available here.
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)
- A solid box is
- 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)
- Functions
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)
- Starting at
- Problem 6 - 10: 3.75
- Problem 10 - 12: 4.5
- 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)
- Let
- Problem 12 - 15: 5
- 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)
ARML
- Individuals, Problem 1: 2
- Individuals, Problems 3, 5, 7, and 9: 3
- Individuals, Problems 2 and 4: 3
- Individuals, Problems 6 and 8: 4
- Individuals, Problem 10: 6
- Team/power, Problem 1-5: 3.5
- Team/power, Problem 6-10: 5
HMMT (November)
- Individual Round, Problem 6-8: 4
- Individual Round, Problem 10: 4.5
- Team Round: 5
- Guts: 6
CEMC Euclid
- Problem 1-6: 1-3
- Problem 7-10: 3-6
Purple Comet
- Problems 1-10 (MS): 2-3
- Problems 11-20 (MS): 3-5
- Problems 1-10 (HS): 2-4
- Problems 11-20 (HS): 4-5
- Problems 21-30 (HS): 5-6
Philippine Mathematical Olympiad Qualifying Round
- Problem 1-15: 2
- Problem 16-25: 3
- Problem 26-30: 4
Beginner Olympiad Competitions
This category consists of beginning Olympiad math competitions. Most junior and first stage Olympiads fall under this category. The range from the difficulty scale would be around 4 to 6. A full list is available here.
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)
- Call a positive real number groovy if it can be written in the form
Indonesia 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>
- In a drawer, there are at most
- 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
- A pair of integers
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
.
- Given an acute triangle
(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>
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>)
- Find all three-digit numbers
- Problem 2,4,5: 5-6
- Show that the equation
has no integer solutions. (<url>viewtopic.php?p=291301#291301 Solution</url>)
- Show that the equation
- 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>
- Let
JBMO
- Problem 1: 4
- Find all real numbers
such that
- Find all real numbers
- Problem 2: 5
- Let
be a convex quadrilateral with
,
and
. The diagonals intersect at point
. Determine the measure of
.
- Let
- Problem 3: 5
- Find all prime numbers
, such that
.
- Find all prime numbers
- 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
.
- A
Olympiad Competitions
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5.5 to 8. A full list is available here.
USAJMO
- Problem 1/4: 5.5
- Problem 2/5: 6.5
- Problem 3/6: 8
HMMT (February)
- Individual Round, Problem 1-5: 5
- Individual Round, Problem 6-10: 6.5
- Team Round: 7.5
- HMIC: 8
Canadian MO
- Problem 1: 5.5
- Problem 2: 6
- Problem 3: 6.5
- Problem 4: 7-7.5
- Problem 5: 7.5-8
Austrian MO
- Regional Competition for Advanced Students, Problems 1-4: 5
- Federal Competition for Advanced Students, Part 1. Problems 1-4: 6
- Federal Competition for Advanced Students, Part 2, Problems 1-6: 7
Ibero American Olympiad
- Problem 1/4: 5.5
- Problem 2/5: 6.5
- Problem 3/6: 7.5
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.
- Solve the equation
- 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
.
- Let
- 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
- A
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
- Denote by
for all
. '
Hard Olympiad Competitions
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available here.
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)
- Let
- 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)
- Three nonnegative real numbers
- 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)
- Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree
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
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
) (Solution)
- Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola
- 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
. (Solution)
- Let
- 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>)
- For any
China TST
- Problem 1/4: 7
- Given an integer
prove that there exist odd integers
and a positive integer
such that
- Given an integer
- Problem 2/5: 8.5
- Given a positive integer
and real numbers
such that
prove that for any positive real number
- Given a positive integer
- Problem 3/6: 10
- Let
be an integer and let
be non-negative real numbers. Define
for
. Prove that
- Let
IMO
- Problem 1/4: 6.5
- Find all functions
(so that
is a function from the positive real numbers) such that
- Find all functions
![$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$](http://latex.artofproblemsolving.com/9/a/b/9abfd05b9258aa6b964529540de233fce9a2edd6.png)
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)
- Let
- 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>)
- Assign to each side
IMO Shortlist
- Problem 1-2: 5.5-7
- Problem 3-4: 7-8
- Problem 5+: 8-10