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 Scale
- 2 Competitions
- 2.1 Introductory Competitions
- 2.1.1 MOEMS
- 2.1.2 AMC 8
- 2.1.3 Mathcounts
- 2.1.4 AMC 10
- 2.1.5 CEMC Multiple Choice Tests
- 2.1.6 CEMC Fryer/Galois/Hypatia
- 2.1.7 Problem Solving Books for Introductory Students
- 2.1.8 Prealgebra by AoPS
- 2.1.9 Introduction to Algebra by AoPS
- 2.1.10 Introduction to Counting and Probability by AoPS
- 2.1.11 Introduction to Number Theory by AoPS
- 2.1.12 Introduction to Geometry by AoPS
- 2.2 Intermediate Competitions
- 2.3 Beginner Olympiad Competitions
- 2.4 Olympiad Competitions
- 2.5 Hard Olympiad Competitions
- 2.1 Introductory Competitions
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, #1-10 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, #11-20 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, easier #1-5 on AIMEs, etc.)
- Intermediate-leveled problems, the most difficult questions on AMC 12s (#21-25s), more difficult AIME-styled questions such as #6-9.
- More difficult AIME problems (#10-12), simple proof-based problems (JBMO), etc
- High-leveled AIME-styled questions (#13-15). 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 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).
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.
- 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)
- Problem 1 - Problem 12: 1
- The of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? (Solution)
- Problem 13 - Problem 25: 1.5-2
- How many positive factors does have? (Solution)
- Countdown: 1-2.
- Sprint: 1-1.5 (school/chapter), 1.5-2 (State), 2-2.5 (National)
- Target: 1-2 (school/chapter), 1.5-2.5 (State), 2-3 (National)
- Problem 1 - 10: 1-2
- A rectangular box has integer side lengths in the ratio . Which of the following could be the volume of the box? (Solution)
- Problem 11 - 20: 2-3
- For some positive integer , the repeating base- representation of the (base-ten) fraction is . What is ? (Solution)
- Problem 21 - 25: 3.5-4.5
- 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)
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)
- 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)
- 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)
- Problem 1-2: 1-2
- Problem 3-4 (early parts): 2-3
- Problem 3-4 (later parts): 3-5
Problem Solving Books for Introductory Students
Remark: There are many other problem books for Introductory Students that are not published by AoPS. Typically the rating on the left side is equivalent to the difficulty of the easiest review problems and the difficulty on the right side is the difficulty of the hardest challenge problems. The difficulty may vary greatly between sections of a book.
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.
- Problem 1-10: 1.5-2
- What is the value of (Solution)
- Problem 11-20: 2.5-3.5
- Points and lie on a circle centered at , and . A second circle is internally tangent to the first and tangent to both and . What is the ratio of the area of the smaller circle to that of the larger circle? (Solution)
- Problem 21-25: 4-5.5
- 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)
- Problem 1 - 5: 3-3.5
- Consider the integer Find the sum of the digits of . (Solution)
- Problem 6 - 9: 4-4.5
- Define to be for odd and for even. When is expressed as a fraction in lowest terms, its denominator is with odd. Find . (Solution)
- Problem 10 - 12: 5-5.5
- Let be the set of all possible remainders when a number , a nonnegative integer, is divided by .Let be the sum of all elements in . Find the remainder when is divided by (Solution)
- Problem 13 - 15: 6-6.5
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)
- Individuals, Problem 1: 2
- Individuals, Problems 2, 3, 4, 5, 7, and 9: 3
- Individuals, Problems 6 and 8: 4
- Individuals, Problem 10: 5.5
- Team/power, Problem 1-5: 3.5
- Team/power, Problem 6-10: 5
- Individual Round, Problem 6-8: 4
- Individual Round, Problem 10: 4.5
- Team Round: 4-5
- Guts: 3.5-5.25
- Problem 1-6: 1-3
- Problem 7-10: 3-5
- Problems 1-10 (MS): 1.5-2.5
- Problems 11-20 (MS): 2.5-4
- Problems 1-10 (HS): 1.5-2.5
- Problems 11-20 (HS): 2-3.5
- Problems 21-30 (HS): 3.5-4.5
- 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 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: 4-6
- 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)
- 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>
- 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>
- Problem 1: 4
- Find all real numbers such that
- Problem 2: 4.5-5
- 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 .
This category consists of standard Olympiad competitions, usually ones from national Olympiads. Average difficulty is from 5 to 8. A full list is available here.
- Problem 1/4: 5
- There are bowls arranged in a row, numbered through , where and are given positive integers. Initially, each of the first bowls contains an apple, and each of the last bowls contains a pear.
A legal move consists of moving an apple from bowl to bowl and a pear from bowl to bowl , provided that the difference is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first bowls each containing a pear and the last bowls each containing an apple. Show that this is possible if and only if the product is even. (Solution)
- Problem 2/5: 6
- Let be positive real numbers such that . Prove that (Solution)
- Problem 3/6: 7
- Two rational numbers and are written on a blackboard, where and are relatively prime positive integers. At any point, Evan may pick two of the numbers and written on the board and write either their arithmetic mean or their harmonic mean on the board as well. Find all pairs such that Evan can write on the board in finitely many steps. (Solution)
- Individual Round, Problem 1-5: 5
- Individual Round, Problem 6-10: 5.5-6
- Team Round: 7.5
- HMIC: 8
- Problem 1: 5.5
- Problem 2: 6
- Problem 3: 6.5
- Problem 4: 7-7.5
- Problem 5: 7.5-8
- 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
- Problem 1/4: 5.5
- Problem 2/5: 6.5
- Problem 3/6: 7.5
- Problem 1: 6
- Problem 2: 7
- Problem 3: 7
- Problem 4: 7.5
- Problem 5: 8.5
- 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 . '
Hard Olympiad Competitions
This category consists of harder Olympiad contests. Difficulty is usually from 7 to 10. A full list is available here.
- Problem 1/4: 6-7
- 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: 7-8
- 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: 8-9
- 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)
- Problem 1/4/7: 6.5-7
- Problem 2/5/8: 7.5-8
- Problem 3/6/9: 8.5-9
- 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)
- 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)
- 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>)
- Problem 1/4: 8-8.5
- Given an integer prove that there exist odd integers and a positive integer such that
- Problem 2/5: 9
- Given a positive integer and real numbers such that prove that for any positive real number
- Problem 3/6: 9.5-10
- Let be an integer and let be non-negative real numbers. Define for . Prove that
- Problem 1/4: 5.5-7
- Let be the circumcircle of acute triangle . Points and are on segments and respectively such that . The perpendicular bisectors of and intersect minor arcs and of at points and respectively. Prove that lines and are either parallel or they are the same line. (Solution)
- Problem 2/5: 7-8
- 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: 9-10
- 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>)
- Problem 1-2: 5.5-7
- Problem 3-4: 7-8
- Problem 5+: 8-10