![]() ![]() What is the smallest value that can be? ( Solution) leaves a remainder of when divided by or. Division E: 1 The whole number is divisible by.Problems in these competitions are usually ranked from 1 to 3. Most middle school and first-stage high school competitions would fall under this category. Prove that for any integers at least numbers from the set cannot be represented as. Suppose there exists a point satisfying for all integers. Define as the intersection of diagonals and for all integers. (Hardest IMO #3/#6).įor reference, here are problems from each of the difficulty levels 1-10:ġ0: Let be a cyclic -gon and let for all. ![]() very few students are capable of solving, even on a worldwide basis). Introductory-leveled Olympiad-level questions (USAJMO #2/#5, easier USAMO #1/#4 and IMO #1/#4).ħ: Tougher Olympiad-level questions, may require more technical knowledge (USAJMO #3/#6, easier USAMO #2/#5 and IMO #2/#5).Ĩ: High-level Olympiad-level questions (easier USAMO #3/#6).ĩ: Expert Olympiad-level questions (hard USAMO #3/#6, common IMO #3/#6).ġ0: Super Expert problems, problems occasionally even unsuitable for very hard competitions (such as the IMO) due to being exceedingly tedious/long/difficult (e.g. For these the first half of the test (#1-3) is similar difficulty to the second half (#4-6).ġ: Problems strictly for beginner, on the easiest elementary school or middle school levels (MOEMS, MathCounts chapter, AMC 8 #1-20, AMC 10 #1-10, AMC 12 #1-5, and others that involve standard techniques introduced up to the middle school level), most traditional middle/high school word problemsĢ: For motivated beginners, harder questions from the previous categories (AMC 8 #21-25, MathCounts State harder items, AMC 10 #11-20, AMC 12 #5-15, AIME #1), traditional middle/high school word problems with extremely complex problem solving.ģ: Advanced Beginner problems that require more creative thinking (MathCounts National harder items, AMC 10 #21-25, AMC 12 #15-20, AIME #1-5).Ĥ: Intermediate-level problems (AMC 12 #21-25, AIME #6-9).ĥ: More difficult AIME problems (#10-12), simple proof-based Olympad-style problems (JBMO, USAJMO #1/#4).Ħ: High-leveled AIME-styled questions (#13-15). Some Olympiads are taken in 2 sessions, with 2 similarly-difficult sets of questions, numbered as one set.Test-takers can use the answer choices as hints, and so correctly answer more AMC questions than Mathcounts or AIME problems of similar difficulty. Multiple choice tests like AMC are rated as though they are free-response.Other contests can be interpolated against this. The following is a rough standard based on the USA tier system AMC 8 – AMC 10 – AMC 12 – AIME – USAMO/USAJMO - IMO, representing Middle School – Junior High – High School – Challenging High School – Olympiad levels. 2.4.2 Intermediate Counting & Probability by AoPSĪll levels are estimated and refer to averages.2.4 Problem Solving Books for Intermediate Students.2.2.4 Introduction to Number Theory by AoPS.2.2.3 Introduction to Counting and Probability by AoPS.2.2 Problem Solving Books for Introductory Students.When considering problem difficulty put more emphasis on problem-solving aspects and less so on technical skill requirements. early AMC problems and 10 is hardest level, e.g. Currently, the system is on a scale from 1 to 10 where 1 is the easiest level, e.g. If you have some experience with mathematical competitions, we hope that you can help us make the difficulty rankings more accurate. A sample problem is provided with each entry, with a link to a solution. Also, due to variances within a contest, ranges shown may overlap. 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. 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.Įach 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). This page contains an approximate estimation of the difficulty level of various competitions.
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