Imo problem 6. Emanouil Atanassov, famously said to have completed the "hardest" IMO problem in a single paragraph and went on to receive the special prize, gave the proof quoted below, Question: Let a The proof relies on reducing the problem to the specific case where n = k and all lines must be sunny. Let C(k) be the assertion that Pk can be covered by k distinct sunny lines. 6 Contributing Countries 2006 IMO Problems/Problem 6 Problem Assign to each side of a convex polygon the maximum area of a triangle that has as a side and is contained in . Let a, b ∈ N, with b ≥ a. 1987 IMO Problems/Problem 6 Problem Let be an integer greater than or equal to 2. In particular, the sum of the roots of the equation x2 – kbx + b2 – k = 0 is kb, and the product of the roots is b2 – k. Prove that is not prime. This included the hardest problem in the competition, solved by only five contestants at this year’s IMO. The problem is considered extremely difficult to solve - most solutions require a high level of mathematical sophistication or are long and tedious. Prove that if is prime for all integers such that , then is prime for all integers such that . youtube. Show that is the square of an integer. #IMO #IMO1988 #MathOlympiad Here is the solution to the Legendary Problem 6 of IMO 1988!! We rearrange the equation and complete the square for the term to transform the problem into the study of a Generalized Pell Equation. 2024 IMO problems and solutions. There exists a very elegant way to prove it that lends itself nic This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Entire Test Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 See Also IMO Problems and Solutions, with authors Mathematics IMO Problem 6 is the last and traditionally the hardest of the IMO problems. So does a = 125, b = 3120, x = 25 Exercise 1. 📍 Host Country and Venue • Host country: Czechoslovakia Our systems solved one problem within minutes and took up to three days to solve the others. 1988 IMO Problems/Problem 6 Contents 1 Problem 2 Video Solution 3 Video Solution 4 Solution 1 5 Solution 2 (Sort of Root Jumping) 6 Solution 3 7 Video Solution For example, in geometry problems I typically use directed angles without further comment, rather than awkwardly work around configuration issues. The test took place in July 2024 in Bath, United Kingdom. In this video, we provide a complete, step-by-step walkthrough of the entire solution. See also IMO problems statistics (eternal) DONATE TO HURRICANE HARVEY RELIEF FUND https://www. #imo #oly #olympiadmathematics #geometryproblems #combinatorics #boards #grids #combi #imo_2025 #imo #imo_2025_problem_6 #imo2025_problem_6 #imo2025 #imo_2025 2009 IMO Problems/Problem 6 Problem Let be distinct positive integers and let be a set of positive integers not containing . 1998 IMO Problems/Problem 6 Problem Determine the least possible value of where is a function such that for all , Video Solution https://www. 37 likes, 0 comments - internationalmathsolympiad on February 11, 2026: "IMO 1962 1962 International Mathematical Olympiad (IMO 1962) The 4th International Mathematical Olympiad (IMO) was held in 1962 and marked an important stage in the early development of the world’s most prestigious school-level mathematics competition. Traditionally, the last problem (Problem 6) is signific… Problem 6 A permutation of the set where is a positive integer, is said to have property if for at least one in Show that, for each , there are more permutations with property than without. My walkthrough of this famously difficult problem from the 1988 International Mathematics Olympiad. Label . Solution See Also Similarly, problem 2020/3 was proposed by Hungary with one Hungarian and one non-Hungarian problem author. Download IMO 1988 Problem 6 page 1 of 1 Reminder: If each of a, b, and c is a real number, and a 0, then the sum of the roots of the equation ax2 + bx + c = 0 is -b/a, and the product of the roots is c/a. be/wqCdEE1Ueh0 Solution 1 Choose integers such that Now, for fixed , out of all pairs choose the one with the lowest value of . Solution First observe that if is relatively prime to , , , , then is relatively prime to any number less than . The first IMO was held in 1959 in Romania, with 7 countries participating. For instance, a = 3, b = 27, x = 9 works. This is the solution to Problem 6 of the 63rd international Mathematical Olympiad (IMO) 2022 by one of our instructors, Onah pius, at Special Maths Academy. After the break, I’ll take you through my solution (which relied on a big hint from the video). The final problem of the International Mathematics Olympiad (IMO) 1988 is considered to be the most difficult problem on the contest. Thus, is a quadratic in . This year was no exception, with the vast majority of students scoring 0 and only 6 students (out of over 600 Problem 6 are positive integers such that . Similarly, sentences like “let R denote the set of real numbers” are typically omitted entirely. The IMO 1988 problem #6 is described as follows: 2009 IMO Problems/Problem 6 Problem Let be distinct positive integers and let be a set of positive integers not containing . IMO Problems Play all Solving an IMO Problem in 6 Minutes!! | International Mathematical Olympiad 1979 Problem 1 letsthinkcritically 618K views4 years ago Solution https://www. This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Video Solution https://youtu. Prove that ⁠a2 + b2 ab + 1 ⁠ is a perfect square. org/donate/hurricane-harvey. Assume that Jan 5, 2020 · Using no more than high school algebra, here’s how to solve the infamous question 6 from the 1988 International Mathematics Olympiad. #mathematics #olympiad #mathInternational Mathematical Olympiad (IMO) 2025 Day 2Solutions and discussion of problem 666th International Mathematical Olympiad Google also provided Gemini with access to a curated corpus of high-quality solutions to mathematics problems, and added some general hints and tips on how to approach IMO problems to its 1988 IMO Problems/Problem 6 Contents 1 Problem 2 Video Solution 3 Solution 1 4 Solution 2 (Sort of Root Jumping) 5 Video Solution 6 Solution 3 International Math Olympiad (IMO) is one of the most prestigious exams conducted by the Science Olympiad Foundation. Am I missing something? Please let me know in the comments. It popularized a proof technique known as Vieta Jumping which is a type of proof by contradiction. imo-problem-solution-1959-2009 (1). Entire Test Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 See Also IMO Problems and Solutions, with authors Mathematics competition resources 1988 IMO Problem 6 This unassuming prompt from the 1988 IMO became instantly infamous. Prove that she can always find out by asking at most 4n questions. The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. Edit (4/10/2020) - I have realised this proof is actuall Problem 6 on the 1988 International Math Olympiad is notorious for its difficulty to prove. Every cell that is adjacent only to cells containing larger numbers is called a valley. Show that the sum of the areas assigned to the sides of is at least twice the area of . It was the hardest IMO problem ever posed for many years. Statement of the problem IMO proof 1988, question 6 Prove that if x, a, b are all integers then x is a square. redcross. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Here, at Practice Olympiad, IMO Sample Papers have been provided for all classes 1 to 5. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in . It has gradually expanded to over 100 countries from 5 continents. The first link contains the full set of test problems. In my opinion this is not the hardest problem you can see IMO 2019 problem 6 I think it is harder than the problem that you just solved. pdf - Google Drive Loading… It became a famous problem because Emanouil Atanassov from Bulgaria easily solved the problem by Vieta jumping in a short paragraph that can be easily understood by middle school students, and received a special prize (the other 10 kids used more standard or cumbersome approaches to solve it). It’s a two-asterisk IMO question. 6 of IMO 1988. This problem could be posed with an explicit statement about points being awarded for weaker bounds cn for some c ą 4, in the style of IMO 2014 Problem 6. AlphaProof solved two algebra problems and one number theory problem by determining the answer and proving it was correct. Show that $$\\frac Problem 6 of IMO 1988 was called “The Legend Problem 6 of IMO” (see [1]) since only 11 among 268 participants answered it correctly (that means that they obtained 7 points, the highest score, for this question), which is significantly lower than the correct ratios of other problems of this Olympiad and also the problems of other years. IMO Problem 6 is the last and traditionally the hardest of the IMO problems. For many problems, the composers do not have the nationality of the proposing country. Problem Let and be positive integers such that divides . A Nordic square is an board containing all the integers from to so that each cell contains exactly one number. After about 20 hours, I think I solved the legendary Problem No. Comment. Can you find any other examples of three integers which work? Can you spot any patterns? westsidecoug 12:25pm Imo it might not be a must win , but it is a win that helps BYU get a 6 seed acougfan 12:47pm The proof relies on reducing the problem to the specific case where n = k and all lines must be sunny. IMO 1988 Problem 6 Sachin Kumar Univeristy of Waterloo, Faculty of Mathematics Abstract No words can explain its beauty, elegance and non-triviality! Question. A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths in some order. This year was no exception, with the vast majority of students scoring 0 and only 6 students (out of over 600 We decide to provide here a collections of past papers and solutions for those who wish to practice the math problems. com/watch?v=vOExNCV8jGQ See Also Categories: Olympiad Algebra Problems Functional Equation Problems If you’d like to have a go at it, please do! Obviously, don’t get disheartened if you struggle. International Mathematical Olympiad. In IMO 2017, only 7 out of 615 participants secured a non-zero score in problem number 3 with 3 participants scoring 1 mark each, 2 participants scoring 4 and 5 marks respectively and only 2 participants out of all were able to solve this problem perfectly scoring full 7 marks. The International Mathematical Olympiad (IMO) consists of a set of six problems, to be solved in two sessions of four and a half hours each. To the current moment, there is only a single IMO problem that has two distinct proposing countries: The if-part of problem 1994/2 was proposed by Australia and its only-if part by Armenia. By multiplying by 4 and grouping terms, we arrive at . Don't forget to share the video and Inspired by this Numberphile video, I decided to try to solve problem six of the 1988 International Mathematics Olympiad. And you can solve it only using high school algebra. An uphill path is a sequence of one or more cells such that: (i) the first cell in the sequence Problem 6 A permutation of the set where is a positive integer, is said to have property if for at least one in Show that, for each , there are more permutations with property than without. This is a compilation of solutions for the 2006 IMO. Let , , and , which yields the following equation. And in case you missed the implication, there are spoilers below the line. We'll start by building intuition with examples, set up the proof by infinite descent, execute the famous This page lists the authors and the proposing countries of the problems of the IMO. Entire Test Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 See Also IMO Problems and Solutions, with authors Mathematics Problem 6 of the 1988 International Math Olympiad is notorious for its difficulty to prove. This problem has a reputation for being one of the hardest, and perhaps the hardest, IMO problem of all time. Theorem 1. 1988 IMO problems and solutions. 94 likes, 1 comments - internationalmathsolympiad on February 11, 2026: "📘 International Mathematical Olympiad 1963 (IMO 1963) The 1963 International Mathematical Olympiad was a milestone in the early history of the world’s most prestigious school-level mathematics competition. 2025 IMO Problems/Problem 6 Consider a 2025 x 2025 grid of unit squares. Two different cells are considered adjacent if they share an edge. The IMO Board ensures that the competition takes place each year and that each host Recall Question 6 of the 1988 Math Olympiad Question 6 is as follows: Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Problem 6 in the 1988 International Mathematical Olympiad paper has almost reached a legendry status. It’s widely regarded as one of the hardest problems ever posed in Olympiad history — and for good reason. The rest contain each individual problem and its solution. Matlida wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile. com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems, it was created by a man known as Najeeb Abdullah, who qualified in the IMO from the time he was 6 years old] 2020 IMO Problems/Problem 6 Contents 1 Problem 2 Solution 3 Video solution 4 See Also Problem Let be a positive integer. Problem 6 of IMO 1988 was called “The Legend Problem 6 of IMO” (see [1]) since only 11 among 268 participants answered it correctly (that means that they obtained 7 points, the highest score, for this question), which is significantly lower than the correct ratios of other problems of this Olympiad and also the problems of other years. IMO General Regulations §6. The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students and is held annually in a different country. fi90, dppi, frud11, vex5d, vf9k, rxpi1, aj17z, oflgl7, xtybf, 5frc,