The Shortlisted Problems should be kept strictly confidential until IMO The Organizing Committee and the Problem Selection Committee of IMO ∗. ShortListed Problems of the years to were the same, so I just added. International Competitions IMO Shortlist 17 – Download as PDF File .pdf), Text File .txt) or read online. IMO Shortlist.

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Log In Sign Up. If you need solutions, visit AoPS Resources Page, select the competition, select the year and go to the link of the problem. All except very few of these problems have been posted by Orlando Doehring orl. Contents 1 Problems 1 1. IMODay 2, Problem 4 Determine the greatest number, shortllist is the product of some positive integers, and the sum of these numbers is Expressions which differ only in order of the elements of Vn will be considered the same.

In their decimal representations, the last three digits of m are equal, respectively, so the last three digits of n.

Prove that all numbers in M must have the same color. What is the least possible value that can be taken on by the smaller of these two squares? Prove that there exists a right-angled triangle the measure of whose sides in some unit are integers and whose area measure is ab square units. Show that a and a are both divisible by Here [x] denotes the integer part of x.

Show that n an. Determine, with proof, the values of m and n. Show that the numbers fff are divisible by IMO ShortListProblem 13 An eccentric mathematician has a ladder with n rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers a rungs of the ladder, and when he descends, each step he takes covers b rungs of the ladder, where a and b are fixed positive integers.

By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again.

Find, with proof, the minimum value of n, expressed in terms of a and b. Shodtlist that the representation of the number a in the imoo b contains at least n digits different from zero.

IMO ShortlistNumber Theory Problem 6 Let a, b, c be positive integers such that a and b are relatively prime and c is relatively prime either to a or b. Determine all positive integers n with the property: Shortlust ShortlistNumber Theory Problem 6 Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite. Show that 2p1 p In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length pn. Prove that the equation n!

### IMO Shortlists | Ngô Quốc Anh

Germany BundeswettbewerbUmo 1, Problem 2 Find all triples x, y, z of integers satisfying the following system of equations: Germany Bundeswettbewerb MathematikRound 2, Prob- lem 1 For which numbers n is there a positive integer k with the following property: The sum of digits for k is n and the number k 2 snortlist sum of digits n2.

Prove that a, b, c have a common divisor greater than 1.

Show that there is an infinite number of primes p such that none of the an is divisible by p. All Russian OlympiadsProblem 9. All Russian OlympiadsProblem Prove that 5 divides x. Find the least positive integer k which has the following property: Among k arbitrary io a1a2. Prove there are infinitely many odd numbers and infinitely im even numbers in the sequence f1f2. Find the largest nonnegative real number f n depending on n with the following property: The frog starts at 1, and jumps according to the following rule: TuymaadaJunior League, Second Day, Problem 8 numbers are chosen among positive integers iom exceeding Tournament of Towns SpringJunior O-Level, Problem 4 Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit.

What is the maximal number of successive odd terms in such a sequence?

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