IMO 2011 Shortlist

30 shortlist problems from IMO 2011.

30 items

IMO 2011 Shortlist

30 problems across categories.

Problem	Category
A1	Algebra
A2	Algebra
A3	Algebra
A4	Algebra
A5	Algebra
A6	Algebra
A7	Algebra
C1	Combinatorics
C2	Combinatorics
C3	Combinatorics
C4	Combinatorics
C5	Combinatorics
C6	Combinatorics
C7	Combinatorics
G1	Geometry
G2	Geometry
G3	Geometry
G4	Geometry
G5	Geometry
G6	Geometry
G7	Geometry
G8	Geometry
N1	Number Theory
N2	Number Theory
N3	Number Theory
N4	Number Theory
N5	Number Theory
N6	Number Theory
N7	Number Theory
N8	Number Theory

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A1

For any set A = {a1,a2,a3,a4} of four distinct positive integers with sum sA = a1 +a2 +a3 +a4, let pA denote the number ...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A2

Determine all sequences (x1,x2,...,x2011) of positive integers such that for every positive inte- ger n there is an inte...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A3

Determine all pairs (f,g) of functions from the set of real numbers to itself that satisfy g(f(x + y)) = f(x) + (2x + y)...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A4

Determine all pairs (f,g) of functions from the set of positive integers to itself that satisfy fg(n)+1 (n) + gf(n) (n) ...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A5

Prove that for every positive integer n, the set {2,3,4,...,3n + 1} can be partitioned into n triples in such a way that...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A6

Let f be a function from the set of real numbers to itself that satisfies f(x + y) ≤ yf(x) + f(f(x)) for all real number...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A7

Let a, b, and c be positive real numbers satisfying min(a+b,b+c,c+a) > √ 2 and a2 +b2 +c2 = 3. Prove that a (b + c − a)2...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C1

Let n > 0 be an integer. We are given a balance and n weights of weight 20 , 21 ,...,2n−1 . In a sequence of n moves we ...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C2

Suppose that 1000 students are standing in a circle. Prove that there exists an integer k with 100 ≤ k ≤ 300 such that i...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C3

Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C4

Determine the greatest positive integer k that satisfies the following property: The set of positive integers can be par...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C5

Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of t...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C6

Let n be a positive integer and let W = ...x−1x0x1x2 ... be an infinite periodic word consisting of the letters a and b....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C7

On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. ...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G1

Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC. Suppose that ω is tangent to AB at B...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G2

Let A1A2A3A4 be a non-cyclic quadrilateral. Let O1 and r1 be the circumcenter and the circumradius of the triangle A2A3A...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G3

Let ABCD be a convex quadrilateral whose sides AD and BC are not parallel. Suppose that the circles with diameters AB an...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G4

Let ABC be an acute triangle with circumcircle Ω. Let B0 be the midpoint of AC and let C0 be the midpoint of AB. Let D b...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G5

Let ABC be a triangle with incenter I and circumcircle ω. Let D and E be the second intersection points of ω with the li...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G6

Let ABC be a triangle with AB = AC, and let D be the midpoint of AC. The angle bisector of ∠BAC intersects the circle th...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G7

Let ABCDEF be a convex hexagon all of whose sides are tangent to a circle ω with center O. Suppose that the circumcircle...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G8

Let ABC be an acute triangle with circumcircle ω. Let t be a tangent line to ω. Let ta, tb, and tc be the lines obtained...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N1

For any integer d > 0, let f(d) be the smallest positive integer that has exactly d positive divisors (so for example we...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N2

Consider a polynomial P(x) = (x + d1)(x + d2) · ... · (x + d9), where d1, d2,..., d9 are nine distinct integers. Prove t...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N3

Let n ≥ 1 be an odd integer. Determine all functions f from the set of integers to itself such that for all integers x a...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N4

For each positive integer k, let t(k) be the largest odd divisor of k. Determine all positive integers a for which there...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N5

Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n,...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N6

Let P(x) and Q(x) be two polynomials with integer coefficients such that no nonconstant polynomial with rational coeffic...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N7

Let p be an odd prime number. For every integer a, define the number Sa = a + a2 + ··· + ap−1 p − 1 . Let m and n be int...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N8

Let k be a positive integer and set n = 2k + 1. Prove that n is a prime number if and only if the following holds: there...