Subcontests
(24)IMO Shortlist 2012, Number Theory 4
An integer a is called friendly if the equation (m2+n)(n2+m)=a(m−n)3 has a solution over the positive integers.
a) Prove that there are at least 500 friendly integers in the set {1,2,…,2012}.
b) Decide whether a=2 is friendly. IMO Shortlist 2012, Geometry 8
Let ABC be a triangle with circumcircle ω and ℓ a line without common points with ω. Denote by P the foot of the perpendicular from the center of ω to ℓ. The side-lines BC,CA,AB intersect ℓ at the points X,Y,Z different from P. Prove that the circumcircles of the triangles AXP, BYP and CZP have a common point different from P or are mutually tangent at P.Proposed by Cosmin Pohoata, Romania IMO Shortlist 2012, Geometry 2
Let ABCD be a cyclic quadrilateral whose diagonals AC and BD meet at E. The extensions of the sides AD and BC beyond A and B meet at F. Let G be the point such that ECGD is a parallelogram, and let H be the image of E under reflection in AD. Prove that D,H,F,G are concyclic. IMO Shortlist 2012, Combinatorics 7
There are given 2500 points on a circle labeled 1,2,…,2500 in some order. Prove that one can choose 100 pairwise disjoint chords joining some of theses points so that the 100 sums of the pairs of numbers at the endpoints of the chosen chord are equal. IMO Shortlist 2012, Combinatorics 5
The columns and the row of a 3n×3n square board are numbered 1,2,…,3n. Every square (x,y) with 1≤x,y≤3n is colored asparagus, byzantium or citrine according as the modulo 3 remainder of x+y is 0,1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are 3n2 tokens of each color.
Suppose that one can permute the tokens so that each token is moved to a distance of at most d from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most d+2 from its original position, and each square contains a token with the same color as the square. IMO Shortlist 2012, Combinatorics 4
Players A and B play a game with N≥2012 coins and 2012 boxes arranged around a circle. Initially A distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order B,A,B,A,… by the following rules:
(a) On every move of his B passes 1 coin from every box to an adjacent box.
(b) On every move of hers A chooses several coins that were not involved in B's previous move and are in different boxes. She passes every coin to an adjacent box.
Player A's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how B plays and how many moves are made. Find the least N that enables her to succeed. IMO Shortlist 2012, Combinatorics 3
In a 999×999 square table some cells are white and the remaining ones are red. Let T be the number of triples (C1,C2,C3) of cells, the first two in the same row and the last two in the same column, with C1,C3 white and C2 red. Find the maximum value T can attain.Proposed by Merlijn Staps, The Netherlands IMO Shortlist 2012, Combinatorics 2
Let n≥1 be an integer. What is the maximum number of disjoint pairs of elements of the set {1,2,…,n} such that the sums of the different pairs are different integers not exceeding n? IMO Shortlist 2012, Combinatorics 1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers x and y such that x>y and x is to the left of y, and replaces the pair (x,y) by either (y+1,x) or (x−1,x). Prove that she can perform only finitely many such iterations.Proposed by Warut Suksompong, Thailand IMO Shortlist 2012, Algebra 7
We say that a function f:Rk→R is a metapolynomial if, for some positive integers m and n, it can be represented in the form
f(x1,⋯,xk)=i=1,⋯,mmaxj=1,⋯,nminPi,j(x1,⋯,xk),
where Pi,j are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial. IMO Shortlist 2012, Algebra 6
Let f:N→N be a function, and let fm be f applied m times. Suppose that for every n∈N there exists a k∈N such that f2k(n)=n+k, and let kn be the smallest such k. Prove that the sequence k1,k2,… is unbounded.Proposed by Palmer Mebane, United States IMO Shortlist 2012, Algebra 2
Let Z and Q be the sets of integers and rationals respectively.
a) Does there exist a partition of Z into three non-empty subsets A,B,C such that the sets A+B,B+C,C+A are disjoint?
b) Does there exist a partition of Q into three non-empty subsets A,B,C such that the sets A+B,B+C,C+A are disjoint?Here X+Y denotes the set {x+y:x∈X,y∈Y}, for X,Y⊆Z and for X,Y⊆Q.