Subcontests
(26)Wot n' Minimization
Let n⩾3 be a positive integer and let (a1,a2,…,an) be a strictly increasing sequence of n positive real numbers with sum equal to 2. Let X be a subset of {1,2,…,n} such that the value of
1−i∈X∑ai
is minimised. Prove that there exists a strictly increasing sequence of n positive real numbers (b1,b2,…,bn) with sum equal to 2 such that
i∈X∑bi=1. is it true c shortlist has 9 problems?
Let H={⌊i2⌋:i∈Z>0}={1,2,4,5,7,…} and let n be a positive integer. Prove that there exists a constant C such that, if A⊆{1,2,…,n} satisfies ∣A∣≥Cn, then there exist a,b∈A such that a−b∈H. (Here Z>0 is the set of positive integers, and ⌊z⌋ denotes the greatest integer less than or equal to z.) why are there 9 problems on combo sl
The infinite sequence a0,a1,a2,… of (not necessarily distinct) integers has the following properties: 0≤ai≤i for all integers i≥0, and (a0k)+(a1k)+⋯+(akk)=2k for all integers k≥0. Prove that all integers N≥0 occur in the sequence (that is, for all N≥0, there exists i≥0 with ai=N). Rootiful sets
We say that a set S of integers is rootiful if, for any positive integer n and any a0,a1,⋯,an∈S, all integer roots of the polynomial a0+a1x+⋯+anxn are also in S. Find all rootiful sets of integers that contain all numbers of the form 2a−2b for positive integers a and b. Merlin and painted labyrinth
On a flat plane in Camelot, King Arthur builds a labyrinth L consisting of n walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet.After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.Let k(L) be the largest number k such that, no matter how Merlin paints the labyrinth L, Morgana can always place at least k knights such that no two of them can ever meet. For each n, what are all possible values for k(L), where L is a labyrinth with n walls? Pairwise distance-one products
Let n⩾2 be a positive integer and a1,a2,…,an be real numbers such that a1+a2+⋯+an=0.
Define the set A by
A={(i,j)∣1⩽i<j⩽n,∣ai−aj∣⩾1}
Prove that, if A is not empty, then
(i,j)∈A∑aiaj<0. Angle sums are equal
Let n>1 be an integer. Suppose we are given 2n points in the plane such that no three of them are collinear. The points are to be labelled A1,A2,…,A2n in some order. We then consider the 2n angles ∠A1A2A3,∠A2A3A4,…,∠A2n−2A2n−1A2n,∠A2n−1A2nA1,∠A2nA1A2. We measure each angle in the way that gives the smallest positive value (i.e. between 0∘ and 180∘). Prove that there exists an ordering of the given points such that the resulting 2n angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group. Minimum times maximum
Let u1,u2,…,u2019 be real numbers satisfying u_{1}+u_{2}+\cdots+u_{2019}=0 \text { and } u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1. Let a=min(u1,u2,…,u2019) and b=max(u1,u2,…,u2019). Prove that
ab⩽−20191. <DPA+ <AQD =< QIP wanted, incircle circumcircle related
Let I be the incentre of acute-angled triangle ABC. Let the incircle meet BC,CA, and AB at D,E, and F, respectively. Let line EF intersect the circumcircle of the triangle at P and Q, such that F lies between E and P. Prove that ∠DPA+∠AQD=∠QIP.(Slovakia) Sum of products is n mod 2
Let x1,x2,…,xn be different real numbers. Prove that
1⩽i⩽n∑j=i∏xi−xj1−xixj={0,1, if n is even; if n is odd. A_2, B_2,C_2 cannot all lie strictly inside the circumcircle of triangle ABC
Let P be a point inside triangle ABC. Let AP meet BC at A1, let BP meet CA at B1, and let CP meet AB at C1. Let A2 be the point such that A1 is the midpoint of PA2, let B2 be the point such that B1 is the midpoint of PB2, and let C2 be the point such that C1 is the midpoint of PC2. Prove that points A2,B2, and C2 cannot all lie strictly inside the circumcircle of triangle ABC.(Australia) Functional Geometry
Let L be the set of all lines in the plane and let f be a function that assigns to each line ℓ∈L a point f(ℓ) on ℓ. Suppose that for any point X, and for any three lines ℓ1,ℓ2,ℓ3 passing through X, the points f(ℓ1),f(ℓ2),f(ℓ3), and X lie on a circle.
Prove that there is a unique point P such that f(ℓ)=P for any line ℓ passing through P.Australia MP = NQ wanted, incircles related
Let ABC be an acute-angled triangle and let D,E, and F be the feet of altitudes from A,B, and C to sides BC,CA, and AB, respectively. Denote by ωB and ωC the incircles of triangles BDF and CDE, and let these circles be tangent to segments DF and DE at M and N, respectively. Let line MN meet circles ωB and ωC again at P=M and Q=N, respectively. Prove that MP=NQ.(Vietnam) Three-variable polynomial
A polynomial P(x,y,z) in three variables with real coefficients satisfies the identitiesP(x,y,z)=P(x,y,xy−z)=P(x,zx−y,z)=P(yz−x,y,z).Prove that there exists a polynomial F(t) in one variable such thatP(x,y,z)=F(x2+y2+z2−xyz). Alice and Bob playing with Pebbles
There are 60 empty boxes B1,…,B60 in a row on a table and an unlimited supply of pebbles. Given a positive integer n, Alice and Bob play the following game.
In the first round, Alice takes n pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer k with 1≤k≤59 and splits the boxes into the two groups B1,…,Bk and Bk+1,…,B60.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest n such that Alice can prevent Bob from winning.Czech Republic