Subcontests
(23)All numbers f(0),f(1),...,f(p-2) are primes(IMO SL 1987-P20)
Let n≥2 be an integer. Prove that if k2+k+n is prime for all integers k such that 0≤k≤3n, then k2+k+n is prime for all integers k such that 0≤k≤n−2.(IMO Problem 6)Original FormulationLet f(x)=x2+x+p, p∈N. Prove that if the numbers f(0), f(1), \cdots , f( \sqrt{p\over 3} ) are primes, then all the numbers f(0),f(1),⋯,f(p−2) are primes.Proposed by Soviet Union. Construct a triangle with area ≤ A (IMO SL 1987-P19)
Let α,β,γ be positive real numbers such that α+β+γ<π, α+β>γ,β+γ>α, γ+α>β. Prove that with the segments of lengths sinα,sinβ,sinγ we can construct a triangle and that its area is not greater than
A=81(sin2α+sin2β+sin2γ).Proposed by Soviet Union Determine the smallest h(r) (IMO SL 1987-P18)
For any integer r≥1, determine the smallest integer h(r)≥1 such that for any partition of the set {1,2,⋯,h(r)} into r classes, there are integers a≥0 ;1≤x≤y, such that a+x,a+y,a+x+y belong to the same class.Proposed by Romania A four-coloring of the set M (IMO SL 1987-P17)
Prove that there exists a four-coloring of the set M={1,2,⋯,1987} such that any arithmetic progression with 10 terms in the set M is not monochromatic. Alternative formulationLet M={1,2,⋯,1987}. Prove that there is a function f:M→{1,2,3,4} that is not constant on every set of 10 terms from M that form an arithmetic progression.Proposed by Romania Number of permutations
Let pn(k) be the number of permutations of the set {1,2,3,…,n} which have exactly k fixed points. Prove that ∑k=0nkpn(k)=n!.(IMO Problem 1)Original formulation Let S be a set of n elements. We denote the number of all permutations of S that have exactly k fixed points by pn(k). Prove:(a) ∑k=0nkpn(k)=n! ; (b) ∑k=0n(k−1)2pn(k)=n!Proposed by Germany, FR Ineq with n variables (IMO SL 1987-P15)
Let x1,x2,…,xn be real numbers satisfying x12+x22+…+xn2=1. Prove that for every integer k≥2 there are integers a1,a2,…,an, not all zero, such that ∣ai∣≤k−1 for all i, and ∣a1x1+a2x2+…+anxn∣≤kn−1(k−1)n. (IMO Problem 3)Proposed by Germany, FR Number of words with n digits (IMO SL 1987-P14)
How many words with n digits can be formed from the alphabet {0,1,2,3,4}, if neighboring digits must differ by exactly one?Proposed by Germany, FR. A,B', C'; A',B, C'; A',B', C are collinear (IMO SL 1987-P12)
Given a nonequilateral triangle ABC, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles A′B′C′ (the vertices listed counterclockwise) for which the triples of points A,B′,C′;A′,B,C′; and A′,B′,C are collinear.Proposed by Poland. Number of partitions (IMO SL 1987-P11)
Find the number of partitions of the set {1,2,⋯,n} into three subsets A1,A2,A3, some of which may be empty, such that the following conditions are satisfied:(i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.(ii) If A1,A2,A3 are all nonempty, then in exactly one of them the minimal number is even .Proposed by Poland. a_i . b_j mod mk (IMO SL 1987-P8)
(a) Let gcd(m,k)=1. Prove that there exist integers a1,a2,...,am and b1,b2,...,bk such that each product aibj (i=1,2,⋯,m; j=1,2,⋯,k) gives a different residue when divided by mk.(b) Let gcd(m,k)>1. Prove that for any integers a1,a2,...,am and b1,b2,...,bk there must be two products aibj and asbt ((i,j)=(s,t)) that give the same residue when divided by mk.Proposed by Hungary. It's possible to find v_i (IMO SL 1987-P7)
Given five real numbers u0,u1,u2,u3,u4, prove that it is always possible to find five real numbers v0,v1,v2,v3,v4 that satisfy the following conditions:(i) u_i-v_i \in \mathbb N, 0 \leq i \leq 4(ii) ∑0≤i<j≤4(vi−vj)2<4.Proposed by Netherlands. Find the point P (IMO SL 1987-P5)
Find, with proof, the point P in the interior of an acute-angled triangle ABC for which BL2+CM2+AN2 is a minimum, where L,M,N are the feet of the perpendiculars from P to BC,CA,AB respectively.Proposed by United Kingdom. If n ≥ 4, then k ≥ 2n - (IMO SL 1987-P2)
At a party attended by n married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques C1,C2,⋯,Ck with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if n≥4, then k≥2n.Proposed by USA.