Subcontests
(11)Prove that we can find a,b,c
Prove that for any positive integers x,y,z with xy−z2=1 one can find non-negative integers a,b,c,d such that x=a2+b2,y=c2+d2,z=ac+bd.
Set z=(2q)! to deduce that for any prime number p=4q+1, p can be represented as the sum of squares of two integers. Determine a,b,c such that system of equations is compatible
Determine all the triples (a,b,c) of positive real numbers such that the system
ax+by−cz=0,a1−x2+b1−y2−c1−z2=0,
is compatible in the set of real numbers, and then find all its real solutions. Prove that for each n there exist B such that B(n)=n
Let p be a prime and A={a1,…,ap−1} an arbitrary subset of the set of natural numbers such that none of its elements is divisible by p. Let us define a mapping f from P(A) (the set of all subsets of A) to the set P={0,1,…,p−1} in the following way:(i) if B={ai1,…,aik}⊂A and ∑j=1kaij≡n(modp), then f(B)=n,(ii) f(∅)=0, ∅ being the empty set.Prove that for each n∈P there exists B⊂A such that f(B)=n. Prove that the function is concave
A function f:I→R, defined on an interval I, is called concave if f(θx+(1−θ)y)≥θf(x)+(1−θ)f(y) for all x,y∈I and 0≤θ≤1. Assume that the functions f1,…,fn, having all nonnegative values, are concave. Prove that the function (f1f2⋯fn)1/n is concave. Prove the existence and uniqueness of three points
We consider three distinct half-lines Ox,Oy,Oz in a plane. Prove the existence and uniqueness of three points A∈Ox,B∈Oy,C∈Oz such that the perimeters of the triangles OAB,OBC,OCA are all equal to a given number 2p>0. Geometric Inequality on the length sides - ISL 1978
Let T1 be a triangle having a,b,c as lengths of its sides and let T2 be another triangle having u,v,w as lengths of its sides. If P,Q are the areas of the two triangles, prove that
16PQ≤a2(−u2+v2+w2)+b2(u2−v2+w2)+c2(u2+v2−w2).
When does equality hold? Partition into k nonintersecting subsets ISL 1978
The set M={1,2,...,2n} is partitioned into k nonintersecting subsets M1,M2,…,Mk, where n≥k3+k. Prove that there exist even numbers 2j1,2j2,…,2jk+1 in M that are in one and the same subset Mi (1≤i≤k) such that the numbers 2j1−1,2j2−1,…,2jk+1−1 are also in one and the same subset Mj(1≤j≤k).