MathDB

1

Part of 2015 Belarus Team Selection Test

Problems(6)

any pair of (1,2),(2,4),...,(1000,2000) contains 1 from set {1,2,...,2000}

Source: 2015 Belarus TST 2.1

11/5/2020
N numbers are marked in the set {1,2,...,2000}\{1,2,...,2000\} so that any pair of the numbers (1,2),(2,4),...,(1000,2000)(1,2),(2,4),...,(1000,2000) contains at least one marked number. Find the least possible value of NN.
I.Gorodnin
combinatorics
3^a+2^b+2015=3 c! for integers a,b,c>=0

Source: 2015 Belarus TST 1.1

11/5/2020
Solve the equation in nonnegative integers a,b,ca,b,c:
3a+2b+2015=3c!3^a+2^b+2015=3c!
I.Gorodnin
number theoryDiophantine equationdiophantine
f(f(x)) = bx f(x) +a, where f surjective

Source: 2015 Belarus TST 3.1

11/5/2020
Do there exist numbers a,b∈Ra,b \in R and surjective function f:R→Rf: R \to R such that f(f(x))=bxf(x)+af(f(x)) = bx f(x) +a for all real xx?
I.Voronovich
functional equationalgebrafunctional
segment of arc midpoints perpendicular to parabola axis, circle related

Source: Belarus TST 2015 4.1

6/9/2020
A circle intersects a parabola at four distinct points. Let MM and NN be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line MNMN is perpendicular to the axis of the parabola.
I. Voronovich
conicsparabolaperpendiculararc midpointgeometryconic section
n=q(q^2-q-1)=r(2r+1) for some primes q and r

Source: 2015 Belarus TST 5.1

11/5/2020
Find all positive integers nn such that n=q(q2−q−1)=r(2r+1)n=q(q^2-q-1)=r(2r+1) for some primes qq and rr.
B.Gilevich
number theoryprimes
exist distinct n primes p_i such that M+k is divisible by p_k for any k=,1...,n

Source: 2015 Belarus TST 8.1

11/7/2020
Given m,n∈Nm,n \in N such that M>nn−1M>n^{n-1} and the numbers m+1,m+2,...,m+nm+1, m+2, ..., m+n are composite. Prove that exist distinct primes p1,p2,...,pnp_1,p_2,...,p_n such that M+kM+k is divisible by pkp_k for any k=1,2,...,nk=1,2,...,n.
Tuymaada Olympiad 2004, C.A.Grimm. USA
number theoryprimesdivisibleconsecutive