MathDB

1

Part of 2011 Romania Team Selection Test

Problems(5)

Square inside unit square

Source:

5/31/2011
Suppose a square of sidelengh ll is inside an unit square and does not contain its centre. Show that l≤1/2.l\le 1/2.
Marius Cavachi
rotationgeometry proposedgeometry
2f(x)=f(x+y)+f(x+2y)

Source: Romanian TST 2011

2/4/2012
Determine all real-valued functions ff on the set of real numbers satisfying 2f(x)=f(x+y)+f(x+2y)2f(x)=f(x+y)+f(x+2y) for all real numbers xx and all non-negative real numbers yy.
functionLaTeXalgebra proposedalgebra
Four points on a circle

Source:

9/6/2011
Let ABCDABCD be a cyclic quadrilateral. The lines BCBC and ADAD meet at a point PP. Let QQ be the point on the line BPBP, different from BB, such that PQ=BPPQ=BP. Consider the parallelograms CAQRCAQR and DBCSDBCS. Prove that the points C,Q,R,SC,Q,R,S lie on a circle.
geometryparallelogramgeometric transformationreflectiongeometry unsolved
Infinitely many positive integers n

Source: Romania BMO/IMO TST 2011 P12.

1/29/2012
Show that there are infinitely many positive integer numbers nn such that n2+1n^2+1 has two positive divisors whose difference is nn.
number theory unsolvednumber theory
Preimage is finite non-empty set of consecutive naturals

Source: American Mathematical Monthly

4/9/2012
Given a positive integer number kk, define the function ff on the set of all positive integer numbers to itself by f(n)={1,if n≤k+1f(f(n−1))+f(n−f(n−1)),if n>k+1f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases} Show that the preimage of every positive integer number under ff is a finite non-empty set of consecutive positive integers.
functioninductionstrong inductionalgebra proposedalgebra