MathDB

3

Part of 2005 Romania Team Selection Test

Problems(5)

all distances between n points are less than 1 - max area

Source: Romanian IMO TST 2005 - day 3, problem 3

4/19/2005
Let PP be a polygon (not necessarily convex) with nn vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than 32\dfrac {\sqrt 3} 2.
geometryintegrationperimeteranalytic geometryinequalitiescalculusgeometry proposed
another hard solid geometry - at first look anyway

Source: Romanian IMO TST - day 1, problem 3

3/31/2005
Prove that if the distance from a point inside a convex polyhedra with nn faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than n−2n-2. Calin Popescu
geometry3D geometrycircumcircleinequalitiestetrahedronspheregeometry proposed
bounded bs sequence

Source: Romanian IMO TST 2005 - day 2, problem 3

4/1/2005
A sequence of real numbers {an}n\{a_n\}_n is called a bs sequence if an=∣an+1−an+2∣a_n = |a_{n+1} - a_{n+2}|, for all n≥0n\geq 0. Prove that a bs sequence is bounded if and only if the function ff given by f(n,k)=anak(an−ak)f(n,k)=a_na_k(a_n-a_k), for all n,k≥0n,k\geq 0 is the null function. Mihai Baluna - ISL 2004
functioninductionalgebra proposedalgebra
prime of the form 8k+7 and fractional parts sum

Source: Romanian IMO TST 2005 - day 4, problem 3

4/23/2005
Let n≥0n\geq 0 be an integer and let p≡7(mod8)p \equiv 7 \pmod 8 be a prime number. Prove that ∑k=1p−1{k2np−12}=p−12. \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . Călin Popescu
modular arithmeticfloor functionquadraticsinductionalgebrafunctional equationnumber theory proposed
hard functional equation with divisibility

Source: Romanian IMO TST 2005 - day 5, problem 3

4/24/2005
Let N={1,2,…}\mathbb{N}=\{1,2,\ldots\}. Find all functions f:N→Nf: \mathbb{N}\to\mathbb{N} such that for all m,n∈Nm,n\in \mathbb{N} the number f2(m)+f(n)f^2(m)+f(n) is a divisor of (m2+n)2(m^2+n)^2.
functionlinear algebramatrixalgebra proposedalgebra