MathDB

2

Part of 2001 District Olympiad

Problems(6)

Romania District Olympiad 2001 - VII Grade

Source:

3/12/2011
Consider the number n=123456789101112…99100101n=123456789101112\ldots 99100101.
a)Find the first three digits of the number n\sqrt{n}. b)Compute the sum of the digits of nn. c)Prove that n\sqrt{n} isn't rational.
Valer Pop
number theory proposednumber theory
Romania District Olympiad 2001 - Grade IX

Source:

3/12/2011
In the xOyxOy system consider the lines d1 : 2x−y−2=0, d2 : x+y−4=0, d3 : y=2d_1\ :\ 2x-y-2=0,\ d_2\ :\ x+y-4=0,\ d_3\ :\ y=2 and d4 : x−4y+3=0d_4\ :\ x-4y+3=0. Find the vertices of the triangles whom medians are d1,d2,d3d_1,d_2,d_3 and d4d_4 is one of their altitudes.
Lucian Dragomir
geometry proposedgeometry
Romania District Olympiad 2001 - VIII Grade

Source:

3/12/2011
Let x,y,z∈R∗x,y,z\in \mathbb{R}^* such that xy,yz,zx∈Qxy,yz,zx\in \mathbb{Q}.
a) Prove that x2+y2+z2x^2+y^2+z^2 is rational; b) If x3+y3+z3x^3+y^3+z^3 is rational, prove that x,y,zx,y,z are rational.
Marius Ghergu
number theory proposednumber theory
Romania District Olympiad 2001 - Grade X

Source:

3/16/2011
Two numbers (z1,z2)∈C∗×C∗(z_1,z_2)\in \mathbb{C}^*\times \mathbb{C}^* have the property (P)(P) if there is a real number a∈[−2,2]a\in [-2,2] such that z12−az1z2+z22=0z_1^2-az_1z_2+z_2^2=0. Prove that if (z1,z2)(z_1,z_2) have the property (P)(P), then (z1n,z2n)(z_1^n,z_2^n) satisfy this property, for any positive integer nn.
Dorin Andrica
algebra proposedalgebra
Romania District Olympiad 2001 - Grade XI

Source:

3/16/2011
Let n∈N, n≥2n\in \mathbb{N},\ n\ge 2. For any matrix A∈Mn(C)A\in \mathcal{M}_n(\mathbb{C}), let m(A)m(A) be the number of non-zero minors of AA. Prove that:
a)m(In)=2n−1m(I_n)=2^n-1; b)If A∈Mn(C)A\in \mathcal{M}_n(\mathbb{C}) is non-singular, then m(A)≥2n−1m(A)\ge 2^n-1.
Marius Ghergu
linear algebramatrixlinear algebra unsolved
Romania District Olympiad 2001 - Grade XII

Source:

3/16/2011
Let KK commutative field with 88 elements. Prove that (∃)a∈K(\exists)a\in K such that a3=a+1a^3=a+1.
Mircea Becheanu
algebrapolynomialsuperior algebrasuperior algebra unsolved