MathDB

Problem 7

Part of 2001 Moldova National Olympiad

Problems(6)

(n-1)^2<=x<(n+1)^2 set

Source: 2001 Moldova MO Grade 7 P7

4/12/2021
Let nn be a positive integer. We denote by SS the sum of elements of the set M={x∈N∣(n−1)2≤x<(n+1)2}M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}. (a) Show that SS is divisible by 66. (b) Find all n∈Nn\in\mathbb N for which S+(1−n)(1+n)=2001S+(1-n)(1+n)=2001.
setnumber theory
incenter-related concurrency

Source: 2001 Moldova MO Grade 8 P7

4/12/2021
The incircle of a triangle ABCABC is centered at II and touches AC,ABAC,AB and BCBC at points K,L,MK,L,M, respectively. The median BB1BB_1 of △ABC\triangle ABC intersects MNMN at DD. Prove that the points I,D,KI,D,K are collinear.
geometryTriangles
proof of concurrency in squares with same size

Source: 2001 Moldova MO Grade 10 P7

4/13/2021
Let ABCDABCD and AB’C’D’AB’C’D’ be equally oriented squares. Prove that the lines BB1,CC1,DD1BB_1,CC_1,DD_1 are concurrent.
geometry
line cutting middle lines in the same ratio

Source: 2001 Moldova MO Grade 9 P7

4/12/2021
A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.
geometryratio
limit of sum of a sequence

Source: 2001 Moldova MO Grade 11 P7

4/13/2021
Set an=2nn4+3n2+4,n∈Na_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N. Prove that the sequence Sn=a1+a2+…+anS_n=a_1+a_2+\ldots+a_n is upperbounded and lowerbounded and find its limit as n→∞n\to\infty.
Sequencealgebra
integral inequality

Source: 2001 Moldova MO Grade 12 P7

4/13/2021
Let f:[0,1]→Rf:[0,1]\to\mathbb R be a continuously differentiable function such that f(x0)=0f(x_0)=0 for some x0∈[0,1]x_0\in[0,1]. Prove that ∫01f(x)2dx≤4∫01f’(x)2dx.\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.
calculusintegrationinequalities