MathDB

Problem 3

Part of 2000 Moldova National Olympiad

Problems(6)

prove concyclicity of projections in quadrilateral if diagonals orthogonal

Source: Moldova 2000 Grade 9 P3

4/25/2021
The diagonals of a convex quadrilateral ABCDABCD are orthogonal and intersect at a point EE. Prove that the projections of EE on AB,BC,CD,DAAB,BC,CD,DA are concyclic.
geometry
show number is prime, {1},{2,3,4},{5,6,7,8,9}

Source: Moldova MO 2000 Grade 7 P3

4/23/2021
Consider the sets A1={1}A_1=\{1\}, A2={2,3,4}A_2=\{2,3,4\}, A3={5,6,7,8,9}A_3=\{5,6,7,8,9\}, etc. Let bnb_n be the arithmetic mean of the smallest and the greatest element in AnA_n. Show that the number 2000b1−1+2000b2−1+…+2000b2000−1\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1} is a prime integer.
number theory
if m+n-1|m^2+n^2-1 then m+n-1 is nonprime

Source: Moldova 2000 Grade 8 P3

4/24/2021
Suppose that m,n≥2m,n\ge2 are integers such that m+n−1m+n-1 divides m2+n2−1m^2+n^2-1. Prove that the number m+n−1m+n-1 is not prime.
number theory
average of sum of min and max of subsets of [2000]

Source: Moldova 2000 Grade 10 P3

4/26/2021
For every nonempty subset XX of M={1,2,…,2000}M=\{1,2,\ldots,2000\}, aXa_X denotes the sum of the minimum and maximum element of XX. Compute the arithmetic mean of the numbers aXa_X when XX goes over all nonempty subsets XX of MM.
combinatorics
excircle and incircle

Source: Moldova 2000 Grade 11 P3

4/26/2021
The excircle of a triangle ABCABC corresponding to AA touches the side BCBC at MM, and the point on the incircle diametrically opposite to its point of tangency with BCBC is denoted by NN. Prove that A,M,A,M, and NN are collinear.
geometryTriangle
2+22+222+...

Source: Moldova 2000 Grade 12 P3

4/27/2021
For any n∈Nn\in\mathbb N, denote by ana_n the sum 2+22+222+⋯+22…22+22+222+\cdots+22\ldots2, where the last summand consists of nn digits of 22. Determine the greatest nn for which ana_n contains exactly 222222 digits of 22.
algebra