MathDB

Problem 3

Part of 2001 Moldova National Olympiad

Problems(5)

proof in a square, arbitrary line (2001 Moldova MO Grade 7 P3)

Source:

4/12/2021
A line di (i=1,2,3)d_i~(i=1,2,3) intersects two opposite sides of a square ABCDABCD at points MiM_i and NiN_i. Prove that if M1N1=M2N2=M3N3M_1N_1=M_2N_2=M_3N_3, then two of the lines did_i are either parallel or perpendicular.
geometry
prove line tangent to circumcircle in triangle

Source: 2001 Moldova MO Grade 8 P3

4/12/2021
In a triangle ABCABC, the line symmetric to the median through AA with respect to the bisector of the angle at AA intersects BCBC at MM. Points PP on ABAB and QQ on ACAC are chosen such that MP∥ACMP\parallel AC and MQ∥ABMQ\parallel AB. Prove that the circumcircle of the triangle MPQMPQ is tangent to the line BCBC.
geometry
roosterfight, arranging roosters by conditions

Source: Moldova MO 2001 Grade 10 P3

4/22/2021
During a fight, each of the 20012001 roosters has torn out exactly one feather of another rooster, and each rooster has lost a feather. It turned out that among any three roosters there is one who hasn’t torn out a feather from any of the other two roosters. Find the smallest kk with the following property: It is always possible to kill kk roosters and place the rest into two henhouses in such a way that no two roosters, one of which has torn out a feather from the other one, stay in the same henhouse.
gamecombinatorics
P(x^2)=P(x)P(x-1) over R

Source: 2001 Moldova MO Grade 12 P3

4/13/2021
Find all polynomials P(x)P(x) with real coefficieints such that P(x2)=P(x)P(x−1)P\left(x^2\right)=P(x)P(x-1) for all x∈Rx\in\mathbb R.
functional equationfePolynomialsalgebra
find locus in triangle

Source: 2001 Moldova MO Grade 11 P3

4/13/2021
For an arbitrary point DD on side BCBC of an acute-angled triangle ABCABC, let O1O_1 and O2O_2 be the circumcenters of the triangles ABDABD and ACDACD, and OO be the circumcenter of the triangle AO1O2AO_1O_2. Find the locus of OO when DD moves across BCBC.
geometry