MathDB

3

Part of 2022 Kosovo National Mathematical Olympiad

Problems(4)

EA,FC and BD are concurrent

Source: Kosovo Mathematical Olympiad 2022, Grade 9, Problem 3

3/6/2022
Let ABCDABCD be a parallelogram and ll the line parallel to ACAC which passes through DD. Let EE and FF points on ll such that DE=DF=DBDE=DF=DB. Show that EA,FCEA,FC and BDBD are concurrent.
geometryparallelogram
a!+b+c, b!+c+a and c!+a+b prime numbers

Source: Kosovo Mathematical Olympiad 2022, Grade 10, Problem 3

3/6/2022
Let a,ba,b and cc be positive integers such that a!+b+c,b!+c+aa!+b+c,b!+c+a and c!+a+bc!+a+b are prime numbers. Show that a+b+c+12\frac{a+b+c+1}{2} is also a prime number.
number theoryprime numbers
Again a circle with center A from Balkans

Source: Kosovo MO 2022 Grade 12 Problem 3

3/5/2022
Let △ABC\bigtriangleup ABC be a triangle and DD be a point in line BCBC such that ADAD bisects ∠BAC\angle BAC. Furthermore, let FF and GG be points on the circumcircle of △ABC\bigtriangleup ABC and E≠DE\neq D point in line BCBC such that AF=AE=AD=AGAF=AE=AD=AG. If XX and YY are the feet of perpendiculars from DD to EFEF and EG,EG, respectively. Prove that XY∥ADXY\parallel AD.
geometryKosovosimsonparallelism
10^n+3^n+2

Source: Kosovo Mathematical Olympiad 2022, Grade 11, Problem 3

3/6/2022
Find all positive integers nn such that 10n+3n+210^n+3^n+2 is a palindrom number.
number theory