MathDB

3

Part of 2011 Czech-Polish-Slovak Match

Problems(2)

Three of four points on a circle determine a 108° angle

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
Points AA, BB, CC, DD lie on a circle (in that order) where ABAB and CDCD are not parallel. The length of arc ABAB (which contains the points DD and CC) is twice the length of arc CDCD (which does not contain the points AA and BB). Let EE be a point where AC=AEAC=AE and BD=BEBD=BE. Prove that if the perpendicular line from point EE to the line ABAB passes through the center of the arc CDCD (which does not contain the points AA and BB), then ∠ACB=108∘\angle ACB = 108^\circ.
geometrygeometric transformationreflectionperpendicular bisectorgeometry unsolved
For a in Z, there are ∞ primes p s.t. p | n^2+3 & p | m^3-a

Source: Czech-Polish-Slovak Match, 2011

8/9/2011
Let aa be any integer. Prove that there are infinitely many primes pp such that p ∣ n2+3andp ∣ m3−a p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a for some integers nn and mm.
number theoryprime numbersrelatively primenumber theory unsolved