Drzavno takmicenje (National round) 2010 · Lemma

1GeometryProofIn a triangle \(ABC\) the angle at \(B\) equals \(80^\circ\). Three further points are marked: the point \(D\) on the side \(BC\) with \(AB = AD = CD\); the point \(F\) on the side \(AB\) with \(AF = BD\); …2GeometryProofA circle can be inscribed in the trapezoid \(ABCD\), whose parallel sides are \(AB\) and \(CD\). Prove that the circle having \(BC\) as a diameter and the circle having \(AD\) as a diameter touch each …3CombinatoricsFree answerLet \(n > 1\) be a natural number. How many \(n\)-digit numbers are palindromes and divisible by \(9\)? (A number is a palindrome when its decimal representation is symmetric, that is, it reads the same …4Number theoryProofLet \(P(x)\) be a polynomial with integer coefficients for which there exist prime numbers \(p < q < r\) with \[ \{P(p),\, P(q),\, P(r)\} = \{20,\, 3,\, 2010\} . \] Prove that \(P(p+q) = 2010\).5CombinatoricsProofCan nine points, no three of them collinear, be placed inside the cross-shaped figure below (its boundary included) in such a way that whenever three of them span a triangle lying inside the figure, that …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.