VTMO 2013 — Grade 7 — final

Replace the letters in MATEMATIKA by digits or+/- signs to obtain $2014$. Different letters use different symbols and repeated letters the same symbol. Give one assignment.

Minimize $C = \frac{6}{| x | - 3}$ for integer $x$ where defined.

Positive integers $a , b$ satisfy exactly three of: $\left(a + 1\right)$ divisible by $b$; $a = 2 b + 5$; $a + b$ divisible by $3$; $a + 7 b$ prime. Find $a , b$.

Triangle $ABC$ has angle $BAC = 120 ^{\circ}$. On its internal bisector ray choose $D$ with $AD = AB + AC$. Prove $BCD$ equilateral.

Triangle $ABC$ has angle $ABC = 30 ^{\circ}$. Construct equilateral $ACD$ externally. Prove $AB^{2} + BC^{2} = BD^{2}$.