MathDB

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Part of 2018 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2018 Grade 9 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
if aa, bb and cc are real numbers such that (a−b)(b−c)(c−a)≠0(a-b)(b-c)(c-a) \neq 0, prove the equality: b2c2(a−b)(a−c)+c2a2(b−c)(b−a)+a2b2(c−a)(c−b)=ab+bc+ca\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca
algebraidentityreal numbers
Regional Olympiad - FBH 2018 Grade 10 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Show that system of equations 2ab=6(a+b)−132ab=6(a+b)-13 a2+b2=4a^2+b^2=4 has not solutions in set of real numbers.
algebrasystem of equations
Regional Olympiad - FBH 2018 Grade 11 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Find all values of real parameter aa for which equation 2sin⁡4(x)+cos⁡4(x)=a2{\sin}^4(x)+{\cos}^4(x)=a has real solutions
parameterizationalgebraequation
Regional Olympiad - FBH 2018 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
a)a) Prove that for all positive integers n≥3n \geq 3 holds: (n1)+(n2)+...+(nn−1)=2n−2\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2 where (nk)\binom{n}{k} , with integer kk such that n≥k≥0n \geq k \geq 0, is binomial coefficent
b)b) Let n≥3n \geq 3 be an odd positive integer. Prove that set A={(n1),(n2),...,(nn−12)}A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\} has odd number of odd numbers
Setsnumber theorybinomial coefficients