MathDB

3

Part of 2018 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2018 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Let pp and qq be prime numbers such that p2+pq+q2p^2+pq+q^2 is perfect square. Prove that p2−pq+q2p^2-pq+q^2 is prime
number theoryprime numbersPerfect Square
Regional Olympiad - FBH 2018 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Solve equation x⌊x⌋+{x}=2018x \lfloor{x}\rfloor+\{x\}=2018, where xx is real number
algebraequationfloor functionfrac function
Regional Olympiad - FBH 2018 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
In triangle ABCABC given is point PP such that ∠ACP=∠ABP=10∘\angle ACP = \angle ABP = 10^{\circ}, ∠CAP=20∘\angle CAP = 20^{\circ} and ∠BAP=30∘\angle BAP = 30^{\circ}. Prove that AC=BCAC=BC
geometryTriangleisosceles
Regional Olympiad - FBH 2018 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
If numbers x1x_1, x2x_2,...,xnx_n are from interval (14,1)\left( \frac{1}{4},1 \right) prove the inequality: log⁡x1(x2−14)+log⁡x2(x3−14)+...+log⁡xn−1(xn−14)+log⁡xn(x1−14)≥2n\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n
inequalitiesalgebralogarithms