MathDB

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Part of 2024 Romania National Olympiad

Problems(4)

Romanian National Olympiad 2024 - Grade 9 - Problem 1

Source: Romanian National Olympiad 2024 - Grade 9 - Problem 1

4/6/2024
The points DD and EE lie on the side (BC)(BC) of the triangle ABCABC such that DD is between BB and E.E. A point RR on the segment (AE)(AE) is called remarkable if the lines PQPQ and BCBC are parallel, where {P}=DR∩AC\{P\}=DR \cap AC and {Q}=CR∩AB.\{Q\}=CR \cap AB. A point R′R' on the segment (AD)(AD) is called remarkable if the lines P′Q′P'Q' and BCBC are parallel, where {P′}=BR′∩AC\{P'\}=BR' \cap AC and {Q′}=ER′∩AB.\{Q'\}=ER' \cap AB.
a) If there exists a remarkable point on the segment (AE),(AE), prove that any point of the segment (AE)(AE) is remarkable. b) If each of the segments (AD)(AD) and (AE)(AE) contains a remarkable point, prove that BD=CE=φ⋅DE,BD=CE=\varphi \cdot DE, where φ=1+52\varphi= \frac{1+\sqrt{5}}{2} is the golden ratio.
geometryGolden Ratio
Romanian National Olympiad 2024 - Grade 10 - Problem 1

Source: Romanian National Olympiad 2024 - Grade 10 - Problem 1

4/4/2024
Solve over the real numbers the equation 3log⁡5(5x−10)−2=5−1+log⁡3x.3^{\log_5(5x-10)}-2=5^{-1+\log_3x}.
algebra
Romanian National Olympiad 2024 - Grade 11 - Problem 1

Source: Romanian National Olympiad 2024 - Grade 11 - Problem 1

4/4/2024
Let I⊂RI \subset \mathbb{R} be an open interval and f:I→Rf:I \to \mathbb{R} a twice differentiable function such that f(x)f′′(x)=0,f(x)f''(x)=0, for any x∈I.x \in I. Prove that f′′(x)=0,f''(x)=0, for any x∈I.x \in I.
functionreal analysis
Recycled from 2018 and even before

Source: Romanian National Olympiad 2024 - Grade 12 - Problem 1

4/3/2024
Let f:R→Rf: \mathbb{R} \to \mathbb{R} be a continuous function such that f(x)+sin⁡(f(x))≥x,f(x)+\sin(f(x)) \ge x, for all x∈R.x \in \mathbb{R}. Prove that ∫0πf(x)dx≥π22−2.\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.
real analysiscalculusintegrationfunction