MathDB

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Part of 2016 Romania National Olympiad

Problems(6)

\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} is integer (2016 Romanian NMO grade VII P1)

Source:

9/3/2024
Find all non-negative integers nn so that n+3+n+n+3\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} is an integer.
number theoryInteger
2 colors for vertices of prism, no of segments with diff. colors

Source: 2016 Romanian NMO grade VIII P1

9/3/2024
The vertices of a prism are colored using two colors, so that each lateral edge has its vertices differently colored. Consider all the segments that join vertices of the prism and are not lateral edges. Prove that the number of such segments with endpoints differently colored is equal to the number of such segments with endpoints of the same color.
geometry3D geometryprismcombinatoricscombinatorial geometryColoring
Problem with circumcenters and orthocenter

Source: Romania National Olympiad 2016, grade ix, p.1

8/23/2019
The orthocenter H H of a triangle ABC ABC is distinct from its vertices and its circumcenter O. O. M,N,P M,N,P are the circumcenters of the triangles HBC,HCA, HBC,HCA, respectively, HAB. HAB. Prove that AM,BN,CP AM,BN,CP and OH OH are concurrent.
geometrycircumcircleorthocenterPure geometry
This function is nondecreasing (knowledge of calculus not required)

Source: Romania National Olympiad 2016, grade x, p.1

8/25/2019
Let be a natural number n≥2 n\ge 2 and n n positive real numbers a1,a2,…,an a_1,a_2,\ldots ,a_n whose product is 1. 1. Prove that the function f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,  f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , is nondecreasing.
functioncalculusalgebra
Too easy limit of sequence of mean values

Source: Romanian National Olympiad 2016, grade xii. p.1

8/25/2019
Prove that there exists an unique sequence (cn)n≥1 \left( c_n \right)_{n\ge 1} of real numbers from the interval (0,1) (0,1) such that∫01dx1+xm=11+cmm, \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , for all natural numbers m, m, and calculate lim⁡k→∞kckk. \lim_{k\to\infty } kc_k^k.
Radu Pop
real analysislimits
linear algebra problem with equality of determinants

Source: Romanian National Olympiad 2016, grade xi, p.1

8/25/2019
Let be a 2×2 2\times 2 real matrix A A that has the property that ∣Ad−I2∣=∣Ad+I2∣, \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , for all d∈{2014,2016}. d\in\{ 2014,2016 \} . Prove that ∣An−I2∣=∣An+I2∣, \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , for any natural number n. n.
linear algebramatrix