MathDB

4

Part of 2010 District Olympiad

Problems(6)

angle bisector wanted, AD=CD=CB, AB // CD (2010 Romania District VII P4)

Source:

5/19/2020
We consider the quadrilateral ABCDABCD, with AD=CD=CBAD = CD = CB and AB∥CDAB \parallel CD. Points EE and FF belong to the segments CDCD and CBCB so that angles ∠ADE=∠AEF\angle ADE = \angle AEF. Prove that: a) 4CF≤CB4CF \le CB , b) if 4CF=CB4CF = CB, then AEAE is the bisector of the angle ∠DAF\angle DAF.
geometryangle bisectorequal anglesequal segments
a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|} N diphantine

Source: 2010 Romania District VIII p4

9/1/2024
Find all non negative integers (a,b)(a, b) such that a+2b−b2=2a+a2+∣2a+1−2b∣.a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.
Diophantine equationnumber theorydiophantine
Romania District Olympiad 2010

Source: Grade IX

3/13/2010
Determine all the functions f:N→N f: \mathbb{N}\rightarrow \mathbb{N} such that f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1,   \forall n\in \mathbb{N}.
functioninductionalgebra proposedalgebra
Romania District Olympiad 2010

Source: Grade X

3/13/2010
Consider the sequence a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1, where z∈C∗ z\in \mathbb{C}^* is given. i) Prove that if a1>2 a_1>2, then: a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^* ii) Prove that if there is a k∈N∗ k\in \mathbb{N}^* such that ak≤2 a_k\le 2, then a1≤2 a_1\le 2.
searchfunctionalgebra proposedalgebra
Romanian District Olympiad

Source: Grade XI

3/17/2010
Prove that exists sequences (an)n≥0 (a_n)_{n\ge 0} with a_n\in \{\minus{}1,\plus{}1\}, for any n∈N n\in \mathbb{N}, such that: \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}
limitcalculuscalculus computations
Romanian District Olympiad

Source: Grade XII

3/17/2010
Let f:[0,1]→R f: [0,1]\rightarrow \mathbb{R} a derivable function such that f(0)\equal{}f(1), \int_0^1f(x)dx\equal{}0 and f′(x)≠1 , (∀)x∈[0,1] f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]. i)Prove that the function g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x is strictly decreasing. ii)Prove that for each integer number n≥1 n\ge 1, we have: \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}
functionintegrationinequalitiesreal analysisreal analysis unsolved