MathDB

1

Part of 2011 District Olympiad

Problems(6)

121 points in a sqaure - 2011 Romania District VII p1

Source:

9/1/2024
In a square of side length 6060, 121121 distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding 3030.
combinatoricsgeometrycombinatorial geometry
(x^2 -x +1)(3y^2-2y + 3) -2=0 2011 Romania District VIII p1

Source:

9/1/2024
Find the real numbers xx and yy such that (x2−x+1)(3y2−2y+3)−2=0.(x^2 -x +1)(3y^2-2y + 3) -2=0.
algebrainequalities
Knowing a sum of vectors, prove a sum of vectors

Source: Romanian District Olympiad 2011, Grade IX, Problem 1

10/8/2018
On the sides AB,BC,CD,DA AB,BC,CD,DA of the parallelogram ABCD, ABCD, consider the points M,N,P, M,N,P, respectively, Q, Q, such that MN→+QP→=AC→. \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . Show that PN→+QM→=DB→. \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} .
geometryparallelogramvector
a^x=b^x+c^x has only one real solution (collection purposes)

Source:

10/8/2018
Let a,b,c a,b,c be three positive numbers. Show that the equation ax+bx=cx a^x+b^x=c^x has, at most, one real solution.
Increasing functionalgebraequationseasydistrict olympiad
Romania District Olympiad 2011 - Grade XI

Source:

3/12/2011
a) Prove that {x+y}−{y}\{x+y\}-\{y\} can only be equal to {x}\{x\} or {x}−1\{x\}-1 for any x,y∈Rx,y\in \mathbb{R}.
b) Let α∈R\Q\alpha\in \mathbb{R}\backslash \mathbb{Q}. Denote an={nα}a_n=\{n\alpha\} for all n∈N∗n\in \mathbb{N}^* and define the sequence (xn)n≥1(x_n)_{n\ge 1} by
xn=(a2−a1)(a3−a2)⋅…⋅(an+1−an)x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)
Prove that the sequence (xn)n≥1(x_n)_{n\ge 1} is convergent and find it's limit.
functionlimitcontinued fractionreal analysisreal analysis unsolved
Show that a particularly easy primitivable function is irrational

Source: Romanian District Olympiad 2011, Grade XII, Problem 1

10/8/2018
Prove the rationality of the number 1π∫sin⁡π13cos⁡π131−x2dx. \frac{1}{\pi }\int_{\sin\frac{\pi }{13}}^{\cos\frac{\pi }{13}} \sqrt{1-x^2} dx.
functionIntegralprimitivesFTCcalculus