MathDB

2

Part of 2014 District Olympiad

Problems(8)

Palindromic Numbers

Source: Romanian District Olympiad 2014, Grade 5, P2

6/15/2014
Let MM be the set of palindromic integers of the form 5n+45n+4 where n≥0n\ge 0 is an integer.
[*]If we write the elements of MM in increasing order, what is the 50th50^{\text{th}} number? [*]Among all numbers in MM with nonzero digits which sum up to 20142014 which is the largest and smallest one?
floor functionnumber theory proposednumber theory
Good sets

Source: Romanian District Olympiad 2014, Grade 6, P2

6/15/2014
We call a nonempty set MM good if its elements are positive integers, each having exactly 44 divisors. If the good set MM has nn elements, we denote by SMS_{M} the sum of all 4n4n divisors of its members (the sum may contain repeating terms).
a) Prove that A={2⋅37,19⋅37,29⋅37}A=\{2\cdot37,19\cdot37,29\cdot37\} is good and SA=2014S_{A}=2014.
b) Prove that if the set BB is good and 8∈B8\in B, then SB≠2014S_{B}\neq2014.
combinatorics proposedcombinatorics
|a-b|

Source: Romania District Olympiad 2014,grade VII(problem 2)

3/8/2014
Let real numbers a,b,ca,b,c such that ∣a−b∣≥∣c∣,∣b−c∣≥∣a∣,∣c−a∣≥∣b∣.\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|. Prove that a=b+ca=b+c or b=c+ab=c+a or c=a+b.c=a+b.
inequalitiesinequalities proposedalgebra
Two triangles having the same centroid

Source: Romanian District Olympiad 2014, Grade 9, P2

6/15/2014
Let ABCABC be a triangle and let the points D∈BC,E∈AC,F∈ABD\in BC, E\in AC, F\in AB, such that DBDC=ECEA=FAFB \frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB} The half-lines AD,BE,AD, BE, and CFCF intersect the circumcircle of ABCABC at points M,NM,N and PP. Prove that the triangles ABCABC and MNPMNP share the same centroid if and only if the areas of the triangles BMC,CNABMC, CNA and APBAPB are equal.
geometrycircumcirclegeometry proposed
The square preceeding n

Source: Romanian District Olympiad 2014, Grade 8, P2

6/15/2014
For each positive integer nn we denote by p(n)p(n) the greatest square less than or equal to nn.
[*]Find all pairs of positive integers (m,n)( m,n), with m≤nm\leq n, for which p(2m+1)⋅p(2n+1)=400 p( 2m+1) \cdot p( 2n+1) =400 [*]Determine the set P={n∈N∗∣n≤100 and p(n+1)p(n)∉N∗}\mathcal{P}=\{ n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}
number theory proposednumber theory
Another equation

Source: Romanian District Olympiad 2014, Grade 10, P2

6/15/2014
Solve in real numbers the equation x+log⁡2(1+5x3x+4x)=4+log⁡1/2(1+25x7x+24x) x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right)
logarithmsalgebra proposedalgebra
Continuous functions

Source: Romanian District Olympiad 2014, Grade 11, P2

6/15/2014
[*]Let f ⁣:R→Rf\colon\mathbb{R}\rightarrow\mathbb{R} be a function such that g ⁣:R→Rg\colon\mathbb{R}\rightarrow\mathbb{R}, g(x)=f(x)+f(2x)g(x)=f(x)+f(2x), and h ⁣:R→Rh\colon\mathbb{R}\rightarrow\mathbb{R}, h(x)=f(x)+f(4x)h(x)=f(x)+f(4x), are continuous functions. Prove that ff is also continuous. [*]Give an example of a discontinuous function f ⁣:R→Rf\colon\mathbb{R} \rightarrow\mathbb{R}, with the following property: there exists an interval I⊂RI\subset\mathbb{R}, such that, for any aa in II, the function ga ⁣:R→Rg_{a} \colon\mathbb{R}\rightarrow\mathbb{R}, ga(x)=f(x)+f(ax)g_{a}(x)=f(x)+f(ax), is continuous.
functionreal analysisreal analysis unsolved
Convergence of a sequence

Source: Romanian District Olympiad 2014, Grade 12, P2

6/15/2014
Let f:[0,1]→Rf:[0,1]\rightarrow{\mathbb{R}} be a differentiable function, with continuous derivative, and let sn=∑k=1nf(kn) s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) Prove that the sequence (sn+1−sn)n∈N∗(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}} converges to ∫01f(x)dx\int_{0}^{1}f(x)\mathrm{d}x.
functioncalculusderivativeintegrationreal analysisreal analysis unsolved