MathDB

4

Part of 2014 District Olympiad

Problems(8)

Cute numbers

Source: Romanian District Olympiad 2014, Grade 5, P4

6/15/2014
A 1010 digit positive integer is called a \emph{cute} number if its digits are from the set {1,2,3}\{1,2,3\} and every two consecutive digits differ by 11. [*]Prove that exactly 55 digits of a cute number are equal to 22. [*]Find the total number of cute numbers. [*]Prove that the sum of all cute numbers is divisible by 14081408.
modular arithmeticnumber theorycombinatorics proposedcombinatorics
Proving a perpendicularity

Source: Romanian District Olympiad 2014, Grade 7, P4

6/15/2014
Let ABCDABCD be a square and consider the points K∈AB,L∈BC,K\in AB, L\in BC, and M∈CDM\in CD such that ΔKLM\Delta KLM is a right isosceles triangle, with the right angle at LL. Prove that the lines ALAL and DKDK are perpendicular to each other.
number theoryleast common multiplegeometry proposedgeometry
Number of integers

Source: Romanian District Olympiad 2014, Grade 6, P4

6/15/2014
Determine all positive integers aa for which there exist exactly 20142014 positive integers bb such that 2≤ab≤5\displaystyle2\leq\frac{a}{b}\leq5.
floor functionceiling functionnumber theory proposednumber theory
All possible values of the sum

Source: Romanian District Olympiad 2014, Grade 8, P4

6/15/2014
Let n≥2n\geq2 be a positive integer. Determine all possible values of the sum S=⌊x2−x1⌋+⌊x3−x2⌋+...+⌊xn−xn−1⌋ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor where xi∈Rx_i\in \mathbb{R} satisfying ⌊xi⌋=i\lfloor{x_i}\rfloor=i for i=1,2,…ni=1,2,\ldots n.
floor functionalgebra proposedalgebra
A functional equation

Source: Romanian District Olympiad 2014, Grade 9, P4

6/15/2014
Find all functions f:N∗→N∗f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast} with the properties:
[*] f(m+n) -1 \mid f(m)+f(n),  \forall m,n\in\mathbb{N}^{\ast} [*]n2−f(n) is a square   ∀n∈N∗ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast}
functionalgebrafunctional equationDivisibility
Another functional equation

Source: Romanian District Olympiad 2014, Grade 10, P4

6/15/2014
Find all functions f:Q→Qf:\mathbb{Q}\to \mathbb{Q} such that f(x+3f(y))=f(x)+f(y)+2y   \forall x,y\in \mathbb{Q}
functioninductionalgebra solvedalgebra
A group and a group morphism

Source: Romanian District Olympiad 2014, Grade 12, P4

6/15/2014
Let (G,⋅)(G,\cdot) be a group with no elements of order 4, and let f:G→Gf:G\rightarrow G be a group morphism such that f(x)∈{x,x−1}f(x)\in\{x,x^{-1}\}, for all x∈Gx\in G. Prove that either f(x)=xf(x)=x for all x∈Gx\in G, or f(x)=x−1f(x)=x^{-1} for all x∈Gx\in G.
superior algebrasuperior algebra unsolved
Sequences converging to zero

Source: Romanian District Olympiad 2014, Grade 11, P4

6/15/2014
Let f ⁣:N→N∗f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast} be a strictly increasing function. Prove that:
[*]There exists a decreasing sequence of positive real numbers, (yn)n∈N(y_{n})_{n\in\mathbb{N}}, converging to 00, such that yn≤2yf(n)y_{n}\leq2y_{f(n)}, for all n∈Nn\in\mathbb{N}. [*]If (xn)n∈N(x_{n})_{n\in\mathbb{N}} is a decreasing sequence of real numbers, converging to 00, then there exists a decreasing sequence of real numbers (yn)n∈N(y_{n})_{n\in\mathbb{N}}, converging to 00, such that xn≤yn≤2yf(n)x_{n}\leq y_{n} \leq2y_{f(n)}, for all n∈Nn\in\mathbb{N}.
inductionreal analysisreal analysis unsolved