MathDB

2

Part of 2005 Romania National Olympiad

Problems(6)

divisibility with 13

Source: Romania Nationals RMO 2005 - grade 7, problem 2

3/31/2005
Let a,ba,b be two integers. Prove that a) 13∣2a+3b13 \mid 2a+3b if and only if 13∣2b−3a13 \mid 2b-3a; b) If 13∣a2+b213 \mid a^2+b^2 then 13∣(2a+3b)(2b+3a)13 \mid (2a+3b)(2b+3a). Mircea Fianu
modular arithmetic
product of the digits of positive integers

Source: Romanian Nationals RMO 2005 - grade 8, problem 2

3/31/2005
For a positive integer nn, written in decimal base, we denote by p(n)p(n) the product of its digits. a) Prove that p(n)≤np(n) \leq n; b) Find all positive integers nn such that 10p(n)=n2+4n−2005. 10p(n) = n^2+ 4n - 2005. Eugen Păltănea
functional equation from R to R

Source: Romanian Nationals RMO 2005 - grade 9, problem 2

3/31/2005
Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} for which x(f(x+1)−f(x))=f(x), x(f(x+1)-f(x)) = f(x), for all x∈Rx\in\mathbb{R} and ∣f(x)−f(y)∣≤∣x−y∣, | f(x) - f(y) | \leq |x-y| , for all x,y∈Rx,y\in\mathbb{R}. Mihai Piticari
functionLaTeXalgebra proposedalgebra
well known but yet very hard solid geometry problem

Source: Romanian Nationals RMO 2005 - grade 10, problem 2, [Arthur Engel, Problem-Solving St., problem 3.30]

3/31/2005
The base A1A2…AnA_{1}A_{2}\ldots A_{n} of the pyramid VA1A2…AnVA_{1}A_{2}\ldots A_{n} is a regular polygon. Prove that if ∠VA1A2≡∠VA2A3≡⋯≡∠VAn−1An≡∠VAnA1,\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1}, then the pyramid is regular.
geometry3D geometrypyramidtrigonometrygeometric transformationrotationconics
continous onto function

Source: Romanian Nationals RMO 2005 - grade 11, problem 2

3/31/2005
Let f:[0,1)→(0,1)f:[0,1)\to (0,1) a continous onto (surjective) function. a) Prove that, for all a∈(0,1)a\in(0,1), the function fa:(a,1)→(0,1)f_a:(a,1)\to (0,1), given by fa(x)=f(x)f_a(x) = f(x), for all x∈(a,1)x\in(a,1) is onto; b) Give an example of such a function.
functiontrigonometryalgebradomainlimitreal analysisreal analysis solved
groups and proper subgroups

Source: Romanian Nationals RMO 2005 - grade 12, problem 2

3/31/2005
Let GG be a group with mm elements and let HH be a proper subgroup of GG with nn elements. For each x∈Gx\in G we denote Hx={xhx−1∣h∈H}H^x = \{ xhx^{-1} \mid h \in H \} and we suppose that Hx∩H={e}H^x \cap H = \{e\}, for all x∈G−Hx\in G - H (where by ee we denoted the neutral element of the group GG). a) Prove that Hx=HyH^x=H^y if and only if x−1y∈Hx^{-1}y \in H; b) Find the number of elements of the set ⋃x∈GHx\bigcup_{x\in G} H^x as a function of mm and nn. Calin Popescu
group theoryabstract algebrafunctionLaTeXsuperior algebrasuperior algebra unsolved