MathDB

1

Part of 2005 District Olympiad

Problems(6)

digits

Source:

3/5/2005
Prove that for all a∈{0,1,2,…,9}a\in\{0,1,2,\ldots,9\} the following sum is divisible by 10: Sa=a‾2005+1a‾2005+2a‾2005+⋯+9a‾2005. S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}.
AMCAIMEUSA(J)MOUSAMOmodular arithmeticLaTeXbinomial coefficients
decimal points, periodic fraction representation

Source:

3/5/2005
Let MM be the set of the positive rational numbers less than 1, which can be expressed with a 10-distinct digits period in decimal representation. a) Find the arithmetic mean of all the elements in MM; b) Prove that there exists a positive integer nn, 1<n<10101<n<10^{10}, such that n⋅a−an\cdot a - a is a non-negative integer, for all a∈Ma\in M.
easy and classical inequality

Source: RMO District 2005, 9th Grade, Problem 1

3/5/2005
a) Prove that if x,y>0x,y>0 then xy2+yx2≥1x+1y. \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. b) Prove that if a,b,ca,b,c are positive real numbers, then a+bc2+b+ca2+c+ab2≥2(1a+1b+1c). \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right).
inequalitiesalgebrapolynomialAMCUSA(J)MOUSAMOrearrangement inequality
monotonous exponential functions

Source: RMO District 2005, 10th Grade, Problem 1

3/5/2005
Let a,b>1a,b>1 be two real numbers. Prove that a>ba>b if and only if there exists a function f:(0,∞)→Rf: (0,\infty)\to\mathbb{R} such that i) the function g:R→Rg:\mathbb{R}\to\mathbb{R}, g(x)=f(ax)−xg(x)=f(a^x)-x is increasing; ii) the function h:R→Rh:\mathbb{R}\to\mathbb{R}, h(x)=f(bx)−xh(x)=f(b^x)-x is decreasing.
functionalgebra proposedalgebra
Romania District Olympiad 2005 - Grade XI

Source:

4/10/2011
Let HH denote the set of the matrices from Mn(N)\mathcal{M}_n(\mathbb{N}) and let PP the set of matrices from HH for which the sum of the entries from any row or any column is equal to 11.
a)If A∈PA\in P, prove that det⁡A=±1\det A=\pm 1.
b)If A1,A2,…,Ap∈HA_1,A_2,\ldots,A_p\in H and A1A2⋅…⋅Ap∈PA_1A_2\cdot \ldots\cdot A_p\in P, prove that A1,A2,…,Ap∈PA_1,A_2,\ldots,A_p\in P.
linear algebramatrixlinear algebra unsolved
coloring nice sets

Source: RMO District 2005, 12th Grade, Problem 1

3/5/2005
Let A1A_1, A2A_2, …\ldots, AnA_n, n≥2n\geq 2 be nn finite sets with the properties i) ∣Ai∣≥2|A_i| \geq 2, for all 1≤i≤n1\leq i \leq n; ii) ∣Ai∩Aj∣≠1|A_i\cap A_j| \neq 1, for all 1≤i<j≤n1\leq i<j\leq n. Prove that the elements of the set ⋃i=1nAi\displaystyle \bigcup_{i=1}^n A_i can be colored with 2 colors, such that all the sets AiA_i are bi-color, for all 1≤i≤n1\leq i \leq n.
inductioncombinatorics proposedcombinatorics