MathDB

Problem 2

Part of 2000 Moldova National Olympiad

Problems(6)

algebraic identity (Moldova MO 2000 Grade 7 P2)

Source:

4/23/2021
Prove that if real numbers a,b,c,da,b,c,d satisfy a2+b2+(a+b)2=c2+d2+(c+d)2a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2, then they also satisfy a4+b4+(a+b)4=c4+d4+(c+d)4a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4.
algebra
arranging thirty numbers on a circle, sum given

Source: Moldova 2000 Grade 8 P2

4/24/2021
Thirty numbers are arranged on a circle in such a way that each number equals the absolute difference of its two neighbors. Given that the sum of the numbers is 20002000, determine the numbers.
number theoryalgebra
2a^4+2b^4+2c^4 square if a+b+c=0

Source: Moldova 2000 Grade 9 P2

4/25/2021
Prove that if a,b,c are integers with a+b+c=0a+b+c=0, then 2a4+2b4+2c42a^4+2b^4+2c^4 is a perfect square.
number theory
syseq, 2 eqns, 2 variables (Moldova 2000 Grade 11 P2)

Source:

4/26/2021
Solve the system \begin{align*} 36x^2y-27y^3&~=~8,\\ 4x^3-27xy^2&~=~4.\end{align*}
algebrasystem of equations
log inequality

Source: Moldova 2000 Grade 10 P2

4/26/2021
Show that if real numbers x<1<yx<1<y satisfy the inequality 2log⁡x+log⁡(1−x)≥3log⁡y+log⁡(y−1),2\log x+\log(1-x)\ge3\log y+\log(y-1),then x3+y3<2x^3+y^3<2.
Inequalityinequalities
limit of binomial sequence

Source: Moldova 2000 Grade 12 P2

4/27/2021
For n∈Nn\in\mathbb N, define an=1(n1)+1(n2)+…+1(nn).a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}. (a) Prove that the sequence bn=annb_n=a_n^n is convergent and determine the limit. (b) Show that lim⁡n→∞bn>(32)3+2\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}.
limitreal analysis