Olympiad problems

2000 Moldova National Olympiad, Problem 7

The Fibonacci sequence is defined by $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$. Prove that the sum of $2000$ consecutive terms of the Fibonacci sequence is never a term of the sequence.

2020 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , Inequality , 3-variable inequality , Moldova

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

2019 Junior Balkan Team Selection Tests - Moldova, 10

Tags: algebra , Moldova

Positive real numbers $a$ and $b$ verify $a^5+b^5=a^3+b^3$. Find the greatest possible value of the expression $E=a^2-ab+b^2$.

2020 Moldova Team Selection Test, 10

Tags: maximum value , algebra , inequalities , Moldova

Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$

2019 Moldova Team Selection Test, 6

Tags: Inequality , inequalities , Moldova

Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$. Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$

2020 Moldova Team Selection Test, 7

Tags: Inequality , inequalities , Moldova , algebra

Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

2021 Moldova Team Selection Test, 6

Tags: combinatorics , game , Moldova

There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)

2016 Moldova Team Selection Test, 1

Tags: Moldova

If $x_1,x_2,...,x_n>0 $ and $x_1^2+x_2^2+...+x_n^2=\dfrac{1}{n}$,prove that $\sum x_i+\sum \dfrac{1}{x_i \cdot x_{i+1}} \ge n^3+1.$