Olympiad problems

2015 Peru IMO TST, 1

Tags: algebra , Peru

Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions: (i) $-1 <x_i <1,$ for all $1\leq i \leq n.$ (ii) $ x_1 + x_2 + ... + x_n = 0.$ (iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

2018 Peru MO (ONEM), 2

Tags: Peru , national olympiad , algebra , inequalities , High School Olympiads

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

2017 Peru MO (ONEM), 3

Tags: algebra , rational number , Peru

The infinity sequence $r_{1},r_{2},...$ of rational numbers it satisfies that: $\prod_{i=1}^ {k}r_{i}=\sum_{i=1}^{k} r_{i}$. For all natural k. Show that $\frac{1}{r_{n}}-\frac{3}{4}$ is a square of rationale number for all natural $n\geq3$

2018 Peru MO (ONEM), 4

Tags: Peru , national olympiad

4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.

2020 Peru IMO TST, 1

Tags: Peru , TST , number theory , Perfect Powers , prime numbers

Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2016 Peru Cono Sur TST, P4

Tags: combinatorics , Peru

Let $n$ be a positive integer. Andrés has $n+1$ cards and each of them has a positive integer written, in such a way that the sum of the $n+1$ numbers is $3n$. Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in the red box is equal to twice the sum of the numbers of the cards in the blue box. Clarification: Some of Andrés's letters can be left out of the boxes.

2018 Peru MO (ONEM), 1

Tags: Peru , national olympiad

1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.