Olympiad problems

AND:

OR:

NO:

Found problems: 2

First
Previous
1
Next
Last

2019 Tuymaada Olympiad, 6

Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$ and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$

2022 BMT, 9

Tags: fun , algebra

We define a sequence $x_1 = \sqrt{3}, x_2 =-1, x_3 =2 - \sqrt{3},$ and for all $n \geq 4$ $$(x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}).$$ Suppose $m$ is the smallest positive integer for which $x_m$ is undefined. Compute $m.$

First
Previous
1
Next
Last