Olympiad problems

2014 CIIM, Problem 2

Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$

2017 CIIM, Problem 2

Tags: CIIM 2017 , undergraduate , college math , real analysis

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.

2011 CIIM, Problem 2

Tags: CIIM 2011 , undergraduate

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$. Prove that $(a + 1)^6-1$ is also a multiple of $7^k$.

2011 CIIM, Problem 1

Tags: CIIM 2011 , undergraduate

Find all real numbers $a$ for which there exist different real numbers $b, c, d$ different from $a$ such that the four tangents drawn to the curve $y = \sin (x)$ at the points $(a, \sin (a)), (b, \sin (b)), (c, \sin (c))$ and $(d, \sin (d))$ form a rectangle.