Olympiad problems

1970 Miklós Schweitzer, 3

The traffic rules in a regular triangle allow one to move only along segments parallel to one of the altitudes of the triangle. We define the distance between two points of the triangle to be the length of the shortest such path between them. Put $ \binom{n\plus{}1}{2}$ points into the triangle in such a way that the minimum distance between pairs of points is maximal. [i]L. Fejes-Toth[/i]

2023 Myanmar IMO Training, 1

Tags: algebra , functional equation , divisibility

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$m+f(n) \mid f(m)^2 - nf(n)$$ for all positive integers $m$ and $n$. (Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)

1990 IMO Longlists, 59

Tags: inequalities

Given eight real numbers $a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8$. Let $x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}$, $y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}$. Prove that \[2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.\]

1986 IMO Shortlist, 15

Tags: geometry , circumcircle , ratio , trigonometry

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ [b](a)[/b] Prove that $ABCD$ and $A''B''C''D''$ are similar. [b](b) [/b]The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

2003 China Western Mathematical Olympiad, 1

Tags: calculus , integration , induction , function , quadratic , algebra

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2023 Chile Classification NMO Seniors, 4

Tags: sfft , algebra

When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product. Find the six-digit number.

Brazil L2 Finals (OBM) - geometry, 2006.5

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$, $N$ and $R$ be the midpoints of $AB$, $BC$ an $AH$, respectively. If $A\hat{B}C=70^\large\circ$, compute $M\hat{N}R$.

2020 USMCA, 17

Tags:

Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b\in \{0,\ldots,2016\}$ and $(a,b)\neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$.

2021 Peru Cono Sur TST., P7

Tags: algebra

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

2002 Canada National Olympiad, 2

Tags: number theory

Call a positive integer $n$ [b]practical[/b] if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$. For example, the divisors of 6 are 1, 2, 3, and 6. Since \[ \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} \] we see that 6 is practical. Prove that the product of two practical numbers is also practical.