Olympiad problems

2019 Swedish Mathematical Competition, 5

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

1999 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra

Let be a subset of the interval $ (0,1) $ that contains $ 1/2 $ and has the property that if a number is in this subset, then, both its half and its successor's inverse are in the same subset. Prove that this subset contains all the rational numbers of the interval $ (0,1). $

2005 Denmark MO - Mohr Contest, 2

Tags: algebra , parameterization , system of equations , system

Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations $$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$ where $x$ and $y$ are real numbers.

2004 Germany Team Selection Test, 1

Tags: algebra , system of equations , polynomial , linear algebra , complex number , matrix

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

1987 Romania Team Selection Test, 7

Tags: algorithm , algebra

Determine all positive integers $n$ such that $n$ divides $3^n - 2^n$.

LMT Speed Rounds, 20

Tags: algebra

The remainder when $x^{100} -x^{99} +... -x +1$ is divided by $x^2 -1$ can be written in the form $ax +b$. Find $2a +b$. [i]Proposed by Calvin Garces[/i]

1990 Swedish Mathematical Competition, 5

Tags: functional , algebra , functional equation

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2012 Brazil Team Selection Test, 4

Tags: trigonometry , triangle inequality , algebra , triangle

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

1969 All Soviet Union Mathematical Olympiad, 125

Tags: algebra , cubic

Given an equation $$x^3 + ?x^2 + ?x + ? = 0$$ First player substitutes an integer on the place of one of the interrogative marks, than the same do the second with one of the two remained marks, and, finally, the first puts the integer instead of the last mark. Explain how can the first provide the existence of three integer roots in the obtained equation. (The roots may coincide.)

2021 BMT, 14

Tags: algebra , number theory

Given an integer $c$, the sequence $a_0, a_1, a_2, ...$ is generated using the recurrence relation $a_0 = c$ and $a_i = a^i_{i-1} + 2021a_{i-1}$ for all $i \ge 1$. Given that $a_0 = c$, let $f(c)$ be the smallest positive integer $n$ such that $a_n - 1$ is a multiple of $47$. Compute $$\sum^{46}_{k=1} f(k).$$

2018-IMOC, N6

Tags: algebra , polynomial , number theory

If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that $$a_1f(1)+\ldots+a_tf(t)=k.$$

2013 IFYM, Sozopol, 4

Tags: number theory , algebra , sum

Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find $\max S=\sum_{i\mid j}a_i a_j $.

1996 All-Russian Olympiad, 4

Tags: algebra , polynomial

Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

2012 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Prove that for any integer $n\geq 2$ it holds that $\dbinom {2n}{n}>\frac {4^n}{2n}$.

1992 Nordic, 1

Tags: equation , algebra

Determine all real numbers $x > 1, y > 1$, and $z > 1$,satisfying the equation $x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2})$

2019 Romania Team Selection Test, 1

Tags: algebra , inequalities

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

2025 Euler Olympiad, Round 2, 6

Tags: number theory , combinatorics , graph theory , iteration , algebra

For any subset $S \subseteq \mathbb{Z}^+$, a function $f : S \to S$ is called [i]interesting[/i] if the following two conditions hold: [b]1.[/b] There is no element $a \in S$ such that $f(a) = a$. [b]2.[/b] For every $a \in S$, we have $f^{f(a) + 1}(a) = a$ (where $f^{k}$ denotes the $k$-th iteration of $f$). Prove that: [b]a) [/b]There exist infinitely many interesting functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$. [b]b) [/b]There exist infinitely many positive integers $n$ for which there is no interesting function $$ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}. $$ [i]Proposed by Giorgi Kekenadze, Georgia[/i]

2008 Germany Team Selection Test, 1

Tags: algebra , functional inequality

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

2009 Paraguay Mathematical Olympiad, 1

Tags: algebra

Find the value of the following expression: $2 + 33 + 6 + 35 + 10 + 37 + \ldots + 1194 + 629 + 1198 + 631$

2017 Brazil Team Selection Test, 4

Tags: algebra

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

2018 ELMO Problems, 5

Tags: algebra

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

1982 All Soviet Union Mathematical Olympiad, 346

Tags: floor function , inequalities , algebra

Prove that the following inequality holds for all real $a$ and natural $n$: $$|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$$ $F(a)$ is the distance from $a$ to the closest integer.

2018 PUMaC Algebra A, 6

Tags: algebra

Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.

2008 AIME Problems, 9

Tags: number theory , polynomial , algebra , relatively prime , function , geometric series , probability

Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.

2011 Laurențiu Duican, 1

Tags: integer part , fractional part , floor , equation , algebra

Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively. [i]Aurel Bârsan[/i]