Olympiad problems

2018 Denmark MO - Mohr Contest, 1

A blackboard contains $2018$ instances of the digit $1$ separated by spaces. Georg and his mother play a game where they take turns filling in one of the spaces between the digits with either a $+$ or a $\times$. Georg begins, and the game ends when all spaces have been filled. Georg wins if the value of the expression is even, and his mother wins if it is odd. Which player may prepare a strategy which secures him/her victory?

2010 Cuba MO, 7

Tags: algebra , inequalities

Let $x, y, z$ be positive real numbers such that $xyz = 1$. Prove that: $$\frac{x^3 + y^3}{x^2 + xy + y^2} +\frac{ y^3 + z^3}{y^2 + yz + z^2} + \frac{z^3 + x^3}{z^2 + zx + x^2} \ge 2.$$

2024 Romania National Olympiad, 1

Tags: algebra

Solve over the real numbers the equation $$3^{\log_5(5x-10)}-2=5^{-1+\log_3x}.$$

2018 Estonia Team Selection Test, 10

Tags: recurrence relation , inequalities , sequence , algebra

A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations $b_1 = a_1$ , $b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ , $b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$. Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$

1978 IMO Shortlist, 11

Tags: concave , mean , algebra , function

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

2001 Czech And Slovak Olympiad IIIA, 3

Tags: algebra , inequalities , radical

Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.

2016 Saint Petersburg Mathematical Olympiad, 7

Tags: integer polynomial , integer , polynomial , algebra

A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.

2020 Candian MO, 5#

Tags: polynomial , induction , first college math post , superior algebra , algebra

If A,B are invertible and the set {Ak - Bk | k is a natural number} is finite , then there exists a natural number m such that Am = Bm.

2024 Korea Junior Math Olympiad (First Round), 14.

Tags: algebra , number theory , number

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

2015 Regional Olympiad of Mexico Southeast, 3

Tags: geometry , perimeter , inequalities , algebra

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

2012 Online Math Open Problems, 38

Tags: polynomial , algebra

Let $S$ denote the sum of the 2011th powers of the roots of the polynomial $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$. How many ones are in the binary expansion of $S$? [i]Author: Alex Zhu[/i]

2014 IMAC Arhimede, 6

Tags: inequalities , cyclic inequality , four variables , algebra

If $a, b, c, d$ are positive numbers, prove that $$\sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0$$

1997 Moldova Team Selection Test, 11

Tags: polynomial , algebra

Let $P(X)$ be a polynomial with real coefficients such that $\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}$, where $\{a\}$ is the fractional part of the number $a$. Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$.

2013 Tournament of Towns, 4

Tags: coloring , sum , product , combinatorics , algebra

On a circle, there are $1000$ nonzero real numbers painted black and white in turn. Each black number is equal to the sum of two white numbers adjacent to it, and each white number is equal to the product of two black numbers adjacent to it. What are the possible values of the total sum of $1000$ numbers?

1987 IMO Shortlist, 1

Tags: function , functional equation , algebra

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

2011 All-Russian Olympiad, 1

Tags: quadratic , algebra

A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.

1960 IMO, 1

Tags: decimal representation , divisibility , number theory , algebra

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

1993 Romania Team Selection Test, 1

Tags: function , arithmetic sequence , increasing , algebra

Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$. Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.

1991 IberoAmerican, 6

Tags: geometry , parallelogram , quadratic , circumcircle , algebra

Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.

1989 Romania Team Selection Test, 2

Tags: polynomial , integer polynomial , algebra

Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.

2014 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

Let $x, y, z > 0$ be real numbers such that $xyz + xy + yz + zx = 4$. Prove that $x + y + z \ge 3$.

2010 ELMO Shortlist, 5

Tags: algebra

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers. [i]Brian Hamrick.[/i]

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: functional equation , algebra , function

Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy: $$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$ for all real numbers $x,y$.

1993 Romania Team Selection Test, 1

Tags: algebra , sequence

Define the sequence ($x_n$) as follows: the first term is $1$, the next two are $2,4$, the next three are $5,7,9$, the next four are $10,12,14,16$, and so on. Express $x_n$ as a function of $n$.

LMT Team Rounds 2021+, A26 B27

Tags: algebra

Chandler the Octopus along with his friends Maisy the Bear and Jeff the Frog are solving LMT problems. It takes Maisy $3$ minutes to solve a problem, Chandler $4$ minutes to solve a problem and Jeff $5$ minutes to solve a problem. They start at $12:00$ pm, and Chandler has a dentist appointment from $12:10$ pm to $12:30$, after which he comes back and continues solving LMT problems. The time it will take for them to finish solving $50$ LMT problems, in hours, is $m/n$ ,where $m$ and $n$ are relatively prime positive integers. Find $m +n$. [b]Note:[/b] they may collaborate on problems. [i]Proposed by Aditya Rao[/i]