Olympiad problems

1973 Spain Mathematical Olympiad, 5

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

1993 Tournament Of Towns, (367) 6

Tags: algebra , geometry

The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a) $0.7$ km? (b) $0.8$ km? (Grigory Kondakov, Moscow) You may assume that the banks consist of segments and arcs of circles.

2019 LIMIT Category B, Problem 9

Tags: algebra

Let $f:\mathbb R\to\mathbb R$ be given by $$f(x)=\left|x^2-1\right|,x\in\mathbb R$$Then $\textbf{(A)}~f\text{ has local minima at }x=\pm1\text{ but no local maxima}$ $\textbf{(B)}~f\text{ has a local maximum at }x=0\text{ but no local minima}$ $\textbf{(C)}~f\text{ has local minima at }x=\pm1\text{ and a local maximum at }x=0$ $\textbf{(D)}~\text{None of the above}$

2002 Taiwan National Olympiad, 5

Tags: polynomial , algebra

Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$ $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$ $...$ $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.

2023 China Western Mathematical Olympiad, 1

Tags: algebra

Are there different integers $a,b,c,d,e,f$ such that they are the $6$ roots of $$(x+a)(x^2+bx+c)(x^3+dx^2+ex+f)=0?$$

2011 QEDMO 10th, 1

Tags: algebra , functional equation , functional

Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.

1974 Polish MO Finals, 3

Tags: algebra , polynomial , quadratic polynomial

Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.

2010 IberoAmerican Olympiad For University Students, 6

Tags: algebra , polynomial , search , number theory

Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

2004 IMO Shortlist, 6

Tags: function , calculus , algebra , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

MOAA Team Rounds, TO4

Tags: algebra , team

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

2023 Iberoamerican, 2

Tags: functional equation in z , algebra

Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that: $$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$ for each integer $x$.

2006 IberoAmerican Olympiad For University Students, 4

Tags: calculus , integration , algebra , polynomial , function , real analysis

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2001 Saint Petersburg Mathematical Olympiad, 10.2

Tags: algebra , combinatorics , operation

The computer "Intel stump-V" can do only one operation with a number: add 1 to it, then rearrange all the zeros in the decimal representation to the end and rearrenge the left digits in any order. (For example from 1004 you could get 1500 or 5100). The number $12345$ was written on the computer and after performing 400 operations, the number 100000 appeared on the screen. How many times has a number with the last digit 0 appeared on the screen?

2023 IFYM, Sozopol, 8

Tags: polynomial , algebra

Do there exist a natural number $n$ and real numbers $a_0, a_1, \dots, a_n$, each equal to $1$ or $-1$, such that the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ is divisible by the polynomial $x^{2023} - 2x^{2022} + c$, where: \\ (a) $c = 1$ \\ (b) $c = -1$? [i] (For polynomials $P(x)$ and $Q(x)$ with real coefficients, we say that $P(x)$ is divisible by $Q(x)$ if there exists a polynomial $R(x)$ with real coefficients such that $P(x) = Q(x)R(x)$.)[/i]

1995 Tournament Of Towns, (452) 1

Tags: algebra , inequalities

Let $a, b, c$ and $d$ be points of the segment $[0,1]$ of the real line (this means numbers $x$ such that $0 \le x \le 1$). Prove that there exists a point $x$ on this segment such that $$\frac{1}{|x-a|}+\frac{1}{|x-b|}+\frac{1}{|x-c|}+\frac{1}{|x-d|}< 40.$$ (LD Kurliandchik)

2005 Abels Math Contest (Norwegian MO), 4a

Tags: algebra , inequalities

Show that for all positive real numbers $a, b$ and $c$, the inequality $(a+b)(a+c)\ge 2\sqrt{abc(a+b+c)}$ is true.

2011 Singapore Senior Math Olympiad, 3

Tags: trigonometry , algebra

Find all positive integers $n$ such that \[\cos\frac{\pi}{n}\cos\frac{2\pi}{n}\cos\frac{3\pi}{n}=\frac{1}{n+1}\]

V Soros Olympiad 1998 - 99 (Russia), 9.1

Tags: algebra

It is known that each of the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ has two different roots and these four roots form an arithmetic progression in some order. Find $a$ and $b$.

1973 Polish MO Finals, 1

Tags: algebra , polynomial

Prove that every polynomial is a difference of two increasing polynomials.

2016 Azerbaijan Team Selection Test, 2

Tags: algebra , polynomial

Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$, the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\]

2005 Kazakhstan National Olympiad, 4

Tags: function , algebra

Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition $f(f(x)+x+y)=2x+f(y)$ for any real $x$ and $y$.

2012 Iran MO (3rd Round), 2

Tags: inequalities , continued fraction , algebra

Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.

2007 Serbia National Math Olympiad, 1

Tags: function , induction , algebra

Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

1973 Bulgaria National Olympiad, Problem 4

Tags: trigonometry , function , fe , functional equation , algebra

Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy $$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$ for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$. [i]L. Davidov[/i]

2011 AIME Problems, 9

Tags: trigonometry , algebra , polynomial , rational root theorem , logarithm

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.