Olympiad problems

2014 Greece JBMO TST, 1

Find all the pairs of real numbers $(x,y)$ that are solutions of the system: $(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $ $| x - y | = 1$

1978 All Soviet Union Mathematical Olympiad, 268

Tags: algebra , analysis , sequence

Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.

2001 China Western Mathematical Olympiad, 3

Tags: trigonometry , algebra

Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$.

2013 District Olympiad, 4

Tags: inequalities , algebra , number theory

For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

V Soros Olympiad 1998 - 99 (Russia), 10.3

Tags: algebra , trigonometry

It is known that $\sin 3x = 3 \sin x - 4 \sin^3x$. It is also easy to prove that $\sin nx$ for odd $n$ can be represented as a polynomial of degree $n$ of $\sin x$. Let $\sin 1999x = P(\sin x)$, where $P(t)$ is a polynomial of the $1999$th degree of $t$. Solve the equation $$P \left(\cos \frac{x}{1999}\right) = \frac12 .$$

2017 Azerbaijan EGMO TST, 2

Tags: algebra , induction , sequence

Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$

2021 IMO Shortlist, N8

Tags: polynomial , algebra , abstract algebra , ceiling function

Find all positive integers $n$ for which there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that for every positive integer $m\geq 1$, the numbers $P^m(1), \ldots, P^m(n)$ leave exactly $\lceil n/2^m\rceil$ distinct remainders when divided by $n$. (Here, $P^m$ means $P$ applied $m$ times.) [i]Proposed by Carl Schildkraut, USA[/i]

2004 Czech-Polish-Slovak Match, 1

Tags: algebra , polynomial , quadratic

Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.

2015 Iran MO (3rd round), 1

Tags: number theory , algebra

Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.

2004 AMC 12/AHSME, 16

Tags: function , trigonometry , polynomial , complex number , algebra

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2004 Vietnam National Olympiad, 1

Tags: algebra , system of equations

Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.

2023 Canada National Olympiad, 4

Tags: algebra

Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$. A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$ Prove that for any such $f$, there are finitely many $f$-cool integers. (The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)

2005 APMO, 2

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

2014 Contests, 4

Tags: function , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

2006 Thailand Mathematical Olympiad, 6

Tags: inequalities , algebra

Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$

2001 Estonia National Olympiad, 1

Tags: algebra , trigonometry , system of equations

Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$

2019 Saudi Arabia Pre-TST + Training Tests, 2.2

Tags: polynomial , algebra

Let be given a positive integer $n > 1$. Find all polynomials $P(x)$ non constant, with real coefficients such that $$P(x)P(x^2) ... P(x^n) = P\left( x^{\frac{n(n+1)}{2}}\right)$$ for all $x \in R$

1993 Chile National Olympiad, 3

Tags: fraction , algebra

Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.

1977 Chisinau City MO, 134

Tags: algebra

Where is the number $35 351$ in the sequence $1, 8, 22, 43,...$?

2009 Hong Kong TST, 1

Tags: algebra

Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants. (a) Find a formula for $ f(x)$, where $ x$ is any integer. (b) If $ f(x) \geq mx^2\plus{}(5m\plus{}1)x\plus{}4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.

1952 Moscow Mathematical Olympiad, 222

Tags: algebra , system of equations

a) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}$ b) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}$ How does the solution vary for distinct values of $n$?

2017 Germany, Landesrunde - Grade 11/12, 6

Tags: algebra , system of equations

Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right), \\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right). \end{align*}

1992 Romania Team Selection Test, 4

Tags: algebra , sequence , set , combinatorics

Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

2016 Postal Coaching, 3

Tags: polynomial , algebra

Call a non-constant polynomial [i]real[/i] if all its coecients are real. Let $P$ and $Q$ be polynomials with complex coefficients such that the composition $P \circ Q$ is real. Show that if the leading coefficient of $Q$ and its constant term are both real, then $P$ and $Q$ are real.

2020 Dutch IMO TST, 3

Tags: functional equation , functional equation in z , functional , algebra

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$