This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1269

1988 IMO Longlists, 55
Tags: inequalities , trigonometry , induction , rearrangement inequality , algebra unsolved , algebra
Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

1978 IMO Longlists, 25
Tags: algebra , polynomial , Vieta , algebra unsolved
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.

1997 China Team Selection Test, 3
Tags: calculus , integration , algebra unsolved , algebra
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

2012 Spain Mathematical Olympiad, 2
Tags: algebra unsolved , algebra
A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence \[a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.\] Prove that all terms of the sequence are integers and find an explicit formula for $a_n$.

2009 All-Russian Olympiad, 2
Tags: function , algebra unsolved , algebra
Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.

2008 Baltic Way, 4
Tags: algebra , polynomial , algebra unsolved
The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

1988 China Team Selection Test, 4
Tags: algebra , polynomial , induction , algebra unsolved
There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.

1989 IMO Longlists, 10
Tags: algebra unsolved , algebra
Given the equation \[ 4x^3 \plus{} 4x^2y \minus{} 15xy^2 \minus{} 18y^3 \minus{} 12x^2 \plus{} 6xy \plus{} 36y^2 \plus{} 5x \minus{} 10y \equal{} 0,\] find all positive integer solutions.

1991 Irish Math Olympiad, 5
Tags: algebra , polynomial , algebra unsolved
Find all polynomials $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties (a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and (b) all the roots of the equation $f(x)=0$ are real.

2006 Regional Competition For Advanced Students, 4
Tags: algebra unsolved , algebra
Let $ <h_n>$ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $ h_n$ is the harmonic mean of its two neighbours $ h_{n\minus{}1}$ and $ h_{n\plus{}1}$ : $ h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}$) Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers.

1988 IMO Longlists, 90
Tags: LaTeX , algebra unsolved , algebra
Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]

1991 Cono Sur Olympiad, 3
Tags: inequalities , number theory , prime numbers , algebra unsolved , algebra
Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$. [b]a[/b] Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$. [b]b[/b] Find all $n$ such that $f(n)=\frac{91}{9}$.

2015 Vietnam National Olympiad, 1
Tags: algebra , polynomial , induction , algebra unsolved
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.

2018 Latvia Baltic Way TST, P4
Tags: function , inequalities , algebra , algebra unsolved
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies $$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$ for all real $x$. Prove for all real $x$: [i](a)[/i] $f(x)\ge 4$; [i](b)[/i] $f(x)\ge 7.$

1976 IMO Longlists, 9
Tags: algebra unsolved , algebra
Find all (real) solutions of the system \[3x_1-x_2-x_3-x_5 = 0,\]\[-x_1+3x_2-x_4-x_6= 0,\]\[-x_1 + 3x_3 - x_4 - x_7 = 0,\]\[-x_2 - x_3 + 3x_4 - x_8 = 0,\]\[-x_1 + 3x_5 - x_6 - x_7 = 0,\]\[-x_2 - x_5 + 3x_6 - x_8 = 0,\]\[-x_3 - x_5 + 3x_7 - x_8 = 0,\]\[-x_4 - x_6 - x_7 + 3x_8 = 0.\]

2010 Germany Team Selection Test, 1
Tags: algebra , polynomial , algebra unsolved
Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

1978 IMO Longlists, 28
Tags: function , logarithms , algebra unsolved , algebra
Let $c, s$ be real functions defined on $\mathbb{R}\setminus\{0\}$ that are nonconstant on any interval and satisfy \[c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0\] Prove that: $(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x)$ for any $x = 0$, and also $c(1) = 1, s(1) = s(-1) = 0$; $(b) c$ and $s$ are either both even or both odd functions (a function $f$ is even if $f(x) = f(-x)$ for all $x$, and odd if $f(x) = -f(-x)$ for all $x$). Find functions $c, s$ that also satisfy $c(x) + s(x) = x^n$ for all $x$, where $n$ is a given positive integer.

2014 Thailand TSTST, 1
Tags: function , algebra unsolved , algebra
Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

1963 Vietnam National Olympiad, 2
Tags: quadratics , algebra , algebra unsolved
For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.

2013 Pan African, 2
Tags: function , algebra unsolved , algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$.

2001 Finnish National High School Mathematics Competition, 2
Tags: conics , parabola , geometry , perpendicular bisector , algebra unsolved , algebra
Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$ Prove that there is a line of the plane which does not meet either of the curves.

1996 Romania Team Selection Test, 7
Tags: function , algebra , polynomial , algebra unsolved
Let $ a\in \mathbb{R} $ and $ f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R} $ are the additive functions such that for every $ x\in \mathbb{R} $ we have $ f_1(x)f_2(x) \cdots f_n(x) =ax^n $. Show that there exists $ b\in \mathbb {R} $ and $ i\in {\{1,2,\ldots,n}\} $ such that for every $ x\in \mathbb{R} $ we have $ f_i(x)=bx $.

2002 All-Russian Olympiad, 3
Tags: algebra unsolved , algebra
Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0<m-n<3\sqrt[4]n$.

2009 German National Olympiad, 1
Tags: geometry , 3D geometry , function , calculus , algebra unsolved , algebra
Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$. For all such $a$, what is $x$?

2003 Bulgaria National Olympiad, 3
Tags: algebra , polynomial , limit , induction , algebra unsolved
Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.