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

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Found problems: 1311

2021 Baltic Way, 3
Tags: algebra , algebra proposed , Sequence , Baltic Way
Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying \[a_{n+1}^2=1+(n+2021)a_n\] for all $n \ge 1$.

2006 Moldova National Olympiad, 12.4
Tags: algebra proposed , algebra
Let $P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}$ , $a_{i} \in C$ , $n\geq 2$ with all roots having same modulo. Prove that $P(-1) \in R$

2024 All-Russian Olympiad, 7
Tags: algebra , algebra proposed
There are $8$ different quadratic trinomials written on the board, among them there are no two that add up to a zero polynomial. It turns out that if we choose any two trinomials $g_1(x), g_2(X)$ from the board, then the remaining $6$ trinomials can be denoted as $g_3(x),g_4(x),\dots,g_8(x)$ so that all four polynomials $g_1(x)+g_2(x),g_3(x)+g_4(x),g_5(x)+g_6(x)$ and $g_7(x)+g_8(x)$ have a common root. Do all trinomials on the board necessarily have a common root? [i]Proposed by S. Berlov[/i]

2010 Contests, 1
Tags: algebra , system of equations , algebra proposed
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

2008 China Western Mathematical Olympiad, 2
Tags: trigonometry , inequalities , geometry , 3D geometry , algebra proposed , algebra
Given $ x,y,z\in (0,1)$ satisfying that $ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$. Find the maximum value of $ xyz$.

1987 IMO Longlists, 64
Tags: algebra proposed , algebra
Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$. Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a [i]base-$r$ expansion[/i] of $x$ we mean a representation of $x$ in the form \[x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots\] where the $a_i$ are integers with $0 \leq a_i < r.$ You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one [i]base-$r$ expansion[/i]. Prove that if $r$ is not an integer, then there exists a number $p$, $0 \leq p \leq \frac{n}{r-1}$, which has infinitely many distinct [i]base-$r$ expansions[/i].

1998 Romania Team Selection Test, 3
Tags: trigonometry , algebra proposed , algebra , Polynomials
The lateral surface of a cylinder of revolution is divided by $n-1$ planes parallel to the base and $m$ parallel generators into $mn$ cases $( n\ge 1,m\ge 3)$. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers $k,l$ such that $n+1$ does not divide $k$ and \[ \cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}\] [i]Ciprian Manolescu[/i]

2005 District Olympiad, 4
Tags: function , induction , algebra , functional equation , algebra proposed
Let $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that \[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]

2013 India IMO Training Camp, 1
Tags: function , induction , limit , algebra proposed , algebra , Cauchy equation
Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

1990 IMO Longlists, 29
Tags: function , floor function , algebra proposed , algebra
Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

2012 Iran MO (3rd Round), 1
Tags: algebra , polynomial , induction , function , modular arithmetic , algebra proposed
Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

1998 Baltic Way, 6
Tags: algebra , polynomial , algebra proposed
Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.

2010 Slovenia National Olympiad, 2
Tags: trigonometry , algebra proposed , algebra
Find all real $x$ in the interval $[0, 2\pi)$ such that \[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\]

2003 District Olympiad, 1
Tags: function , geometry , 3D geometry , algebra proposed , algebra
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$. [i]Dinu Teodorescu[/i]

2010 China Western Mathematical Olympiad, 4
Tags: algebra proposed , algebra
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously: (1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$; (2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$; (3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$. Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.

2012 India IMO Training Camp, 3
Tags: function , inequalities , induction , algebra proposed , algebra
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

2009 China Team Selection Test, 6
Tags: algebra proposed , algebra , number theory , combinatorics , Additive combinatorics
Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.

2003 Moldova National Olympiad, 12.2
Tags: calculus , derivative , algebra , polynomial , algebra proposed
For every natural number $n\geq{2}$ consider the following affirmation $P_n$: "Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots." Are $P_4$ and $P_5$ both true? Justify your answer.

2006 Bulgaria National Olympiad, 2
Tags: function , limit , induction , inequalities , algebra proposed , algebra
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ \[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\] a) Prove that $f(2x)=4f(x)$ for all $x>0$; b) Find all such functions. [i]Nikolai Nikolov, Oleg Mushkarov [/i]

2013 Iran MO (3rd Round), 3
Tags: complex numbers , algebra proposed , algebra
For every positive integer $n \geq 2$, Prove that there is no $n-$tuple of distinct complex numbers $(x_1,x_2,\dots,x_n)$ such that for each $1 \leq k \leq n$ following equality holds. $\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) $ (20 points)

2005 Taiwan National Olympiad, 2
Tags: function , floor function , algebra proposed , algebra
Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

2006 ITAMO, 5
Tags: inequalities , algebra proposed , algebra
Consider the inequality \[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\] a) Find all $n\ge 3$ such that the inequality is true for positive reals. b) Find all $n\ge 3$ such that the inequality is true for reals.

2024 Austrian MO National Competition, 1
Tags: inequalities , inequalities proposed , algebra , algebra proposed
Determine the smallest real constant $C$ such that the inequality \[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\] holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$? [i](Walther Janous)[/i]

1998 Romania National Olympiad, 4
Tags: algebra proposed , algebra
Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that :\[ S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. \] (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.

2009 China Team Selection Test, 6
Tags: algebra proposed , algebra , number theory , combinatorics , Additive combinatorics
Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.