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

2004 Bulgaria Team Selection Test, 1
Tags: algebra proposed , algebra
Find the maximum possible value of the product of distinct positive integers whose sum is $2004$.

1991 Balkan MO, 4
Tags: function , algebra proposed , algebra
Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.

2007 Ukraine Team Selection Test, 3
Tags: algebra , polynomial , logarithms , algebra proposed
It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

2009 Indonesia TST, 3
Tags: function , algebra proposed , algebra
Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

2011 Regional Competition For Advanced Students, 2
Tags: logarithms , algebra , system of equations , algebra proposed
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\ \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}

2006 Indonesia MO, 7
Tags: algebra proposed , algebra
Let $ a,b,c$ be real numbers such that $ ab,bc,ca$ are rational numbers. Prove that there are integers $ x,y,z$, not all of them are $ 0$, such that $ ax\plus{}by\plus{}cz\equal{}0$.

2006 District Olympiad, 4
Tags: quadratics , abstract algebra , vector , group theory , algebra proposed , algebra
a) Find two sets $X,Y$ such that $X\cap Y =\emptyset$, $X\cup Y = \mathbb Q^{\star}_{+}$ and $Y = \{a\cdot b \mid a,b \in X \}$. b) Find two sets $U,V$ such that $U\cap V =\emptyset$, $U\cup V = \mathbb R$ and $V = \{x+y \mid x,y \in U \}$.

2005 Vietnam National Olympiad, 1
Tags: function , algebra proposed , algebra
Find all function $ f: \mathbb R\to \mathbb R$ satisfying the condition: \[ f(f(x \minus{} y)) \equal{} f(x)\cdot f(y) \minus{} f(x) \plus{} f(y) \minus{} xy \]

2008 Iran MO (3rd Round), 1
Tags: algebra , polynomial , algebra proposed
Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3\plus{}1)\neq0$.

2012 India IMO Training Camp, 3
Tags: function , algebra proposed , algebra
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.

2005 Iran MO (3rd Round), 4
Tags: function , ratio , algebra proposed , algebra
Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.

2010 Iran MO (3rd Round), 5
Tags: algebra proposed , algebra
[b]interesting sequence[/b] $n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties: it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.) There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have \[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\] Prove that for every natural number $s$ that $s<2^n-1$ we have \[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\] Time allowed for this question was 1 hours and 15 minutes.

2009 Iran MO (2nd Round), 1
Tags: algebra , polynomial , quadratics , algebra proposed
Let $ p(x) $ be a quadratic polynomial for which : \[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \] Prove that: \[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]

2014 International Zhautykov Olympiad, 1
Tags: algebra , polynomial , calculus , integration , algebra proposed
Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

2014 Dutch BxMO/EGMO TST, 2
Tags: function , algebra proposed , algebra
Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

2011 Canadian Mathematical Olympiad Qualification Repechage, 3
Tags: algebra , system of equations , algebra proposed
Determine all solutions to the system of equations: \[x^2 + y^2 + x + y = 12\]\[xy + x + y = 3\] [This is the exact form of problem that appeared on the paper, but I think it means to solve in $\mathbb R.$]

2001 Baltic Way, 15
Tags: inequalities , algebra proposed , algebra
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $ Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.

2005 Romania National Olympiad, 1
Tags: trigonometry , calculus , derivative , algebra proposed , algebra
Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

2009 Indonesia TST, 2
Tags: parameterization , logarithms , inequalities , algebra proposed , algebra
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

1999 Moldova Team Selection Test, 8
Tags: algebra proposed , algebra
Find a function $f: \mathbb N \to \mathbb N$ such that for all positive integers $n$, \[ f(f(n))\equal{}n^2.\]

2012 Kyoto University Entry Examination, 4
Tags: polynomial , algebra , algebra proposed
(1) Prove that $\sqrt[3]{2}$ is irrational. (2) Let $P(x)$ be a polynomoal with rational coefficients such that $P(\sqrt[3]{2})=0$. Prove that $P(x)$ is divisible by $x^3-2$. 35 points

2012 Spain Mathematical Olympiad, 2
Tags: function , algebra proposed , algebra
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\] for all $x,y\in\mathbb{R}$.

2005 Brazil National Olympiad, 2
Tags: inequalities , algebra proposed , algebra
Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.

2008 Iran MO (3rd Round), 3
Tags: algebra , polynomial , algebra proposed
Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5\plus{}b_3x^3\plus{}b_2x^2\plus{}b_1x\plus{}b_0$ is irreducible in $ \mathbb Z[x]$.

2012 Kyoto University Entry Examination, 5
Tags: algebra , function , domain , trigonometry , analytic geometry , algebra proposed
Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane. 30 points