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

2014 Contests, 1
Tags: inequalities , induction , algebra proposed , algebra
Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

1985 Iran MO (2nd round), 3
Tags: geometry , 3D geometry , vector , trigonometry , algebra proposed , algebra
Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$

2009 Indonesia TST, 3
Tags: algebra proposed , algebra
Find all triples $ (x,y,z)$ of positive real numbers which satisfy $ 2x^3 \equal{} 2y(x^2 \plus{} 1) \minus{} (z^2 \plus{} 1)$; $ 2y^4 \equal{} 3z(y^2 \plus{} 1) \minus{} 2(x^2 \plus{} 1)$; $ 2z^5 \equal{} 4x(z^2 \plus{} 1) \minus{} 3(y^2 \plus{} 1)$.

2010 ELMO Shortlist, 5
Tags: algebra proposed , algebra
Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers. [i]Brian Hamrick.[/i]

2013 District Olympiad, 1
Tags: algebra proposed , algebra
Let $a,b\in \mathbb{R}$ and $z\in \mathbb{C}\backslash \mathbb{R}$ so that $\left| a-b \right|=\left| a+b-2z \right|$. a) Prove that the equation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$, has a unique solution. b) Solve the following inequation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$. The Mathematical Gazette

2012 Middle European Mathematical Olympiad, 1
Tags: algebra , system of equations , algebra proposed
Find all triplets $ (x,y,z) $ of real numbers such that \[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]

2005 Greece National Olympiad, 3
Tags: function , algebra proposed , algebra
We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with $x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$; ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.

1985 Vietnam Team Selection Test, 1
Tags: algebra proposed , algebra
The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.

2013 Bosnia Herzegovina Team Selection Test, 2
Tags: algebra , polynomial , induction , algebra proposed
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

2008 China Team Selection Test, 5
Tags: function , inequalities , Cauchy Inequality , algebra proposed , algebra
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

1974 IMO Longlists, 9
Tags: parameterization , algebra proposed , algebra
Solve the following system of linear equations with unknown $x_1,x_2 \ldots, x_n \ (n \geq 2)$ and parameters $c_1,c_2, \ldots , c_n:$ \[2x_1 -x_2 = c_1;\]\[-x_1 +2x_2 -x_3 = c_2;\]\[-x_2 +2x_3 -x_4 = c_3;\]\[\cdots \qquad \cdots \qquad \cdots \qquad\]\[-x_{n-2} +2x_{n-1} -x_n = c_{n-1};\]\[-x_{n-1} +2x_n = c_n.\]

2021 Romania EGMO TST, P1
Tags: algebra proposed , algebra
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

2008 Moldova National Olympiad, 12.5
Tags: algebra , polynomial , algebra proposed
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

2007 Italy TST, 3
Tags: function , algebra , polynomial , algebra proposed , functional equation
Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

1986 IMO Longlists, 22
Tags: induction , algebra proposed , algebra
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$

1999 China National Olympiad, 2
Tags: algebra , polynomial , function , induction , algebra proposed
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$. a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$. b) Find an explicit expression for $f_n(x)$.

2005 Bundeswettbewerb Mathematik, 2
Tags: algebra proposed , algebra
Let be $x$ a rational number. Prove: There are only finitely many triples $(a,b,c)$ of integers with $a<0$ and $b^2-4ac=5$ such that $ax^2+bx+c$ is positive.

2010 APMO, 5
Tags: function , algebra proposed , algebra
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\]

2012 Serbia Team Selection Test, 1
Tags: algebra , polynomial , function , inequalities , absolute value , algebra proposed
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

2008 Moldova MO 11-12, 5
Tags: algebra , polynomial , algebra proposed
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

2010 Regional Competition For Advanced Students, 2
Tags: algebra proposed , algebra
Solve the following in equation in $\mathbb{R}^3$: \[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]

2010 Romania Team Selection Test, 2
Tags: function , parameterization , induction , inequalities , rearrangement inequality , algebra proposed , algebra
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

1991 Romania Team Selection Test, 8
Tags: function , number theory , greatest common divisor , Diophantine equation , algebra proposed , algebra
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

2013 All-Russian Olympiad, 1
Tags: algebra proposed , algebra
Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations \[(x-a)(x-b)=x-c\] \[(x-c)(x-b)=x-a\] \[(x-c)(x-a)=x-b\] have real solutions.

1985 IMO Longlists, 3
Tags: function , algebra proposed , algebra
A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?