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: 15925

2016 Estonia Team Selection Test, 3
Tags: algebra , functional equation , functional
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

2016 APMO, 5
Tags: function , functional equation , algebra
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

Kvant 2022, M2726
Tags: algebra
Let $a_1=1$ and $a_{n+1}=2/(2+a_n)$ for all $n\geqslant 1$. Similarly, $b_1=1$ and $b_{n+1}=3/(3+b_n)$ for all $n\geqslant 1$. Which is greater between $a_{2022}$ and $b_{2022}$? [i]Proposed by P. Kozhevnikov[/i]

2020 BMT Fall, 9
Tags: algebra , diophantine equation
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

2006 Irish Math Olympiad, 3
Tags: algebra , polynomial , inequalities
let x,y are positive and $ \in R$ that : $ x\plus{}2y\equal{}1$.prove that : \[ \frac{1}{x}\plus{}\frac{2}{y} \geq \frac{25}{1\plus{}48xy^2}\]

1988 Poland - Second Round, 2
Tags: algebra , inequalities
Given real numbers $ x_i $, $ y_i $ ($ i = 1, 2, \ldots, n $) such that $$ \qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0,$$ and $$ \prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n.$$ Prove that $$ \sum_{i=1}^n x_i > \sum_{i=1}^n y_i.$$

1987 Romania Team Selection Test, 4
Tags: algebra , polynomial , vector , inequalities
Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and: a) $ a_{0} > 0$, $ a_{n} > 0$; b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$. Prove that $ P(x)\geq 0$ for all real values $ x$. [i]Laurentiu Panaitopol[/i]

2012 Poland - Second Round, 1
Tags: algebra
$a,b,c,d\in\mathbb{R}$, solve the system of equations: \[ \begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases} \]

2024 Caucasus Mathematical Olympiad, 5
Tags: algebra
Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.

1940 Putnam, A6
Tags: algebra , polynomial
Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$. Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.

2011 Romania National Olympiad, 4
Tags: algebra
[b]a)[/b] Show that there exists exactly a sequence $ \left( x_n,y_n \right)_{n\ge 0} $ of pairs of nonnegative integers, that satisfy the property that $ \left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33, $ for all nonegative integers $ n. $ [b]b)[/b] Having in mind the sequence from [b]a),[/b] prove that, for any natural prime $ p, $ at least one of the numbers $ y_{p-1} ,y_p $ and $ y_{p+1} $ are divisible by $ p. $

2003 Czech-Polish-Slovak Match, 6
Tags: function , algebra
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition \[f(f(x) + y) = 2x + f(f(y) - x)\quad \text{ for all } x, y \in\mathbb{R}.\]

Mathley 2014-15, 5
Tags: sequence , recurrence relation , algebra
Given the sequence $(u_n)_{n=1}^{\infty}$, where $u_1 = 1, u_2 = 2$, and $u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2}$ for any positive integers $n$. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers $(u_n)_{n=1}^{\infty}$ Nguyen Duy Thai Son, The University of Danang, Da Nang.

2024 China Girls Math Olympiad, 1
Tags: algebra
Let $\{a_n\}$ be a sequence defined by $a_1=0$ and $$a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k$$ for any positive integer $n$. Find the maximal term of this sequence.

2019 Saudi Arabia Pre-TST + Training Tests, 1.1
Tags: algebra , system of equations
Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$ Find all possible values of $T = x + y + z$

1985 Traian Lălescu, 2.3
Tags: complex number , algebra , inequalities
Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that: $$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$

1991 Swedish Mathematical Competition, 3
Tags: sequence , algebra
The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$, $x_{k+1} = [(n - \sum_0^k x_i)/2]$. Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$.

1973 Chisinau City MO, 67
Tags: algebra , max , sum , product , inequalities
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

1972 Bulgaria National Olympiad, Problem 2
Tags: algebra , system of equations , equation
Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$, $0<y<t$, $0<z<t$. [i]H. Lesov[/i]

PEN E Problems, 11
Tags: euler , algebra , polynomial , modular arithmetic
In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.

1966 Czech and Slovak Olympiad III A, 1
Tags: inequalities , absolute value , algebra
Consider a system of inequalities \begin{align*}y-x&\ge|x+1|-|x-1|, \\ |y&-x|-y+x\ge2.\end{align*} Draw solutions of each inequality in the plane separately and highlight solution of the system.

2000 Harvard-MIT Mathematics Tournament, 11
Tags: algebra
Find all polynomials $f(x)$ with integer coefficients such that the coefficients of both $f(x)$ and $[f(x)]^3$ lie in the set $\{0,1, -1\}$.

PEN H Problems, 16
Tags: diophantine equation , algebra , polynomial , function
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.

2011 District Olympiad, 4
Tags: algebra , integer part
Find all positive integers $m$ such that $$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$

2006 China Western Mathematical Olympiad, 4
Tags: algebra , sum of divisors
Assuming that the positive integer $a$ is not a perfect square, prove that for any positive integer n, the sum ${S_{n}=\sum_{i=1}^{n}\{a^{\frac{1}{2}}\}^{i}}$ is irrational.