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

2022 Indonesia TST, A
Tags: inequality , am-gm , simple , inequalities , algebra
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

1998 Irish Math Olympiad, 4
Tags: algebra
A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.

2017 Romanian Master of Mathematics, 1
Tags: representation , additive combinatorics , algebra
[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers. This number $k$ is called [i]weight[/i] of $n$. [b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.

1987 Yugoslav Team Selection Test, Problem 2
Tags: algebra , function
Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.

2023 Indonesia TST, 2
Tags: inequalities , algebra
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2017 Thailand TSTST, 3
Tags: functional equation , function , algebra
Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.

2018 Greece JBMO TST, 3
Tags: sum , sum of powers , maximum , combinatorics , algebra
$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

1989 Bulgaria National Olympiad, Problem 3
Tags: polynomial , irreducibility , algebra
Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

VII Soros Olympiad 2000 - 01, 8.9
Tags: algebra
It is known about the numbers $a, b$ and $c$ that $\frac{a}{b+c-a}=\frac{b}{a ​​+ c-b}= \frac{c}{a ​​+ b-c}$. What values ​​can an expression take $\frac{(a + b) (b + c) (a + c)}{abc}$ ?

2013 Romania National Olympiad, 1
Tags: arithmetic sequence , induction , algebra
A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

2011 Bogdan Stan, 1
Tags: inequalities , logarithm , algebra
If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then $$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$ [i]Gheorghe Duță[/i]

2012 Princeton University Math Competition, A2 / B4
Tags: algebra
If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

1975 Chisinau City MO, 103
Tags: algebra , inequalities
Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$

2024 TASIMO, 2
Tags: sequence , algebra
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$ \[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\] Proposed by Navid Safaei, Iran

2008 Hanoi Open Mathematics Competitions, 3
Tags: coefficient , combinatorics , polynomial , algebra
Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

2014 Czech-Polish-Slovak Junior Match, 2
Tags: equation , algebra
Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

1995 IMO, 2
Tags: algebra , inequalities
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that \[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}. \]

2012 IFYM, Sozopol, 3
Tags: trigonometric inequality , algebra
Prove the following inequality: $tan \, 1>\frac{3}{2}$.

2019 Tournament Of Towns, 3
Tags: combinatorics , algebra
An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

2015 Turkey Team Selection Test, 7
Tags: sophie germain identity , functional equation , function , algebra
Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2012 India IMO Training Camp, 3
Tags: function , induction , inequalities , 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 $x, y\in \mathbb{R}^{+}$.

1969 Polish MO Finals, 2
Tags: algebra , inequalities
Given distinct real numbers $a_1,a_2,...,a_n$, find the minimum value of the function $$y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.$$

2007 Puerto Rico Team Selection Test, 2
Tags: diophantine , diophantine equation , number theory , algebra , system of equations
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

1999 Harvard-MIT Mathematics Tournament, 7
Tags: algebra
Let $\frac{1}{1-x-x^2-x^3} =\sum^{\infty}_{i=0} a_nx^n$, for what positive integers $n$ does $a_{n-1} = n^2$?

2016 Korea - Final Round, 4
Tags: inequality , inequalities , algebra
If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$