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

Tags were heavily modified to better represent problems.

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Found problems: 15925

2010 IFYM, Sozopol, 6
Tags: algebra , fractional part
Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

1976 Spain Mathematical Olympiad, 5
Tags: algebra , complex number
Show that the equation $$z^4 + 4(i + 1)z + 1 = 0$$ has a root in each quadrant of the complex plane.

2002 Romania Team Selection Test, 2
Tags: number theory , polynomial , algebra , diophantine equation , induction
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$. Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.

1999 Vietnam Team Selection Test, 1
Tags: algebra , modular arithmetic
Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$, where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.

2008 Princeton University Math Competition, A4/B7
Tags: algebra
What's the greatest integer $n$ for which the system $k < x^k < k + 1$ for $k = 1,2,..., n$ has a solution?

1996 China Team Selection Test, 2
Tags: algebra , function , induction , limit , strong induction
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

2016 All-Russian Olympiad, 5
Tags: number theory , algebra , integer polynomial , polynomial
Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

2025 239 Open Mathematical Olympiad, 4
Tags: algebra , inequality
Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]

2010 Today's Calculation Of Integral, 558
Tags: function , calculus , inequalities , algebra , integration , quadratic , calculus computations
For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$. Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$

2001 India IMO Training Camp, 1
Tags: algebra , number theory , polynomial
For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

2023 South East Mathematical Olympiad, 7
Tags: algebra , inequalities
The positive integer number $S$ is called a "[i]line number[/i]". if there is a positive integer $n$ and $2n$ positive integers $a_1$, $a_2$,...,$a_n$, $b_1$,$b_2$,...,$b_n$, such that $S = \sum^n_{i=1} a_ib_i$, $\sum^n_{i=1} (a_i^2-b_1^2)=1$, and $\sum^n_{i=1} (a_i+b_i)=2023$, find: (1) The minimum value of [i]line numbers[/i]. (2)The maximum value of [i]line numbers[/i].

2024 Polish MO Finals, 4
Tags: algebra , system of equations , parameterization
Do there exist real numbers $a,b,c$ such that the system of equations \begin{align*} x+y+z&=a\\ x^2+y^2+z^2&=b\\ x^4+y^4+z^4&=c \end{align*} has infinitely many real solutions $(x,y,z)$?

1993 Tournament Of Towns, (382) 4
Tags: combinatorics , algebra
Three players Alexander, Beverley and Catherine participate in a tournament (all of them play the same number of games with each other). Is it possible that Alexander gets more points than the others, Catherine gets less points than the others, but Alexander has a smaller number of wins than the others and Catherine has a greater number of wins than the others? (A win scores $1$ point, a draw scores $\frac12$.) (A Rubin,)

2013 JBMO Shortlist, 1
Tags: algebra
$\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$

2014 Portugal MO, 1
Tags: algebra
The ship [i]Meridiano do Bacalhau[/i] does its fishing business during $64$ days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails $1$ mile, on the second day sails $2$ miles; generally, on the $n$-th day it sails $n$ miles. After of the $64$-th day, the ship was $2014$ miles north from its initial position. What is the greatest number of days that the ship could have sailed south?

1996 Romania National Olympiad, 1
Tags: algebra
Find all pairs of real numbers $(x, y) $ such that: a) $x\ge y\ge1$ b) $2x^2-xy-5x +y + 4 = 0 $

2016 Iran Team Selection Test, 2
Tags: algebra , inequalities
Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$

2015 India PRMO, 11
Tags: number theory , algebra , integer , diophantine equation
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$

2005 Switzerland - Final Round, 6
Tags: algebra , sum
Let $a, b, c$ be positive real numbers with $abc = 1$. Find all possible values ​​of the expression $$\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}$$ can take.

2001 All-Russian Olympiad Regional Round, 11.5
Tags: algebra , sequence , periodic , trigonometry , recurrence relation
Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.

1977 IMO Longlists, 46
Tags: function , algebra , inequalities
Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\] Assume that the inequalities\[2^{-1} < g(x, t) < 2\] hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise. Show that\[ 14^{-1} < g(x, t) < 14\] for all real $x$ and positive $t.$

2007 Harvard-MIT Mathematics Tournament, 9
Tags: polynomial , algebra , vieta , quadratic , complex number , system of equations
The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]

2002 Moldova National Olympiad, 2
Tags: trigonometry , inequalities , algebra
For every nonnegative integer $ n$ and every real number $ x$ prove the inequality: $ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

2007 Indonesia TST, 3
Tags: algebra , inequalities , polynomial
Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$. Prove that $P(a) + P(b) + P(c) \le -1$.

2020 Canadian Mathematical Olympiad Qualification, 7
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers with $ab + bc + ac = abc$. Prove that $$\frac{bc}{a^{a+1}} +\frac{ac}{b^{b+1 }}+\frac{ab}{c^{c+1}} \ge \frac13$$