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

2019 PUMaC Algebra B, 7
Tags: algebra
Let $\omega=e^{\frac{2\pi i}{2017}}$ and $\zeta = e^{\frac{2\pi i}{2019}}$. Let $S=\{(a,b)\in\mathbb{Z}\,|\,0\leq a \leq 2016, 0 \leq b \leq 2018, (a,b)\neq (0,0)\}$. Compute $$\prod_{(a,b)\in S}(\omega^a-\zeta^b).$$

2020 Australian Mathematical Olympiad, DAY 2
Tags: algebra
Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$, [list] [*] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and [*] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$ Prove that if $a_1~=~0$ , then $a_{2020}~=~1$

1993 National High School Mathematics League, 7
Tags: complex number , quadratic , imaginary numbers , algebra
Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.

2005 Iran MO (3rd Round), 4
Tags: ratio , algebra , function
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.

2007 Belarusian National Olympiad, 1
Tags: polynomial , algebra
Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$

2018 China Western Mathematical Olympiad, 2
Tags: inequalities , fractional part , n-variable inequality , algebra
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

2020 Tuymaada Olympiad, 5
Tags: quadratic , parabola , construction , conic , algebra , analytic geometry , geometry
Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

2010 Princeton University Math Competition, 1
Tags: algebra , polynomial
Find the sum of the coefficients of the polynomial $(63x-61)^4$.

MathLinks Contest 2nd, 3.1
Tags: functional , inequalities , algebra
Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

2010 CentroAmerican, 5
Tags: algebra , triangle inequality , polynomial , inequalities
If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$ is also a rational number.

2014 Saudi Arabia BMO TST, 3
Tags: algebra
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.

2022 Princeton University Math Competition, A5 / B7
Tags: algebra
Suppose that $x,y,z$ are nonnegative real numbers satisfying the equation $$\sqrt{xyz}-\sqrt{(1-x)(1-y)z} - \sqrt{(1-x)y(1-z)}-\sqrt{x(1-y)(1-z)} = -\frac{1}{2}.$$ The largest possible value of $\sqrt{xy}$ equals $\tfrac{a+\sqrt{b}}{c}.$ where $a,b,$ and $c$ are positive integers such that $b$ is not divisible by the square of any prime. Find $a^2+b^2+c^2.$

2018 VJIMC, 2
Tags: integer part , floor function , inequalities , algebra
Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

1985 Austrian-Polish Competition, 4
Tags: system of equations , algebra
Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

2021 LMT Spring, B10
Tags: algebra
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$. [i]Proposed by Zachary Perry[/i]

1987 All Soviet Union Mathematical Olympiad, 442
Tags: weighing , combinatorics , algebra
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.

1999 AMC 12/AHSME, 17
Tags: system of equations , algebra , polynomial
Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$? $ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad \textbf{(B)}\ x \plus{} 80 \qquad \textbf{(C)}\ \minus{}x \plus{} 118 \qquad \textbf{(D)}\ x \plus{} 118 \qquad \textbf{(E)}\ 0$

2022 AMC 10, 21
Tags: algebra
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial? $\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$

1988 Canada National Olympiad, 1
Tags: quadratic , number theory , greatest common divisor , polynomial , algebra
For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?

2017 Swedish Mathematical Competition, 6
Tags: algebra , inequalities
Let $a,b,c,x,y,z$ be real numbers such that $x+y+z=0$, $a+b+c\geq 0$, $ab+bc+ca \ge 0$. Prove that $$ ax^2+by^2+cz^2\ge 0 $$

1999 All-Russian Olympiad, 6
Tags: floor function , inequalities , number theory , algebra , induction
Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.

2012 District Olympiad, 1
Tags: rational , algebra
Let $a$ and $b$ be distinct positive real numbers, such that $a -\sqrt{ab}$ and $b -\sqrt{ab}$ are both rational numbers. Prove that $a$ and $b$ are rational numbers.

2019 China Girls Math Olympiad, 6
Tags: algebra , inequalities
Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

2001 Estonia Team Selection Test, 4
Tags: sum , product , combinatorics , algebra
Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

2018 CHMMC (Fall), 3
Tags: sum , algebra
Compute $$\sum^{\infty}_{n=1} \left( \frac{1}{n^2 + 3n} - \frac{1}{n^2 + 3n + 2}\right)$$