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

2020 MOAA, TO5
Tags: theme , algebra
For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

2008 Romanian Master of Mathematics, 2
Tags: algorithm , absolute value , algebra , function
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.

2001 Austrian-Polish Competition, 6
Tags: sequence , integer , floor function , algebra , number theory
Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

2014 European Mathematical Cup, 1
Tags: algebra
Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample. a) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. b) Let $a,b,c$ be real numbers such that $ a^{2014} + b^{2014} + c^{2014} = 0 $. Then $ a^{2015} + b^{2015} + c^{2015} = 0 $. c) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $ and $ a^{2015} + b^{2015} + c^{2015} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. [i]Proposed by Matko Ljulj[/i]

2000 Turkey MO (2nd round), 2
Tags: polynomial , relatively prime , number theory , algebra
Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

2011 Morocco National Olympiad, 2
Tags: algebra
Solve in $\mathbb{R}$ the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.

2010 Chile National Olympiad, 4
Tags: number theory , algebra
Let $m, n$ integers such that satisfy $$m + n\sqrt2 = \left(1 +\sqrt2\right)^{2010} .$$ Find the remainder that is obtained when dividing $n$ by $5$.

2013 Serbia National Math Olympiad, 1
Tags: algebra
Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$, $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$. Prove that for every $i,j\in \mathbb{Z}$ it stands: \[|f(i)-f(j)|= |i-j|.\]

2002 Tournament Of Towns, 3
Tags: polynomial , algebra , quadratic , modular arithmetic , number theory
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

2001 China Team Selection Test, 1
Tags: algebra
Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying: \[ \alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k). \] Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.

2007 Indonesia TST, 3
Tags: inequalities , algebra , 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 $P(a) + P(b) + P(c) \le -1$.

2007 China National Olympiad, 2
Tags: algebra
Let $\{a_n\}_{n \geq 1}$ be a bounded sequence satisfying \[a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots \] Show that \[a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots\]

2011 Kyrgyzstan National Olympiad, 8
Tags: algebra
Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$, where $(n=1,2,3,...)$. What must be value of $x_1$, so that $x_{100}$ and $x_{1000}$ becomes equal?

1989 IMO Longlists, 5
Tags: trigonometry , limit , induction , algebra , inequalities
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities \[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}} \] and \[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n} \] Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$ \[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n. \]

2012 Indonesia TST, 1
Tags: polynomial , least common multiple , number theory , algebra
Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

2011 Brazil Team Selection Test, 1
Tags: polynomial , trinomial , algebra
Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.

VMEO III 2006 Shortlist, A5
Tags: functional equation , algebra , functional
Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$

2021/2022 Tournament of Towns, P3
Tags: algebra
Let $n$ be a positive integer. Let us call a sequence $a_1,a_2,\dots,a_n$ interesting if for any $i=1,2,\dots,n$ either $a_i=i$ or $a_i=i+1$. Let us call an interesting sequence even if the sum of its members is even, and odd otherwise. Alice has multiplied all numbers in each odd interesting sequence and has written the result in her notebook. Bob, in his notebook, has done the same for each even interesting sequence. In which notebook is the sum of the numbers greater than by how much? (The answer may depend on $n$.)

2005 Slovenia National Olympiad, Problem 2
Tags: algebra , sequence
Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.

2018 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{1}{a}+\frac{3}{b}+\frac{5}{c} \ge 4a^2 + 3b^2 + 2c^2$$ When does the equality hold? Marius Stanean

2009 ISI B.Math Entrance Exam, 4
Tags: trigonometry , algebra
Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.

2010 ISI B.Math Entrance Exam, 7
Tags: polynomial , trigonometry , algebra
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.

2007 Baltic Way, 2
Tags: algebra , induction
A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

1971 IMO Longlists, 16
Tags: polynomial , vieta , system of equations , algebra
Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

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