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

2018 Polish Junior MO Finals, 4
Tags: algebra , system of equations
Real numbers $a, b, c$ are not equal $0$ and are solution of the system: $\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$ Prove that $(a - b)(b - c)(c - a) = 1$.

2023 New Zealand MO, 3
Tags: algebra
Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation. $$9^{x+1} + 2187 = 3^{6x-x^2}.$$

1946 Moscow Mathematical Olympiad, 109
Tags: algebra , system of equations
Solve the system of equations: $\begin{cases} x_1 + x_2 + x_3 = 6 \\ x_2 + x_3 + x_4 = 9 \\ x_3 + x_4 + x_5 = 3 \\ x_4 + x_5 + x_6 = -3 \\ x_5 + x_6 + x_7 = -9 \\ x_6 + x_7 + x_8 = -6 \\ x_7 + x_8 + x_1 = -2 \\ x_8 + x_1 + x_2 = 2 \end{cases}$

1996 Canada National Olympiad, 3
Tags: algebra
We denote an arbitrary permutation of the integers $1$, $2$, $\ldots$, $n$ by $a_1$, $a_2$, $\ldots$, $a_n$. Let $f(n)$ denote the number of these permutations such that: (1) $a_1 = 1$; (2):$|a_i - a_{i+1}| \leq 2$, $i = 1, \ldots, n - 1$. Determine whether $f(1996)$ is divisible by 3.

1965 All Russian Mathematical Olympiad, 069
Tags: algebra
A spy airplane flies on the circle with the centre $A$ and radius $10$ km. Its speed is $1000$ km/h. At a certain moment, a rocket , that has same speed with the airplane, is launched from point $A$ and moves along on the straight line connecting the airplane and point $A$.How long after launch will the rocket hit the plane?

2019 Dutch IMO TST, 2
Tags: function , algebra
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.

2011 Moldova Team Selection Test, 2
Tags: quadratic , polynomial , complex number , algebra
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$

2020 Junior Balkаn MO, 1
Tags: algebra , system of equations , polynomial
Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

1996 Vietnam Team Selection Test, 3
Tags: algebra
Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$ that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.

1972 All Soviet Union Mathematical Olympiad, 172
Tags: minimum , maximum , algebra , inequalities
Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$ What is the minimal possible $s$? What $x_i $correspond it?

1952 Czech and Slovak Olympiad III A, 2
Tags: table , algebra
Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.

2010 Putnam, B6
Tags: linear algebra , matrix , algebra , polynomial , induction , vector
Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

2018 Saudi Arabia IMO TST, 1
Tags: increasing , sequence , number theory , prime , algebra
Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

1994 All-Russian Olympiad, 5
Tags: algebra , equality
Prove the equality $$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$ (R. Zhenodarov)

KoMaL A Problems 2017/2018, A. 718
Tags: algebra , polynomial
Let $\mathbb{R}[x,y]$ denote the set of two-variable polynomials with real coefficients. We say that the pair $(a,b)$ is a [i]zero[/i] of the polynomial $f \in \mathbb{R}[x,y]$ if $f(a,b)=0$. If polynomials $p,q \in \mathbb{R}[x,y]$ have infinitely many common zeros, does it follow that there exists a non-constant polynomial $r \in \mathbb{R}[x,y]$ which is a factor of both $p$ and $q$?

1999 All-Russian Olympiad Regional Round, 11.1
Tags: algebra , continuous , function
The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.

2013 Saudi Arabia Pre-TST, 2.2
Tags: max , quadratic polynomial , algebra
The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.

1999 Vietnam National Olympiad, 3
Tags: function , modular arithmetic , algebra
Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that [b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$ [b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$

Russian TST 2018, P2
Tags: algebra
Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.

1949 Moscow Mathematical Olympiad, 162
Tags: geometric progression , algebra , geometric sequence , combinatorics
Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

1997 Iran MO (3rd Round), 1
Tags: polynomial , algebra
Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.

2004 Croatia National Olympiad, Problem 3
Tags: algebra , sequence
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations $$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$. (a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$; (b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?

1968 All Soviet Union Mathematical Olympiad, 113
Tags: sequence , algebra , inequalities
The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$

1991 Chile National Olympiad, 5
Tags: fibonacci , number theory , algebra , sum
The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$

1982 Spain Mathematical Olympiad, 3
Tags: algebra
A rocket is launched and reaches $120$ m in height; in the fall he loses $60$ m, then it recovers $40$ m, loses $ 30 $ again, gains $24$, loses $20$, etc. If the process continues indefinitely, at what height does it tend to stabilize?