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: 744

KoMaL A Problems 2020/2021, A. 788
Tags: system of equations , algebra
Solve the following system of equations: $$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$

2018 Junior Balkan Team Selection Tests - Romania, 1
Tags: system of equations , algebra
Determine all triples of real numbers $(a,b,c)$ that satisfy simultaneously the equations: $$\begin{cases} a(b^2 + c) = c(c + ab) \\ b(c^2 + a) = a(a + bc) \\ c(a^2 + b) = b(b + ca) \end{cases}$$

2022 Bulgarian Spring Math Competition, Problem 9.3
Tags: algebra , system of equations , prime
Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

1967 IMO, 6
Tags: combinatorics , recurrence relation , system of equations
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

1983 IMO Shortlist, 20
Tags: algebra , system of equations
Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

1989 IMO Shortlist, 15
Tags: system of equations , diophantine equation , additive number theory , number theory
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

1984 IMO Longlists, 17
Tags: algebra , system of equations
Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

2008 Regional Competition For Advanced Students, 2
Tags: system of equations , algebra
For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$. Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations: \[ \{a\}+[b]+\{c\}=2,9\]\[ \{b\}+[c]+\{a\}=5,3\]\[\{c\}+[a]+\{b\}=4,0\]

2005 German National Olympiad, 1
Tags: algebra , system of equations
Find all pairs (x; y) of real numbers satisfying the system of equations $x^3 + 1 - xy^2 - y^2 = 0$; $y^3 - 1 - x^2y + x^2 = 0$. Darij

2004 Estonia National Olympiad, 5
Tags: system of equations , algebra
Real numbers $a, b$ and $c$ satisfy $$\begin{cases} a^2 + b^2 + c^2 = 1 \\ a^3 + b^3 + c^3 = 1. \end{cases}$$ Find $a + b + c$.

2023 Moldova EGMO TST, 4
Tags: number theory , prime number , algebra , system of equations
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

2015 NIMO Summer Contest, 15
Tags: funky , system of equations , algebra
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

1957 Moscow Mathematical Olympiad, 368
Tags: algebra , system of equations
Find all real solutions of the system : (a) $$\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}$$ (b)* $$\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}$$

2021 Junior Balkan Team Selection Tests - Moldova, 6
Tags: algebra , system of equations , system
Solve the system of equations $$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$

2018 IMO, 2
Tags: algebra , sequence , system of equations
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

2014 Regional Competition For Advanced Students, 2
Tags: algebra , system of equations
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

1991 IMO Shortlist, 16
Tags: number theory , arithmetic sequence , system of equations , eulers function , laurentiu panaitopol
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

2014 Contests, 2
Tags: algebra , system of equations
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

2004 China Team Selection Test, 1
Tags: system of equations , algebra
Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$): \[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]

1993 IMO Shortlist, 4
Tags: linear algebra , matrix , algebra , system of equations
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$

2010 Dutch IMO TST, 5
Tags: system of equations , algebra
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.

2010 Mediterranean Mathematics Olympiad, 1
Tags: system of equations , algebra
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

2018 Hanoi Open Mathematics Competitions, 1
Tags: algebra , system of equations
Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$. A. $0$ B. $1$ C. $4$ D. $9$ E.$ -1$

2013 Hanoi Open Mathematics Competitions, 13
Tags: algebra , system of equations
Solve the system of equations $\begin{cases} xy=1 \\ \frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\end{cases}$

2018 Hanoi Open Mathematics Competitions, 6
Tags: system of equations , algebra
Write down all real numbers $(x, y)$ satisfying two conditions: $x^{2018} + y^2 = 2$, and $x^2 + y^{2018} = 2$.