Olympiad problems

2018 CMIMC Number Theory, 1

Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$?

1989 IMO Longlists, 50

Tags: number theory , system of equations , diophantine equation , additive 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.$

2000 Baltic Way, 17

Tags: vieta , polynomial , quadratic , system of equations , algebra

Find all real solutions to the following system of equations: \[\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}\]

1979 IMO Shortlist, 15

Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2011 Dutch Mathematical Olympiad, 4

Tags: algebra , system of equations , radical

Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations: $a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.

1973 Spain Mathematical Olympiad, 2

Tags: inequalities , system of equations , system , algebra

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$ in which the first three are equations and the last one is a linear inequality.

2017 NZMOC Camp Selection Problems, 8

Tags: system of equations , system , algebra

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.

2016 Uzbekistan National Olympiad, 5

Tags: algebra , system of equations

Solve following system equations: \[\left\{ \begin{array}{c} 3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array} \right.\ \ \]

2004 Federal Competition For Advanced Students, P2, 5

Tags: algebra , system of equations

Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$

2023 USAMTS Problems, 4

Tags: algebra , system of equations , vieta

Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]

1936 Moscow Mathematical Olympiad, 027

Tags: algebra , system of equations , parameter

Solve the system $\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}$

1967 IMO Shortlist, 2

Tags: algebra , system of equations , polynomial

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

2013 India PRMO, 11

Tags: algebra , system of equations

Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$?

2022 Lusophon Mathematical Olympiad, 1

Tags: algebra , system of equations

How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system? $$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.

1935 Moscow Mathematical Olympiad, 016

Tags: system of equations , algebra

How many real solutions does the following system have ?$\begin{cases} x+y=2 \\ xy - z^2 = 1 \end{cases}$

2013 Estonia Team Selection Test, 1

Tags: number theory , system of equations , diophantine , diophantine equation

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

2001 Grosman Memorial Mathematical Olympiad, 1

Tags: system of equations , algebra

Find all real solutions of the system $$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$

1971 IMO Longlists, 30

Tags: parameterization , system of equations , algebra

Prove that the system of equations \[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \] $a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?

2004 Purple Comet Problems, 21

Tags: algebra , system of equations

Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$.

2010 Saudi Arabia BMO TST, 4

Tags: system of equations , system , diophantine , number theory

Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$

2007 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation , system of equations , diophantine

Solve the following system \[ \left\{ \begin{array}{l} xyzu-x^3=9 \\ x+yz=\dfrac{3}{2}u \\ \end{array} \right. \] in positive integers $x$, $y$, $z$ and $u$.

2014 Contests, 1

Tags: algebra , system of equations , solve system

Find all the pairs of real numbers $(x,y)$ that are solutions of the system: $(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $ $| x - y | = 1$

2019 Bosnia and Herzegovina EGMO TST, 1

Tags: algebra , system of equations

Let $x_1,x_2, ..., x_n$ be non-negative real numbers. Solve the system of equations: $$x_k+x_{k+1}=x^2_{k+2}\,\,,\,\,\, (k =1,2,...,n),$$ where $x_{n+1} = x_1$, $x_{n+2} = x_2$.

2013 Costa Rica - Final Round, 1

Tags: algebra , system of equations

Determine and justify all solutions $(x,y, z)$ of the system of equations: $x^2 = y + z$ $y^2 = x + z$ $z^2 = x + y$

2009 All-Russian Olympiad Regional Round, 10.7

Tags: algebra , system of equations

Positive numbers $ x_1, x_2, . . ., x_{2009}$ satisfy the equalities $$x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}= x^2_{2009}-x_{2009}x_1+x^2_1$$. Prove that the numbers $ x_1, x_2, . . ., x_{2009}$ are equal.