Olympiad problems

2021 Peru PAGMO TST, P3

Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations: $$x+y=z^2+w^2+6zw$$ $$x+z=y^2+w^2+6yw$$ $$x+w=y^2+z^2+6yz$$ $$y+z=x^2+w^2+6xw$$ $$y+w=x^2+z^2+6xz$$ $$z+w=x^2+y^2+6xy$$

2019 Czech-Polish-Slovak Junior Match, 4

Tags: algebra , system of equations

Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.

1949-56 Chisinau City MO, 15

Tags: algebra , system of equations

Solve the system of equations: $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{12}{5}\\ \\ \dfrac{yz}{y+z}=\dfrac{18}{5} \\ \\ \dfrac{zx}{z+y}=\dfrac{36}{13} \end{cases}$$

2011 Mathcenter Contest + Longlist, 7

Tags: algebra , system , system of equations

Given $k_1,k_2,...,k_n\in R^+$, find all the naturals $n$ such that $$k_1+k_2+...+k_n=2n-3$$ $$\frac{1}{k_1}+\frac{1}{k_2}+...+\frac{1}{k_n}=3$$ [i](Zhuge Liang)[/i]

2017 India PRMO, 18

Tags: algebra , system of equations

If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?

2021 Czech-Polish-Slovak Junior Match, 5

Tags: algebra , system of equations

Find all three real numbers $(x, y, z)$ satisfying the system of equations $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}$$ $$x^2 + y^2 + z^2 = xy + yz + zx + 4$$

1967 IMO Longlists, 6

Tags: matrix , algebra , linear algebra , system of equations

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

2014 Belarus Team Selection Test, 2

Tags: algebra , system of equations

Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$. Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$. (I. Voronovich)

2024 Israel National Olympiad (Gillis), P1

Tags: algebra , system of equations

Solve the following system (over the real numbers): \[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\ 3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]

1963 Poland - Second Round, 3

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

Solve the system of equations in integers $$x + y + z = 3$$ $$x^3 + y^3 + z^3 = 3$$

1967 IMO Shortlist, 6

Tags: linear algebra , matrix , system of equations , algebra

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

2015 Dutch IMO TST, 2

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

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

1991 All Soviet Union Mathematical Olympiad, 535

Tags: system of equations , number theory , diophantine

Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

VI Soros Olympiad 1999 - 2000 (Russia), 8.5

Tags: number theory , system of equations

Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

2005 Argentina National Olympiad, 1

Tags: number theory , system of equations

Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.

1999 Austrian-Polish Competition, 6

Tags: algebra , system of equations

Solve in the nonnegative real numbers the system of equations $$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\ x_0 = x_{1999} \end{cases}$$

2019 Saudi Arabia Pre-TST + Training Tests, 1.1

Tags: algebra , system of equations

Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$ Find all possible values of $T = x + y + z$

1966 German National Olympiad, 4

Tags: algebra , system of equations , system

Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$ $$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$ $$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$ $$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$

1956 Moscow Mathematical Olympiad, 332

Tags: algebra , system of equations , parameterization

Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\ x_3 - x_4 = b \\ x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.

2002 German National Olympiad, 1

Tags: algebra , system of equations

Find all real numbers $a,b$ satisfying the following system of equations \begin{align*} 2a^2 -2ab+b^2 &=a\\ 4a^2 -5ab +2b^2 & =b. \end{align*}

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.

2003 AMC 12-AHSME, 24

Tags: function , algebra , analytic geometry , slope , graphing lines , system of equations , ceiling function

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

1963 IMO, 4

Tags: algebra , parameterization , system of equations

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

1968 IMO Shortlist, 4

Tags: system of equations , discriminant , polynomial , algebra

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

2015 BMT Spring, 3

Tags: system of equations , algebra , number theory

Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}