Olympiad problems

2004 Germany Team Selection Test, 1

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

2024 German National Olympiad, 1

Tags: system of equations , algebra

The five real numbers $v,w,x,y,s$ satisfy the system of equations \begin{align*} v&=wx+ys,\\ v^2&=w^2x+y^2s,\\ v^3&=w^3x+y^3s. \end{align*} Show that at least two of them are equal.

2015 District Olympiad, 2

Tags: exponential equation , algebra , system of equations

Solve in $ \mathbb{Z} $ the following system of equations: $$ \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. . $$

1978 Vietnam National Olympiad, 4

Tags: arithmetic sequence , system of equations , algebra

Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.

2017 CCA Math Bonanza, L5.1

Tags: algebra , system of equations

Find $x+y+z$ when $$a_1x+a_2y+a_3z= a$$$$b_1x+b_2y+b_3z=b$$$$c_1x+c_2y+c_3z=c$$ Given that $$a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9$$$$a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17$$$$a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c-bc_1\right)=-8$$$$a_1\left(b_2c-bc_2\right)-a_2\left(b_1c-bc_1\right)+a\left(b_1c_2-b_2c_1\right)=7.$$ [i]2017 CCA Math Bonanza Lightning Round #5.1[/i]

1992 IMO Longlists, 34

Tags: linear algebra , number theory , system of equations , additive number theory

Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that \[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]

1967 IMO Longlists, 33

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

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

2010 Greece JBMO TST, 2

Tags: system of equations , algebra

Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.

2006 Lithuania National Olympiad, 1

Tags: system of equations , algebra

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

2023 Bosnia and Herzegovina Junior BMO TST, 1.

Tags: algebra , system of equations

Determine all real numbers $a, b, c, d$ for which $ab+cd=6$ $ac+bd=3$ $ad+bc=2$ $a+b+c+d=6$

2022 Bulgaria National Olympiad, 4

Tags: algebra , system of equations , parameter

Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. If there exists an $a>0$ such that \[x_{i}^2=a+x_{i+1}x_{i+2}\quad\forall 1\leq i\leq n\] then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.

1963 Poland - Second Round, 1

Tags: algebra , system of equations , system

Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality $$ p+q + r=1$$ $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$ then for any numbers $ a $, $ b $, $ c $ equality holds $$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$

1949-56 Chisinau City MO, 16

Tags: algebra , system of equations

Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$

2018 Hanoi Open Mathematics Competitions, 8

Tags: system of equations , algebra

Let $a,b, c$ be real numbers with $a+b+c = 2018$. Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$. Compute the value of the following expression $P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$

1994 Austrian-Polish Competition, 6

Tags: system of equations , algebra

Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy: $$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\ (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\ ...\\ (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$ Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.

1972 Swedish Mathematical Competition, 1

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

Find the largest real number $a$ such that \[\left\{ \begin{array}{l} x - 4y = 1 \\ ax + 3y = 1\\ \end{array} \right. \] has an integer solution.

2019 Purple Comet Problems, 18

Tags: system of equations , algebra

Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$.

2021 Canadian Mathematical Olympiad Qualification, 2

Tags: algebra , system of equations

Determine all integer solutions to the system of equations: \begin{align*} xy + yz + zx &= -4 \\ x^2 + y^2 + z^2 &= 24 \\ x^{3} + y^3 + z^3 + 3xyz &= 16 \end{align*}

III Soros Olympiad 1996 - 97 (Russia), 10.5

Tags: system of equations , algebra

Solve the system of equations $$\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}$$

1994 India Regional Mathematical Olympiad, 4

Tags: quadratic , algebra , system of equations , complex number

Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}

2010 Morocco TST, 1

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 ?

2009 Ukraine National Mathematical Olympiad, 1

Tags: function , algebra , system of equations

Solve the system of equations \[\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}\]

2014 Contests, 1

Tags: algebra , system of equations

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

2020 Vietnam National Olympiad, 5

Tags: system of equations , algebra

Let a system of equations: $\left\{\begin{matrix}x-ay=yz\\y-az=zx\\z-ax=xy\end{matrix}\right.$ a)Find (x,y,z) if a=0 b)Prove that: the system have 5 distinct roots $\forall$a>1,a$\in\mathbb{R}.$

2022 Israel TST, 1

Tags: algebra , system of equations , parameterization

Let $n>1$ be an integer. Find all $r\in \mathbb{R}$ so that the system of equations in real variables $x_1, x_2, \dots, x_n$: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers $x_1, x_2, \dots, x_n$ are pairwise distinct.